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J Comb Optim (2009) 18: 124–150DOI 10.1007/s10878-008-9142-4
Using heuristics to find minimal unsatisfiablesubformulas in satisfiability problems
Christian Desrosiers · Philippe Galinier ·Alain Hertz · Sandrine Paroz
Published online: 5 March 2008© Springer Science+Business Media, LLC 2008
Abstract In this paper, we propose efficient algorithms to extract minimal unsatisfi-able subsets of clauses or variables in unsatisfiable propositional formulas. Such sub-sets yield unsatisfiable propositional subformulas that become satisfiable when any oftheir clauses or variables is removed. These subformulas have numerous applications,including proving unsatisfiability and post-infeasibility analysis. The algorithms wepropose are based on heuristics, and thus, can be applied to large instances. Further-more, we show that, in some cases, the minimality of the subformulas can be provenwith these algorithms. We also present an original algorithm to find minimum car-dinality unsatisfiable subformulas in smaller instances. Finally, we report computa-tional experiments on unsatisfiable instances from various sources, that demonstratethe effectiveness of our algorithms.
Keywords Satisfiability · Heuristics · Minimal unsatisfiable subformulas (MUS)
1 Introduction
A propositional formula F in conjunctive normal form (CNF) is the conjunction ofclauses from a set C = {C1, . . . ,Cm}, acting on variable set X = {x1, . . . , xn}. Each
C. Desrosiers · A. Hertz (�) · S. ParozEcole Polytechnique and GERAD, Montreal, Canadae-mail: [email protected]
C. Desrosierse-mail: [email protected]
S. Paroze-mail: [email protected]
P. GalinierEcole Polytechnique and CRT, Montreal, Canadae-mail: [email protected]
J Comb Optim (2009) 18: 124–150 125
clause Ci ∈ C is the disjunction of literals from a set Li , each literal being either avariable x ∈ X or the negation of this variable (¬x). A CNF formula can thus beexpressed in the following way:
F =m∧
i = 1
( ∨
l ∈ Li
l
)
A formula F is satisfiable if and only if there exists a truth assignment that satisfiesall its clauses. The satisfiability problem (SAT) consists in finding a truth assignment(i.e. true or false) for the variables in X such that the formula F evaluates to true, orshowing that no such assignment exist. The first problem to be shown NP-complete,SAT plays a key role in mathematical logic and computing theory, as well as in auto-mated reasoning, machine vision, database, scheduling and integrated circuit design(Franco et al. 1997). In the case of unsatisfiable SAT instances, it is often necessaryto find a truth assignment for which the most clauses are satisfied (i.e. evaluate totrue) in a given CNF formula. This task, known as the maximum satisfiability prob-lem (Max-SAT) was proven NP-hard, and is fundamental to solve many practicalproblems in computer science (Hansen and Jaumard 1990). The Max-SAT is clearlya generalization of the SAT. The weighted Max-SAT, or Max-WSAT, is another ver-sion of the satisfiability problem where each clause Ci ∈ C has a weight wi , and thegoal is to find a truth assignment that maximizes the sum of the weights of satisfiedclauses. These additional weights allow to assign preferences in the satisfaction ofclauses. Since the Max-SAT is a particular instance of its weighted version, for whichall weights are equal, the later problem is thus also NP-hard. A considerable amountof solution methods have been proposed for these problems, which can be roughlydivided in enumeration techniques (Borchers and Furman 1999; Davis et al. 1962;Goldberg and Novikov 2000; Hansen and Jaumard 1990; Madigan et al. 2001;Shang and Wah 1997; Zhang 1997), approximation schemes (Battiti and Protasi1998; Dantsin et al. 2002), and local search algorithms (Battiti and Protasi 1998;Mazure et al. 1997; Levesque et al. 1992; Marques Silva and Sakallah 1999;Mills and Tsang 2000).
It is often the case that a propositional formula, encoding a particular system orapplication, is unsatisfiable. When this occurs, it can be difficult to identify the causesof the inconsistency, and even harder to fix it. In fact, detecting the infeasibility, alone,is NP-hard. However the inconsistency is, most of the time, caused by contradictionslocal to the problem. These contradictions form an inconsistent sub-system explain-ing the inconsistency of the whole system.
Definition 1 Given an unsatisfiable propositional formula F , a minimal unsatisfiablesubset of clauses (MUSC) M ⊆ C is a subset of clauses that is unsatisfiable, and forwhich every proper clause subset is satisfiable.
The extraction of a MUSC from an unsatisfiable formula, referred to as the mini-mal unsatisfiable subformulas selection problem (MUSSP for short), has many ben-efits. The most common is related to helping a user redesign an incoherent system,modeled as a propositional formula, to make it once again coherent. To do so, the
126 J Comb Optim (2009) 18: 124–150
Table 1 Example of a nonsatisfiable instance of the SATproblem
Clause M1 M2 M3
C1 ¬x1 ∨ x2 • •C2 ¬x1 ∨ ¬x2 • •C3 x1 ∨ x3 • • •C4 x1 ∨ ¬x3 •C5 x3 ∨ ¬x5 •C6 ¬x3 ∨ ¬x5 • •C7 ¬x3 ∨ x5 • •C8 ¬x1 ∨ x3 ∨ ¬x4 •C9 ¬x1 ∨ x4 ∨ x5 •
user needs a small, yet, complete set of contradictions that led to the incoherence.A MUSC gives the user sufficient information to identify which design constraints torelax. Another use of MUSCs lies in proving that a given SAT instance is unfeasi-ble. The general idea is to find an unsatisfiable subformula, whether or not minimal,using a heuristic procedure, and then solving this subformula using an exact algo-rithm. Since a problem is feasible only if all its subproblems are, showing that asubformula is unsatisfiable proves that the original formula is also unsatisfiable. Fur-thermore, since the complexity of obtaining such a proof is exponential to the size ofthe problem, an exact algorithm is more likely to find a proof on the MUSC, then onthe original instance. In Mazure et al. (1998), small unsatisfiable subformulas, foundusing a local search strategy, were used to boost exact algorithms.
Table 1 shows a simple CNF instance of 9 clauses operating on 5 variables. ThisCNF problem is not satisfiable, and has three MUSCs, each explaining the incoher-ence of the formula. For example, M1 expresses the impossibility to assign truthvalues to x1, x2, and x3 such that the first four clauses are simultaneously satisfied.
It is often easier to understand why a problem is incoherent by simply looking atthe variables involved in this incoherence. Given a propositional formula F actingon a variable set X , and given a subset Q ⊆ X , the subformula of F induced by Q,denoted FQ, is the formula composed of the clauses Ci which have all the variablesof the literal set Li in Q. We say that a subset Q of variables is satisfiable if andonly if FQ is satisfiable. Hence, given a partial assignment in which only variablesin Q get a value, every clause Ci that contains at least one unassigned variable xj /∈Q in its literal sets is considered as satisfiable, the reason being that it is alwayspossible to assign a value to xj so that Ci evaluates to true. This leads to a variationof the Max-SAT problem where partial assignments are allowed and the goal is tofind an assignment of the largest possible subset Q ⊆ X of variables such that thecorresponding subformula FQ evaluates to true. We can further extend this problemby giving each variable xi ∈ X a weight wi , and having for goal to find an assignmentthat maximizes the sum of the assigned variables weights.
While a MUSC is a minimal unsatisfiable subset of clauses, a similar definitioncan be given for variables subsets.
J Comb Optim (2009) 18: 124–150 127
Definition 2 Given an unsatisfiable propositional formula F , acting on variableset X , a minimal unsatisfiable subset of variables (MUSV) is a subset M ⊆ X ofvariables such that M is unsatisfiable while every proper subset of M is satisfiable.
Notice that the variables composing the MUSCs are MUSVs. However, the op-posite is not necessarily true. Consider, once more, the CNF instance of Table 1,which contains two MUSVs: {x1, x2, x3} and {x1, x3, x4, x5}. The first MUSV givesa subformula composed of clauses C1, C2, C3 and C4, which is also the sub-formula corresponding to M1. However, the second MUSV yields a subformulaformed with clauses C3, C4, C5, C6, C7, C8 and C9, that is not minimal in thesense of MUSCs (i.e. C4 can be removed without making it satisfiable). The no-tions of MUSCs and MUSVs are closely related to that of irreducible infeasi-ble subsystems (IISs) in the context of linear programming (Amaldi et al. 1999;Gleeson and Ryan 1990). In the case of systems of linear inequalities with integervariables, additive and subtractive heuristics were proposed to find constraint IISs(Chinneck 1997). Other heuristic algorithms have been proposed to find vertex andedge critical-subgraphs which are the equivalent of MUSVs and MUSCs for the graphk-coloring problem (Desrosiers et al. 2008). The graph k-coloring problem consistsin coloring, if possible, the vertices of a graph G using at most k colors, and suchthat adjacent vertices have different colors. If such coloring exists, then G is saidk-colorable. A k-vertex (k-edge) critical subgraph is a subgraph G′ of G that is notk-colorable, and for which removing any vertex (edge) makes it k-colorable.
The MUSSP in unsatisfiable CNF formulas has been the subject of numerous re-searches. Recent theoretical works have shown that deciding whether a CNF formulacontains a MUSC of fixed deficiency k (i.e. for which the difference between the num-ber of clauses and variables equals k), for all k ∈ N, is NP-complete, but that efficientalgorithms can be developed for small values of k (Kleine Büning and Zhao 2002;Fleischner et al. 2002). Many methods have been proposed for the general MUSSP.Mazure et al. (1998) use local search to identify the clauses that are most difficult tosatisfy. These literals form a core that is used to guide a branch and bound algorithmfor the SAT problem. Bruni (2003) uses a procedure that ranks clauses based on thenumber of times they are conflicted during the search of an exact algorithm. Startingwith an initial set of clauses, a core is then expanded or contracted by a fixed percent-age of clauses, based on their rank, until the core becomes unsatisfiable. Although thisprocedure works well on small instances, it may take quite a long time to convergeon large real-world instances. Additionally, the cores obtained by such procedure arerarely minimal, for more difficult instances. Zhang and Malik (2003) propose a tech-nique to find unsatisfiable subformulas in large instances. This technique, which usesthe resolution proof of any DLL-based SAT solver to derive an unsatisfiable subset ofclauses, has two drawbacks: it offers no guarantee that the unsatisfiable subformulasfound are minimal, and it relies on the hypothesis that the exact SAT solver is ableto find a proof of satisfiability for a given instance, which is not always the case.The AMUSE algorithm, proposed by Oh et al. (2004), extends a generic DLL-basedSAT solver to implicitly search for an unsatisfiable subformula instead of a satisfyingassignment. This is done by adding to each clause an extra variable that acts as aselector for the clause, such that the problem becomes finding a truth assignment for
128 J Comb Optim (2009) 18: 124–150
these new variables, for which the extended formula is unsatisfiable. Unlike the pre-vious selection method, AMUSE doesn’t need to solve the whole instance, but only theunsatisfiable subformula. However, it still offers no guarantee of minimality. Mneim-neh et al. (2005) present a branch and bound algorithm that uses Max-SAT solutionsto compute lower and upper bounds on the number of clauses of a MUSC. Liffitonand Sakallah (2005) propose a 2-phase algorithm, called CAMUS, aimed at finding allMUSCs. During phase 1, this algorithm builds the set F of all co-MSSs (a MSS is amaximal satisfiable set of clauses, and a co-MSS the complementary set of a MSS).During phase 2, it builds all minimal hitting sets of F, i.e. all MUSCs. This algorithmis shown to perform better than a similar technique proposed by Bailey and Stuckey(2005).
In this paper, we propose original algorithms that guarantee to obtain MUSCs andMUSVs in unsatisfiable CNF formulas. We show that these algorithms can be appliedto relatively large SAT instances, with the use of heuristics. We also present an algo-rithm to select minimum cardinality MUSCs and MUSVs in small SAT instances.Finally, we give some heuristics to help find smaller MUSCs or MUSVs, more use-ful to diagnose incoherent systems. The remainder of this paper is as follows. Theselection procedures and heuristics are presented in Sect. 2. In Sect. 3, we give somecomputational results of the algorithms carried out on generated and real-life bench-marks. Finally, concluding remarks are given in Sect. 4.
2 Selection algorithms and heuristics
The algorithms presented in this section are based on solution methods for the largeset covering problem, proposed in Galinier and Hertz (2007). This paper demon-strates that the problem of finding IISs in unfeasible constraint satisfaction problems(CSPs) can be formulated as a large set covering problem (LSCP). Since the CSP isa generalization of SAT, where variables are not restricted to take on boolean values,these algorithms, which have successfully been applied to find vertex and edge crit-ical subgraphs and solving the graph coloring problem (Desrosiers et al. 2008), canthus be applied to SAT. Furthermore, since the properties, given in Galinier and Hertz(2007), are still valid, we will refer to that paper for their proofs.
To be more succinct, we will present one version of each algorithm, which canbe used to find either MUSCs or MUSVs. This is done by introducing some genericterms in the algorithms, that take on a specific meaning, depending on the task. Thus,if one needs to find MUSCs, S corresponds to the set of clauses C , wi ∈ W is theweight of clause ci ∈ C and MaxWSAT is a procedure that takes a CNF formula F andthe set of clause weights W , and returns a set U of unsatisfied clauses for which thesum f (U) of weights is minimum. On the other hand, if the goal is to select MUSVs,S is the set of variables X , wi ∈ W is the weight of variable xi ∈ X and MaxWSAT is aprocedure that takes a CNF formula F and the set of variable weights W , and returnsa set U of unassigned variables for which the sum f (U) of weights is minimum.Depending on the task, a MUS will refer to a MUSC or a MUSV.
J Comb Optim (2009) 18: 124–150 129
Fig. 1 The Removalalgorithm
2.1 The Removal algorithm
The Removal algorithm is perhaps the simplest of all MUS selection algorithms.Similar approaches have already been proposed, for example, the zMinimal algo-rithm in Zhang and Malik (2003), Chinneck’s deletion algorithm presented in Chin-neck (1997) for linear programming, or the algorithm in Herrmann and Hertz (2002)for vertex critical subgraph detection. Given an unsatisfiable CNF formula F , theRemoval algorithm proceeds in a top-down approach, removing clauses or variablesone by one, and re-inserting those that cause F to be satisfiable. Figure 1 containsthe pseudo-code of this algorithm. The term unexplored refers to the clauses or vari-ables that haven’t been tested for removal yet, and SAT to a procedure that solves theproblem of the same name.
Property 1 Given an unsatisfiable CNF formula F , the Removal algorithm returns,in a finite number of steps, a MUS if SAT is an exact procedure. Otherwise, if SAT isheuristic, the Removal algorithm returns either a MUS or a satisfiable subformula.
To illustrate the Removal algorithm, consider the task of finding a MUSC in theCNF formula of Table 1. Suppose the clauses are removed following their index, wefirst remove C1, “destroying” in the process M1 and M3. Since M2 is the onlyremaining MUSC, it will be returned by the algorithm after the other clauses areremoved, and its clauses re-inserted. Notice that the order in which the clauses orvariables are removed affects the outcome of the algorithm. Accordingly, if we hadremoved the clauses following the reverse order of their indices, M2 and M3 wouldhave been destroyed first, and M1 returned.
2.2 The Insertion algorithm
The number of steps (i.e. calls to procedure SAT) required by the Removal algo-rithm to find a MUSC or MUSV is equal to the number of clauses or variables ofthe original formula, regardless the size of this MUS. This is most inefficient in caseswhere the MUS is much smaller than the original formula F . For such cases, it makesmore sense to use a bottom-up approach, where an initially empty formula is ex-panded until it becomes unsatisfiable. The Insertion algorithm, shown in Fig. 2,is such an approach that selects a MUS in a number of steps equal to the size of that
130 J Comb Optim (2009) 18: 124–150
Fig. 2 The Insertionalgorithm
MUS. The algorithm starts by setting the weights of all clauses or variables of F to 1.It then modifies these weights as follows. When we want to give more importance toa clause or variable s ∈ S, we set its weight to α ≥ |S|, and we say that we harden s.On the other hand, when we do not want to take a clause or variable s into account,we fix its weight to 0, and we say that s is removed. This incites procedure MaxWSATto satisfy all hard clauses or assign all hard variables, and ignore all removed ones.Every iteration, MaxWSAT returns a set U containing at least one clause or variablefrom each remaining MUS of F . It then hardens one clause or variable from U andremoves the other, thus making sure that F remains unsatisfiable. When the set ofhard clauses or variables becomes unsatisfiable, MaxWSAT will obtain f (U) ≥ α
and the algorithm will return this set. If MaxWSAT is sub-optimal, it may happen thatF becomes satisfiable (i.e. f (U) = 0), in which case failure is reported. This errorcan also be repaired by re-inserting clauses or variables (i.e. setting their weight backto 1), until F becomes once again unsatisfiable.
Property 2 Given an unsatisfiable CNF formula F , the Insertion algorithm re-turns, in a finite number of steps, a MUS if MaxWSAT is an exact procedure. Other-wise, if MaxWSAT is not optimal, the Insertion algorithm returns either a MUS1
or a satisfiable subformula.
Let us illustrate the Insertion algorithm on the selection of a MUSV, in theCNF of Table 1. Since the set U , returned by MaxWSAT, must contain a variable fromeach of the two MUSVs {x1, x2, x3} and {x1, x3, x4, x5}, and since f (U) must beminimum, U can either contain x1 or x3, suppose x3. This variable is then hardened,and the next U contains x1, which is, in turn, hardened. The next U then containsx2 and a variable from the second MUSV, say x4. We must then choose to hardenone of those variables and remove the other one. Suppose x2 is hardened, the hardvariables then form a MUSV which is returned. Once again, the choice of the variableto harden determines which MUSV is returned. Thus, if we had chosen to harden x4instead of x2, the previously selected MUSV would have been destroyed and the
1Unless the Repair procedure is used.
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Table 2 An example where theInsertion algorithm cannotfind all MUSCs
Clause M1 M2 M3 M4 M5
C1 • • •C2 • • •C3 • •C4 • •C5 • •C6 • •
other one found. Notice, finally, that the Insertion algorithm may not be able toselect all MUSs of a given formula. Consider, for example, the selection of a MUSCin the CNF of Table 2. Suppose that the clauses all have a weight of 1, the optimalset U , returned by MaxWSAT, clearly contains C1 and C2. Since it contains both theseclauses, M1, which is the smallest MUSC of the CNF, will be destroyed.
Although both Insertion and AMUSE build a MUS step by step by adding newconstraints, there are major differences between these two algorithms. In particular,Insertion iteratively solves a weighted max-SAT problem by using metaheuris-tics.
2.3 The HittingSet algorithm
The HittingSet algorithm differs from the previous selection algorithms in thatit finds minimum cardinality unsatisfiable subformulas (MCUSs). This algorithm isbased on the fact that, given an unsatisfiable formula, an optimal solution to the max-SAT problem, at iteration i, gives a set Ui of unsatisfied clauses or unassigned vari-ables that intersects each MUS. A MUS M is therefore a hitting set (i.e. a set in-tersecting each set of a collection) of U = {U1, . . . ,U|M|}. Evidently, a MCUS is aminimum hitting set (MHS) of collection U . The algorithm’s pseudo-code is shownin Fig. 3. Every iteration, procedure MinHS returns a MHS H of an initially emptycollection U . The clauses or variables of S are then modified so that only those also inH become hard. If the set U returned by procedure MaxWSAT is such that f (U) ≥ α,then H is a MCUS and is returned. Else, U is added to collection U and the sameprocess is repeated.
Property 3 Given an unsatisfiable CNF formula F , the HittingSet algorithmreturns, in a finite number of steps, a MCUS if MaxWSAT and MinHS are exactprocedures. Otherwise, if MinHS returns minimal hitting sets and MaxWSAT is notoptimal, the HittingSet algorithm returns either a MUS or a satisfiable subfor-mula.
Table 3 illustrates the HittingSet algorithm on the selection of a MUSV in theCNF of Table 1. Each row gives the MHS produced by MinHS, as well as the optimaltruth assignment found by MaxWSAT and corresponding set U , at a given iteration.Notice that the sets U of the first three iterations are identical to those obtained inthe example given for the Insertion algorithm. The MHS of iteration 4 should
132 J Comb Optim (2009) 18: 124–150
Fig. 3 The HittingSetalgorithm
Table 3 Illustration of theHittingSet algorithm on theexample of the Table 1
It# H Assignment U
1 ∅ 〈0 0 - 0 1〉 {x3}2 {x3} 〈- 1 0 0 0〉 {x1}3 {x1, x3} 〈1 - 0 - 0〉 {x2, x4}4 {x1, x3, x4} 〈1 - 1 0 -〉 {x2, x5}5 {x1, x2, x3} 〈- 0 0 0 0〉 {x1}
contain either x2 or x4, in this example x4. As with the Insertion algorithm, theMCUS formed of variables x1, x2 and x3 is then destroyed. However, set U shouldonce more contain x2, such that the only possible MHS for iteration 5 is the MCUS{x1, x2, x3}.
Although the HittingSet algorithm produces a MUS of minimum cardinality(which was not necessarily the case for the Removal and Insertion algorithms),it may require an exponential number of steps. Therefore, this algorithm is best suitedfor small instances. Note that one can stop the HittingSet algorithm at any timeand use |H | as a lower bound on the size of the MUSs.
Although HittingSet shares some similarities with the CAMUS algorithm (Lif-fiton and Sakallah 2005), some fundamental differences separate these two methods.First, while CAMUS focuses on finding all MUSs, HittingSet only tries to find onewith a minimum number of variables or clauses. Also, CAMUS requires to computethe set of all co-MSSs, which limits the number of instances that can be solved withthis algorithm. Finally, HittingSet uses metaheuristics instead of an exact solverfor the SAT problem, which allows it to deal with harder SAT instances. Comparativeresults can be found in the section containing computational experiments.
2.4 The PreFiltering algorithm
When dealing with large instances, it can be useful to quickly filter out as manyvariables and clauses as possible, leaving less for the MUS selection algorithm. ThePreFiltering algorithm, shown in Fig. 4, is a variation of the Insertion algo-rithm, where all clauses or variables of U are made hard, every iteration. Since at least
J Comb Optim (2009) 18: 124–150 133
Fig. 4 The PreFilteringalgorithm
one clause or variable of each MUS becomes hard at each iteration, smaller MUSs aremore likely to remain in the filtered formula than bigger ones. The PreFilteringalgorithm thus acts as a heuristic that isolates smaller MUSs.
Consider, once again, the selection of a MUSC in the CNF of Table 2, for whichthe Insertion algorithm is unable to find the MCUS M1. The first set U returnedby MaxWSAT should contain C1 and C2. Both these clauses are made hard, suchthat the algorithm returns M1. In contrast to the Insertion algorithm, the Pre-Filtering algorithm succeeds in finding the MCUS.
2.5 Neighborhood weight heuristic
Recall that, in the Insertion algorithm, the choice of the clause or variable toharden, at each iteration, determines which MUS is obtained. The neighborhoodweight heuristic uses the information in the weights of the clauses or variables tomake choices leading to smaller MUSs. We define the neighbors N (C) of a clauseC ∈ C as the set of clauses, excluding C, that contain at least one of the variablesconstrained by C. The neighborhood weight of C is then the sum of the weights ofclauses in N (C). Correspondingly, the neighbors N (x) of a variable x ∈ X is the setof variables, excluding x, which are constrained by at least one of the clauses con-straining x. The neighborhood weight of x is thus the sum of the weights of variablesin N (x).
When all weights are equal (e.g. after initialization) the neighborhood weight ofa clause or a variable can be considered as a measure of the density of the regionsurrounding this clause or variable. As clauses or variables are made hard, the neigh-borhood weight then evaluates the hardness of the surrounding region. Because theyusually have fewer variables and clauses, it makes sense to build denser MUSs. Bychoosing to harden a clause or variable with the greatest neighborhood weight, wetherefore increase the density of the obtained MUS. This strategy can also be used inthe Removal algorithm by considering the weight of removed clauses or variablesas 0 and the others as 1. Since the selected MUS has a first clause or variable removedlast, we remove those with smallest neighborhood weight first, preserving the denserones.
2.6 Heuristic selection speed-up
This subsection presents a technique that can be used to speed-up the selection ofMUSs when using heuristic versions of the Insertion and PreFiltering al-
134 J Comb Optim (2009) 18: 124–150
gorithms. In those algorithms, procedure MaxWSAT is a heuristic, rather than an ex-act method. Given a formula F , if this procedure returns a set U with f (U) = 1, weknow that it is either optimal, or that F is satisfiable. Suppose that F is not satisfiable,the single clause or variable in U necessarily belongs to all the MUSs in F , and wecan right-away harden it. Furthermore, if MaxWSAT is implemented as a local searchheuristic, it can visit a great number of truth assignments for which f (U) = 1. Wecan thus memorize the unsatisfied clauses or the unassigned variables correspondingto those assignments, and harden all of them once the search is over. This technique isparticularly efficient when F contains only one MUS. In such cases, a single iterationof the selection algorithm is often enough to find the whole MUS. This idea can alsobe used to prove that the unsatisfiable formulas obtained using the Insertion orPreFiltering algorithm are minimal. Hence, if MaxWSAT finds f (U) = 1 everyiteration,2 we know that the subformula is either the only MUS of the problem, or issatisfiable.
2.7 Implementation details
In the selection algorithms, procedures SAT and MaxWSAT can be exact or heuristic.However, since the SAT and Max-WSAT problems are NP-hard, exact algorithmscan have some difficulties solving large instances, and, in those cases, we mustturn to heuristics. To solve these problems efficiently, we have implemented a lo-cal search heuristic based on the tabu search algorithm (Glover and Laguna 1997).In (Desrosiers et al. 2008), we have successfully used such a heuristic to find criticalsubgraphs for large instances. For the detection of MUSCs, we used as solution spacethe set of truth assignments, and as cost function the sum of the weights of unsatisfiedclauses, which we minimize. Given a truth assignment s, a neighbor of s is obtainedby inverting the truth value of a single variable. Furthermore, to avoid cycling andescape local minima, the tabu search algorithm forbids, during τ iterations, changingthe value of the assigned variable, unless it improves the best cost found at that mo-ment. The value of τ is given by the user as a parameter. On the other hand, for thedetection of MUSVs, the solution space of MaxWSAT is the set of partial satisfiableassignments, and the cost function minimizes the sum of the weights of unassignedvariables. A neighbor of a solution s is obtained by giving a truth value to an unas-signed variable of s, and if necessary, by unassigning variables of unsatisfied clauses,starting with the ones that are involved in the most unsatisfied clauses, until the as-signment evaluates to true. Once more, the tabu search algorithm forbids, during τ
iterations, giving to an unassigned variable the value it has just lost, unless it improvesthe best cost found.
We have also implemented a tabu search algorithm for the minimum hitting setproblem. However, for the results presented in this paper, we have used an exactalgorithm for this task. This algorithm, implemented in CPLEX, solves the followinginteger program
minm∑
i=1
xi
2Excluding the last iteration where f (U) ≥ α.
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s.t.∑
i∈Uj
xi ≥ 1, 1 ≤ j ≤ n, (1)
m∑
i=1
xi ≥ K, (2)
xi ∈ {0,1}, 1 ≤ i ≤ m
where Uj are the subsets produced by the MaxWSAT procedure, K is the cardinalityof the previous hitting set found, and where xi equals 1 if and only if the i-th clauseor variable is included in the hitting set. The role of constraint (2) is to speed up thesearch by reducing the size of the solution space, since there can not be any hittingset with strictly less than K elements.
3 Computational experiments
In this section, we present experiments that evaluate our MUS selection algorithmsusing various benchmark random and real-life instances. We test three different meth-ods.
• In the first method, named “REMOVAL”, we first use the exact Removal algorithmto find a MUSV by removing variables in a random order. We then remove clausesof that MUSV, using the MUSC version of the Removal algorithm, to find aMUSC. We solve the SAT problem exactly using the zChaff solver (Madigan etal. 2001).
• In the second method, named “P+Insertion”, we first use the PreFilter-ing algorithm to reduce the number of variables. We then apply the Insertionalgorithm to find a MUSV, and repeat the same process on the MUSV, this timeto find a MUSC. We use the tabu search algorithm, described in the last section,to solve the Max-WSAT problem, the neighborhood weight heuristic to select theclauses and variables to insert in the MUS, and also the selection speed-up tech-nique to accelerate the extraction. Because this method uses a heuristic to solvethe Max-WSAT problem, it is prone to errors that may render the subset of clausesor variables satisfiable. In those cases, we re-insert randomly some previously re-moved clauses or variables until the subset becomes unsatisfiable once again.
• In the third method, we find MCUSs using the HittingSet algorithm with thetabu search algorithm to solve the Max-WSAT problem and the exact algorithmdescribed in the last section to solve the minimum hitting set problem. Unlikethe previous two methods, we find MCUSs of variables and clauses separately,and write “HS-V” and “HS-C”, respectively, the HittingSet method that findsMUSVs and MUSCs.
We compare these three methods with several of the most popular algorithms toextract MUSCs. First of all, zCore (Zhang and Malik 2003) and AMUSE (Oh et al.2004) are algorithms available on the web that produce (non necessarily minimal)
136 J Comb Optim (2009) 18: 124–150
unsatisfiable subformulas. They can be combined with the zMinimal algorithm(Zhang and Malik 2003) (also available on the web) which is similar to our Removalalgorithm for finding MUSCs. We use the output of zCore and AMUSE as input forzMinimal. The combination of zCore and AMUSE with zMinimal is denotedzCore+ and AMUSE+.
We also compare our results with those obtained using the MUP (Huang 2005),SMUS (Mneimneh et al. 2005) and CAMUS (Liffiton and Sakallah 2005) algorithmswhich all produce MUSCs. The SMUS algorithm offers the added guarantee that itsoutput is a MUSC with a minimum number of clauses. Also, since the CAMUS al-gorithm extracts all MUSCs of a CNF formula, it also finds one with a minimumnumber of clauses.
The algorithms proposed by Bruni (2003) and Mazure et al. (1998) find (non nec-essarily minimal) unsatisfiable subformulas. Since they cannot be downloaded on theweb, we could not combine their output with zMinimal.
All experiments were carried out on a 2 GHz Intel Pentium IV PC with 512 kBcache and 1 GB RAM, running Linux CentOS release 4.2. All reported computingtimes are in seconds unless the time values end with “h”, in which case the values arein hours. For our methods, we did 10 runs using different random seeds, and give themean values of the successful runs. All values reported for zCore+ and AMUSE+were obtained on our computer (since these algorithms could be downloaded on theweb). For the other algorithms, the reported solutions and times are taken from thecorresponding papers (when available). Bruni did his experiments on a 450 MHz Pen-tium II PC, Mazure et al. on a 133 MHz Pentium PCs, MUP was tested on a 2.4 GHzPentium IV, SMUS on a 2 GHz Pentium IV and CAMUS on a 2.2 GHz Opteron proces-sor with 8 GB of RAM.
In the following tables of results, “Name” gives the name of the tested instance,“V” and “C” are the number of variables and clauses of the original instance or aMUS. Furthermore, “C’” refers to the number of clauses of the MUSC obtained afterreducing a MUSV, and “t” gives the CPU times.
3.1 DIMACS instances
The first set of benchmark instances, which can be found at (Dimacs 1993), comesfrom the Second DIMACS Challenge Workshop. These instances are of four differenttypes. The AIM instances, provided by Iwama et al., are artificial 3-sat instances.The JNH instances, provided by Hooker are difficult random instances generated byrejecting unit clauses and setting the density (clause to variable ratio) to a hard value.Lastly, the SSA and BF instances, from Van Gelder and Tsuji, are real-life instancesfrom circuit fault analysis.
AIM instances The aim instances are relatively small instances which have onlyone MUS each (excepts two of them which have two MUSs having the same size).For these instances, we found MUSs of variables and clauses with the REMOVAL,P+Insertion, HS-V and HS-C methods. The results are shown in Table 4, andare compared with those obtained by zCore+, AMUSE+ and by Mazure et al. andBruni. We observe that our methods found MUSCs with an equal or lesser number of
J Comb Optim (2009) 18: 124–150 137
Tabl
e4
Com
puta
tiona
lres
ults
for
the
AIM
inst
ance
s
Inst
ance
zCor
e+A
MU
SE+
Maz
ure
Bru
niR
EM
OV
AL
P+In
sert
ion
HS-
CH
S-V
Nam
eV
CV
Ct
VC
tV
Ct
VC
tV
CC
’t
VC
C’
tV
Ct
VC
t
aim
50-1
.6-1
5080
2022
0.0
2022
0.1
2022
0.1
2022
0.1
20.0
22.0
22.0
0.2
20.0
22.0
22.0
1.7
20.0
22.0
0.1
20.0
22.0
0.1
aim
50-1
.6-2
5080
2832
0.0
2832
0.0
2832
0.1
2832
0.1
28.0
34.0
32.0
0.3
28.0
34.0
32.0
2.7
28.0
32.0
0.2
28.0
34.0
0.2
aim
50-1
.6-3
5080
2831
0.0
2831
0.1
2831
0.1
2831
0.1
28.0
33.0
31.0
0.3
28.0
33.0
31.0
3.5
28.0
31.0
0.2
28.0
33.0
0.3
aim
50-1
.6-4
5080
1820
0.0
1820
0.0
1820
0.1
1820
0.0
18.0
21.0
20.0
0.2
18.0
21.0
20.0
1.9
18.0
20.0
0.1
18.0
21.0
0.1
aim
50-2
.0-1
5010
021
220.
021
220.
021
220.
121
220.
121
.026
.022
.00.
321
.026
.022
.01.
821
.022
.00.
121
.026
.00.
2
aim
50-2
.0-2
5010
028
300.
028
300.
128
300.
128
310.
328
.034
.030
.00.
328
.034
.030
.05.
428
.030
.00.
228
.034
.00.
6
aim
50-2
.0-3
5010
022
280.
022
280.
122
280.
122
280.
022
.029
.028
.00.
222
.029
.028
.02.
422
.028
.00.
222
.029
.00.
3
aim
50-2
.0-4
5010
018
210.
018
210.
118
210.
118
210.
318
.023
.021
.00.
218
.023
.021
.01.
718
.021
.00.
118
.023
.00.
2
aim
100-
1.6-
110
016
043
470.
143
470.
143
470.
243
471.
243
.049
.047
.00.
743
.049
.047
.05.
143
.047
.00.
343
.049
.00.
7
aim
100-
1.6-
210
016
046
530.
146
530.
146
520.
346
544.
546
.057
.053
.00.
846
.057
.053
.06.
046
.053
.00.
446
.057
.01.
5
aim
100-
1.6-
310
016
051
570.
151
570.
151
570.
351
574.
651
.059
.057
.00.
851
.059
.057
.05.
151
.057
.00.
451
.059
.00.
8
aim
100-
1.6-
410
016
043
480.
143
480.
143
480.
343
482.
543
.050
.048
.00.
743
.050
.048
.04.
643
.048
.00.
343
.050
.00.
7
aim
100-
2.0-
110
020
018
190.
018
190.
118
190.
118
190.
518
.020
.019
.00.
518
.020
.019
.02.
118
.019
.00.
118
.020
.00.
4
aim
100-
2.0-
210
020
035
390.
035
390.
135
390.
235
390.
935
.041
.039
.00.
735
.041
.039
.04.
935
.039
.00.
335
.041
.01.
4
aim
100-
2.0-
310
020
025
270.
025
270.
125
270.
125
271.
825
.030
.027
.00.
625
.030
.027
.02.
525
.027
.00.
225
.030
.00.
6
aim
100-
2.0-
410
020
026
310.
026
310.
126
310.
226
321.
626
.033
.031
.00.
626
.033
.031
.02.
226
.031
.00.
226
.033
.00.
6
aim
200-
1.6-
120
032
052
550.
152
550.
152
550.
652
552.
652
.055
.055
.01.
952
.055
.055
.08.
752
.055
.00.
852
.055
.05.
1
aim
200-
1.6-
220
032
077
800.
177
800.
277
800.
976
8243
.077
.085
.080
.02.
577
.085
.080
.016
.177
.080
.01.
777
.085
.045
.5
aim
200-
1.6-
320
032
077
830.
277
830.
277
831.
277
8630
0.0
77.0
89.0
83.0
2.7
77.0
89.0
83.0
11.4
77.0
83.0
1.7
77.0
89.0
432.
4
aim
200-
1.6-
420
032
044
460.
144
460.
144
460.
644
462.
344
.049
.046
.01.
744
.049
.046
.03.
444
.046
.00.
544
.049
.00.
9
aim
200-
2.0-
120
040
049
530.
149
530.
149
530.
549
543.
749
.056
.053
.02.
149
.056
.053
.09.
049
.053
.00.
749
.056
.045
.9
aim
200-
2.0-
220
040
046
500.
146
500.
146
500.
746
503.
046
.052
.050
.02.
246
.052
.050
.011
.546
.050
.00.
746
.052
.027
.5
aim
200-
2.0-
320
040
035
370.
135
370.
135
370.
435
370.
435
.038
.037
.01.
935
.038
.037
.06.
935
.037
.00.
635
.038
.010
.9
aim
200-
2.0-
420
040
036
420.
136
420.
136
420.
636
420.
836
.045
.042
.01.
736
.045
.042
.05.
736
.042
.00.
636
.045
.07.
5
138 J Comb Optim (2009) 18: 124–150
variables and clauses as those found by the other algorithms. For the instance aim100-16-2, Mazure et al. found a MUSC with one less clause than ours. We believe howeverthis to be an error since HS-C proved our MUSC with 53 clauses to be minimum.We note that in almost all cases, the MUSs are obtained very rapidly. For example,both the REMOVAL and HS-C methods find MUSCs in less than 3 seconds. TheP+Insertion method is somewhat slower than REMOVAL. This is due to the factthat zChaff solves these instances exactly in a few milliseconds, faster than the timerequired by our tabu search algorithm. In addition, we observe that the zCore+ andAMUSE+ algorithms are significantly faster than the other algorithms.
JNH instances Unlike the AIM instances, the JNH instances have generally manyMUSCs. For these instances, we tested the REMOVAL, P+Insertion, HS-V andHS-C methods with a time limit of one hour, and compared our results with thoseobtained by Bruni and with zCore+ and AMUSE+. The results are shown in Table 5.For HS-V and HS-C, the given values represent the cardinality of the hitting setsfound by each algorithm. When the time limit was reached, the values are precededby a “≥” to indicate that they represent lower bounds on the number of variables orclauses of a MUS. We observe that for these larger instances, HS-V and HS-C couldnot find MUSs with a minimum number of variables or clauses, except for instancejnh2. Also, the P+Insertion algorithm could produce a MUSV but no MUSCwithin one hour CPU time for instances jnh16 and jnh18.
We also observe that REMOVAL and P+Insertion found in many cases MUSscontaining much less variables and clauses than those found by the other algo-rithms. However, zCore+ and AMUSE+ are significantly faster. Although slower,P+Insertion produces MUSs with substantially less variables and clauses thanREMOVAL. Thus, for jnh13, REMOVAL obtained a MUSC of 66 variables and 92clauses (on average 77.3 variables and 92.0 clauses), while P+Insertion found aMUSC of 45 variables and 55 clauses (on average 45.2 variables and 55.0 clauses).This is due to the neighborhood weight heuristic that guides the Insertionmethodtowards smaller MUSs. Lastly, we can see that the bounds obtained by the HS meth-ods are generally far from the values found by the other two methods. Yet, we do notknow how close the actual MCUSs are from these bounds.
SSA and BF instances The SSA and BF instances have much more variables andconstraints than the AIM and JNH instances (a few thousands versus a few hun-dreds). Table 6 shows the results obtained with zCore+, AMUSE+, REMOVAL,P+Insertion and the algorithm of Mazure et al. These instances are too bigfor the HittingSet methods. Again, we observe that REMOVAL obtains in manycases MUSCs with fewer variables and clauses than those obtained by the other algo-rithms, although it is significantly slower than zCore+ and AMUSE+. Note that bothzCore+ and AMUSE+ use the zMinimal algorithm which is similar to our Re-moval algorithm for finding MUSCs. The small number of variables in the MUSCsfound by REMOVAL can be explained by its first phase which is to apply Removalto find a MUSV (to which clauses are then removed to get a MUSC).
Furthermore, we notice that, as expected, the P+Insertion method is fasterthan REMOVAL to find MUSs containing few variables and clauses, but much slower
J Comb Optim (2009) 18: 124–150 139
Tabl
e5
Com
puta
tiona
lres
ults
for
the
JNH
inst
ance
s
Inst
ance
zCor
e+A
MU
SE+
Bru
niR
EM
OV
AL
P+In
sert
ion
HS-
VH
S-C
Nam
eV
CV
Ct
VC
tV
Ct
VC
C’
tV
CC
’t
VC
jnh2
100
850
6169
0.2
4854
0.2
5160
3.2
63.1
181.
345
.01.
244
.576
.145
.054
.941
45
jnh3
100
850
9621
50.
991
190
0.9
9217
329
.789
.051
6.3
168.
02.
982
.047
3.0
148.
519
8.3
≥27
≥74
jnh4
100
850
8612
80.
371
102
0.3
8614
08.
284
.624
7.9
134.
02.
364
.017
7.0
102.
014
6.4
≥29
≥60
jnh5
100
850
6980
0.2
8513
10.
485
125
7.7
74.2
277.
882
.01.
564
.717
4.7
74.3
106.
1≥2
8≥6
0
jnh6
100
850
9518
10.
681
152
0.6
8815
922
.979
.133
2.1
130.
02.
273
.026
3.0
127.
914
5.0
≥31
≥61
jnh8
100
850
6780
0.2
8413
60.
470
910.
674
.326
8.2
86.0
2.2
60.0
150.
379
.210
2.9
≥24
≥39
jnh9
100
850
7910
40.
266
870.
278
118
180
.535
0.0
114.
01.
957
.012
6.5
71.9
114.
6≥2
7≥4
5
jnh1
010
085
081
119
0.3
7299
0.3
9516
10.
175
.630
7.2
77.0
1.8
58.5
137.
873
.096
.5≥2
7≥5
4
jnh1
110
085
088
140
0.3
9016
80.
679
129
1981
.035
6.1
116.
02.
070
.021
1.0
110.
726
3.8
≥28
≥77
jnh1
310
085
046
540.
162
710.
277
106
0.1
77.3
310.
392
.01.
745
.288
.755
.066
.9≥2
8≥5
0
jnh1
410
085
066
780.
270
920.
287
124
0.5
73.1
271.
070
.01.
557
.013
3.8
68.0
96.2
≥26
≥48
jnh1
510
085
086
145
0.4
8213
40.
587
140
1.4
77.0
313.
211
4.0
1.8
66.5
194.
092
.618
8.6
≥25
≥44
jnh1
610
085
010
045
110
.999
384
10.6
100
321
55.8
93.7
634.
628
1.0
7.6
92.0
600.
0–
>1h
≥35
≥161
jnh1
810
085
096
196
0.6
9422
30.
991
168
40.6
88.0
491.
716
6.0
2.9
81.0
364.
5–
>1h
≥25
≥69
jnh1
910
085
090
143
0.4
8615
30.
578
122
7.4
79.3
344.
011
0.0
2.0
72.0
239.
011
6.0
100.
9≥2
7≥5
0
jnh2
010
085
074
950.
279
121
0.3
8112
00.
779
.933
6.5
128.
02.
062
.517
0.5
86.9
142.
2≥2
4≥4
6
140 J Comb Optim (2009) 18: 124–150
Tabl
e6
Com
puta
tiona
lres
ults
for
the
SSA
and
BF
inst
ance
s
Inst
ance
zCor
e+A
MU
SE+
Maz
ure
RE
MO
VA
LP+
Inse
rtio
n
Nam
eV
CV
Ct
VC
tV
Ct
VC
C’
tV
CC
’t
ssa0
432-
003
435
1027
307
323
1.2
303
328
1.3
306
320
0.8
296.
658
6.6
309.
010
.333
5.0
714.
5–
>10
h
ssa2
670-
130
1359
3321
559
787
7.5
558
712
7.2
552
669
2.23
h55
2.4
1280
.466
9.0
83.0
570.
013
46.0
–>
10h
ssa2
670-
141
986
2315
580
1303
17.9
580
1280
26.3
579
1247
1.84
h57
5.0
1366
.012
77.0
87.9
587.
013
86.0
–>
10h
ssa6
288-
047
1041
034
238
2425
1.7
2425
2.9
––
>24
h24
.045
.025
.00.
4h24
.045
.025
.037
3.9
bf04
32-0
0710
4036
6867
113
5525
.566
413
3346
.267
412
5285
.359
7.7
1421
.311
51.0
120.
965
9.0
1616
.0–
>10
h
bf13
55-0
7521
8067
7881
152
0.9
8015
11.
482
185
26.2
79.0
199.
015
0.0
127.
680
.020
3.0
150.
088
.3
bf13
55-6
3821
7747
6882
153
0.9
8115
21.
183
154
32.6
81.0
203.
015
2.0
126.
382
.020
7.0
153.
012
8.1
bf26
70-0
0113
9334
3474
133
0.5
7713
70.
879
139
519.
474
.015
2.0
141.
043
.973
.014
7.0
132.
051
.3
J Comb Optim (2009) 18: 124–150 141
in the other cases. Thus, for instance ssa6288-047, P+Insertion found the sameMUSC as REMOVAL in a mean time of 373.9 seconds, instead of a mean time of1440.0 seconds for REMOVAL.
3.2 Daimler Chrysler instances
The second set of benchmark instances, which can be found at (Sat benchmarks2003), encodes different consistency properties of the configuration database ofDaimler Chrysler’s Mercedes car lines. The results obtained for these instances arepresented in Tables 7, 8, 9 and 10.
In Table 7, we show results obtained with zCore+, AMUSE+, MUP, REMOVAL,P+Insertion, HS-C and HS-V. We have imposed a time limit of 10 hours. Weobserve that REMOVAL is much slower than zCore+, AMUSE+ and MUP, while pro-ducing MUSCs having about the same size. Furthermore, we see once more thatP+Insertion is faster than REMOVAL when the MUSs are small, but fails to findthe bigger MUSs. Finally, we notice that the HittingSet method is better suitedfor finding MCUSs of clauses than of variables. Thus, the HS-C method found,within an hour, a MCUS for each of the instances.
In Table 8, we compare the results obtained with zCore+, AMUSE+, SMUS, Re-moval, HS-C and HS-V, with a time limit of 10 hours. These instances were toodifficult for the Max-WSAT heuristic of our Insertion method and we thereforedo not report any result for P+Insertion. In the cases where SMUS failed to find aMCUS, we instead give lower and upper bounds, separated by “:”, on the number ofclauses of a MUS. As expected, since they both guarantee finding MCUSs, our HS-C algorithm and SMUS find MUSCs with the same number of clauses. Yet, HS-Crequired less computational time than SMUS, even though the experiments involv-ing SMUS were carried out on a more powerful computer. Furthermore, even thoughHS-C finds MUSs that have a minimum number of clauses, it is faster than RE-MOVAL. This is due to the fact that, unlike the REMOVAL algorithm, it builds MCUSsin a constructive approach, and the MCUSs have much less clauses than the originalinstances. Again, zCore+ and AMUSE+ are the fastest methods.
In Tables 9 and 10, we compare the results obtained with zCore+, AMUSE+,CAMUS and HS-C. When CAMUS succeeded in finding all the MUSs of an instance,within a 600 seconds CPU time limit, “#MUS” gives the number of different MUScontained in that instance, while “MUS Size” gives the smallest and largest number ofclauses of these MUSs. Otherwise, if “#MUS” is preceded by “>”, the given valueswere calculated using the MUSs found before the timeout. Finally, if “-” is givenfor all these values, then no MUS was found by their algorithm within the imposedtime limit. We see that, while CAMUS was able to determine the smallest number ofclauses of a MUS for 32 of the 84 instances, within 600 seconds, our HS-C methodfound this value for 79 instances within the same time limit. Again, zCore+ andAMUSE+ are the fastest algorithms but they do not necessarily produce a MUSC witha minimum number of clauses.
3.3 Hard instances from graph coloring
The instances considered so far were all relatively easy for exact algorithms (suchas zChaff), making them tractable both by our Removal algorithms and by tech-
142 J Comb Optim (2009) 18: 124–150
Tabl
e7
Som
eco
mpu
tatio
nalr
esul
tsfo
rD
aim
ler
Chr
ysle
rin
stan
ces
Inst
ance
zCor
e+A
MU
SE+
MU
PR
EM
OV
AL
P+In
sert
ion
HS-
CH
S-V
Nam
eV
CV
Ct
VC
tV
Ct
VC
C’
tV
CC
’t
VC
tV
Ct
168_
FW_S
Z_1
0716
9865
9941
470.
843
501.
141
470.
141
.020
6.0
48.0
98.6
41.6
215.
559
.078
.841
.047
.032
.541
.020
8.2
69.1
168_
FW_U
T_2
468
1909
7487
3235
0.5
3235
0.8
3235
0.1
30.0
125.
033
.010
2.8
31.3
129.
347
.028
.430
.033
.040
.630
.012
5.0
75.3
202_
FS_R
Z_4
417
5061
9912
180.
321
270.
512
180.
117
.014
1.0
32.0
72.4
12.0
59.0
28.7
7.3
12.1
18.0
11.9
12.0
59.0
11.6
202_
FS_S
Z_8
417
5062
7320
421
91.
220
522
01.
220
622
10.
319
9.0
1923
.021
8.0
213.
9–
––
>10
h19
9.0
214.
059
9.3
198.
019
29.0
0.8h
202_
FS_S
Z_9
717
5062
5026
330.
424
340.
626
330.
121
.097
.028
.083
.821
.097
.032
.614
.221
.028
.017
.121
.097
.025
.5
202_
FW_S
Z_1
0017
9987
3822
260.
429
300.
824
280.
121
.010
6.0
29.0
106.
221
.510
9.4
30.7
18.1
22.0
23.0
16.7
21.0
106.
019
.9
202_
FW_S
Z_1
0317
9910
283
139
158
2.5
138
156
3.4
140
159
0.5
134.
016
41.0
156.
00.
3h13
3.0
1757
.033
4.0
6.3h
133.
014
8.0
424.
5≥2
8–
>10
h
202_
FW_S
Z_8
717
9989
4624
337
93.
124
439
83.
624
738
30.
623
4.0
2831
.038
2.0
372.
8–
––
>10
h23
1.0
361.
01.
0h23
1.0
2828
.00.
8h
202_
FW_S
Z_9
617
9988
4920
521
01.
521
522
01.
821
021
50.
320
5.0
2377
.021
0.0
286.
5–
––
>10
h20
4.0
209.
098
3.8
204.
023
92.0
0.9h
210_
FS_S
Z_5
517
5557
8128
450.
327
460.
629
460.
124
.010
9.0
42.0
70.4
24.1
111.
462
.010
.524
.041
.031
.424
.010
9.0
29.1
210_
FW_S
Z_9
017
8979
9421
627
93.
122
829
13.
922
128
40.
520
9.0
2044
.028
7.0
408.
5–
––
>10
h20
9.0
274.
00.
34h
≥64
–>
10h
210_
FW_S
Z_9
117
8977
2121
427
72.
621
627
93.
122
528
80.
520
6.0
2052
.028
0.1
362.
824
0.0
2593
.0–
>10
h20
6.0
271.
00.
3h≥2
8–
>10
h
210_
FW_U
T_8
630
2024
9721
2938
0.6
2538
1.1
2335
0.1
19.0
88.0
31.0
139.
519
.088
.030
.016
.919
.030
.039
.719
.088
.067
.8
220_
FV_S
Z_5
517
2857
5324
030
43.
823
830
54.
624
631
00.
622
9.0
1452
.035
6.0
427.
622
9.0
1452
.049
9.0
5.0h
233.
029
7.0
664.
9≥4
0–
>10
h
220_
FV_S
Z_6
517
2844
9618
230.
218
230.
424
290.
118
.050
.023
.054
.918
.050
.036
.05.
518
.323
.09.
718
.050
.09.
6
J Comb Optim (2009) 18: 124–150 143
Tabl
e8
Mor
eco
mpu
tatio
nalr
esul
tsfo
rD
aim
ler
Chr
ysle
rin
stan
ces
Inst
ance
zCor
e+A
MU
SE+
SMU
SR
EM
OV
AL
HS-
CH
S-V
Nam
eV
CV
Ct
VC
tC
tV
CC
’t
VC
tV
Ct
168_
FW_S
Z_1
0716
9865
9941
470.
843
501.
147
546.
541
.020
6.0
48.0
98.6
41.0
47.0
32.5
41.0
208.
269
.1
168_
FW_S
Z_4
116
9853
8727
300.
325
270.
526
257.
423
.086
.027
.060
.924
.926
.037
.023
.086
.093
.3
168_
FW_S
Z_6
616
9854
0112
160.
212
160.
516
18.8
12.0
44.0
16.0
65.8
12.0
16.0
11.6
12.0
44.0
6.3
168_
FW_U
T_2
463
1909
7489
3538
0.5
3538
0.7
3535
0.4
32.0
165.
037
.210
5.0
32.0
35.0
50.6
32.0
154.
811
2.0
168_
FW_U
T_2
469
1909
7500
3033
0.4
3033
0.7
3283
1.5
29.0
102.
032
.093
.429
.032
.038
.829
.011
9.2
133.
7
168_
FW_U
T_7
1419
0974
876
90.
46
90.
79
14.5
6.0
17.0
10.0
83.3
6.0
9.0
4.0
6.0
17.0
6.4
168_
FW_U
T_8
5119
0974
919
100.
49
100.
78
59.9
7.0
27.0
8.0
83.5
7.0
8.0
5.5
7.0
27.0
9.6
170_
FR_R
Z_3
216
5949
5630
327
0.8
3032
70.
722
712
1.3
30.0
870.
022
7.0
75.3
30.0
227.
090
.430
.087
0.0
16.4
170_
FR_S
Z_5
816
5950
0147
490.
347
490.
446
15.3
43.0
176.
047
.056
.543
.046
.019
.743
.017
6.0
58.1
170_
FR_S
Z_9
216
5950
8246
131
0.5
4613
10.
613
115
.146
.013
7.0
131.
051
.646
.013
1.0
39.7
46.0
137.
026
.5
170_
FR_S
Z_9
616
5949
5522
640.
322
630.
453
322.
821
.030
1.0
62.0
59.5
22.0
53.0
28.6
21.0
301.
011
.8
202_
FS_R
Z_4
417
5061
9912
180.
321
270.
518
131.
017
.014
1.0
32.0
72.4
12.1
18.0
11.9
12.0
59.0
11.6
202_
FS_S
Z_1
0417
5062
0130
350.
329
340.
524
5.0
19.0
116.
024
.076
.219
.024
.015
.319
.011
6.0
26.6
202_
FS_S
Z_1
2117
5061
8120
220.
321
230.
522
2.5
20.0
38.0
22.0
64.9
20.0
22.0
11.4
20.0
38.0
19.9
202_
FS_S
Z_1
2217
5061
7919
330.
319
330.
633
3.8
19.0
216.
033
.076
.419
.033
.010
.219
.021
6.0
15.1
202_
FS_S
Z_7
417
5063
5534
150
0.7
3415
00.
715
036
.434
.023
4.0
150.
079
.534
.015
0.0
88.0
34.0
236.
010
3.3
202_
FS_S
Z_8
417
5062
7320
421
91.
220
522
01.
239
:216
1.1h
199.
019
23.0
218.
021
3.9
199.
021
4.0
599.
319
8.0
1929
.00.
8h
202_
FS_S
Z_9
717
5062
5026
330.
424
340.
628
62.1
21.0
97.0
28.0
83.9
21.0
28.0
17.1
21.0
97.0
25.5
202_
FW_R
Z_5
717
9986
8530
213
1.0
3021
31.
321
358
.330
.081
5.0
213.
011
3.4
30.0
213.
081
.330
.081
5.0
42.3
202_
FW_S
Z_1
0017
9987
3822
260.
429
300.
823
174.
021
.010
6.0
29.0
106.
222
.023
.016
.721
.010
6.0
19.9
202_
FW_S
Z_1
0317
9910
283
139
158
2.5
138
156
3.4
35:1
492.
4h13
4.0
1641
.015
6.0
0.3h
133.
014
8.0
424.
5≥2
8–
>10
h
202_
FW_S
Z_1
2317
9986
8621
370.
521
370.
736
14.7
20.0
248.
036
.010
0.2
20.0
36.0
13.2
20.0
248.
015
.3
202_
FW_S
Z_6
117
9987
4516
180.
416
180.
718
163.
815
.067
.019
.098
.516
.018
.052
.715
.067
.068
.3
202_
FW_S
Z_7
717
9988
6034
156
0.8
3415
61.
015
637
.234
.024
0.0
156.
010
3.8
34.0
156.
013
6.0
34.0
240.
010
0.5
202_
FW_S
Z_9
817
9986
896
80.
416
210.
77
58.2
6.0
20.0
7.1
89.0
6.0
7.0
6.5
6.0
20.0
7.0
144 J Comb Optim (2009) 18: 124–150
Tabl
e8
(Con
tinu
ed)
Inst
ance
zCor
e+A
MU
SE+
SMU
SR
EM
OV
AL
HS-
CH
S-V
Nam
eV
CV
Ct
VC
tC
tV
CC
’t
VC
tV
Ct
208_
FA_R
Z_4
316
0852
978
90.
26
80.
48
76.9
7.0
19.0
9.0
54.2
6.0
8.0
9.2
6.0
19.0
3.9
208_
FA_S
Z_1
2016
0852
7819
340.
219
340.
434
3.9
19.0
247.
034
.059
.019
.034
.09.
619
.024
7.0
9.2
208_
FA_S
Z_8
716
0852
9917
200.
217
190.
418
15.1
17.0
136.
021
.054
.717
.018
.015
.117
.013
6.0
12.7
208_
FA_U
T_3
254
1876
7334
4068
0.5
4170
0.8
4095
.338
.088
3.0
40.0
109.
938
.040
.018
.638
.088
3.0
35.2
208_
FA_U
T_3
255
1876
7337
4068
0.5
4170
0.8
4094
.638
.088
3.0
40.0
109.
838
.040
.018
.738
.088
3.0
38.0
210_
FS_R
Z_2
317
5557
7822
350.
323
410.
631
266.
317
.071
.033
.068
.617
.031
.024
.217
.071
.024
.1
210_
FS_R
Z_3
817
7557
6317
280.
319
340.
525
261.
913
.057
.025
.066
.313
.025
.015
.913
.057
.018
.0
210_
FS_R
Z_4
017
7557
5229
140
0.6
2914
00.
614
036
.229
.065
2.0
140.
081
.529
.014
0.0
47.9
29.0
652.
041
.9
210_
FS_S
Z_1
0317
7557
7535
450.
339
500.
645
386.
435
.014
4.0
45.0
71.1
35.0
45.0
31.9
35.0
144.
051
3.5
210_
FS_S
Z_1
0717
7557
6211
150.
311
150.
515
25.3
11.0
35.0
15.0
65.0
11.0
15.0
6.9
11.0
35.0
11.5
210_
FS_S
Z_1
2317
7559
2149
176
0.7
4917
60.
817
60.
4h49
.031
8.0
176.
075
.549
.017
6.0
83.9
49.0
318.
034
7.5
210_
FS_S
Z_7
817
7559
3037
170
0.7
3717
00.
817
056
.734
.023
9.0
180.
074
.837
.017
0.0
116.
534
.023
9.0
114.
9
210_
FW_R
Z_5
717
8974
0514
260.
417
320.
725
355.
013
.057
.025
.093
.113
.025
.017
.313
.057
.027
.7
210_
FW_R
Z_5
917
8973
9429
140
0.6
2914
00.
814
056
.729
.065
2.0
140.
010
7.7
29.0
140.
051
.329
.065
2.0
96.7
210_
FW_S
Z_1
0617
8974
1739
550.
442
530.
849
789.
638
.020
5.0
51.0
97.6
38.0
49.0
37.2
37.0
201.
074
1.6
210_
FW_S
Z_1
1117
8974
0411
150.
411
150.
615
35.2
11.0
36.0
15.0
99.5
11.0
15.0
7.8
11.0
36.0
29.2
210_
FW_S
Z_1
2817
8974
1214
230.
413
230.
722
151.
312
.030
.023
.097
.313
.022
.020
.012
.030
.035
.0
210_
FW_S
Z_9
017
8979
9421
627
93.
122
829
13.
937
:275
1.8h
209.
020
44.0
287.
040
8.5
209.
027
4.0
0.34
h≥6
4–
>10
h
210_
FW_S
Z_9
117
8977
2121
427
72.
621
627
93.
141
:271
1.8h
206.
020
52.0
280.
136
2.7
206.
027
1.0
0.3h
≥28
–>
10h
220_
FV_R
Z_1
217
2845
1210
110.
213
140.
411
50.6
16.0
81.0
17.0
50.6
10.0
11.0
5.6
10.0
40.0
11.1
220_
FV_R
Z_1
317
2845
099
100.
213
140.
310
33.4
13.0
42.0
14.0
49.2
9.0
10.0
5.6
9.0
31.0
8.0
220_
FV_R
Z_1
417
2845
0810
110.
210
110.
411
33.9
10.0
40.0
11.0
53.7
10.0
11.0
3.8
10.0
40.0
7.3
220_
FV_S
Z_4
617
2844
9816
170.
231
320.
417
78.4
16.0
78.0
17.0
50.4
16.0
17.0
10.7
16.0
78.0
31.8
220_
FV_S
Z_6
517
2844
9618
230.
218
230.
423
18.9
18.0
50.0
23.0
54.9
18.3
23.0
9.7
18.0
50.0
16.4
J Comb Optim (2009) 18: 124–150 145
Table 9 More computational results for Daimler Chrysler instances
Instance zCore+ AMUSE+ CAMUS HS-C
Name V C V C t V C t t #MUS MUS Size V C t
Min Max
C168_FW_SZ_107 1698 6599 41 47 0.8 43 50 1.1 – – – – 41.0 47.0 32.5
C168_FW_SZ_128 1698 5425 30 96 0.6 29 96 0.7 – – – – 28.0 92.0 52.9
C168_FW_SZ_41 1698 5387 27 30 0.3 25 27 0.5 – – – – 24.9 26.0 37.0
C168_FW_SZ_66 1698 5401 12 16 0.2 12 16 0.5 – – – – 12.0 16.0 11.6
C168_FW_SZ_75 1698 5422 47 48 0.3 47 48 0.5 – – – – 47.0 48.0 29.1
C168_FW_UT_2463 1909 7489 35 38 0.5 35 38 0.7 – – – – 32.0 35.0 50.7
C168_FW_UT_2468 1909 7487 32 35 0.5 32 35 0.8 – – – – 30.0 33.0 40.6
C168_FW_UT_2469 1909 7500 30 33 0.4 30 33 0.7 – – – – 29.0 32.0 38.8
C168_FW_UT_714 1909 7487 6 9 0.4 6 9 0.7 – – – – 6.0 9.0 4.0
C168_FW_UT_851 1909 7491 9 10 0.4 9 10 0.7 0.3 102 8 16 7.0 8.0 5.5
C168_FW_UT_852 1909 7489 9 10 0.4 9 10 0.7 0.3 102 8 16 7.0 8.0 5.4
C168_FW_UT_854 1909 7486 9 10 0.4 9 10 0.7 0.3 102 8 16 7.0 8.0 5.1
C168_FW_UT_855 1909 7485 9 10 0.4 9 10 0.7 0.3 102 8 16 7.0 8.0 5.2
C170_FR_RZ_32 1659 4956 30 327 0.8 30 327 0.7 0.8 32768 227 228 30.0 227.0 90.4
C170_FR_SZ_58 1659 5001 47 49 0.3 47 49 0.4 7.5 218692 46 63 43.0 46.0 19.7
C170_FR_SZ_92 1659 5082 46 131 0.5 46 131 0.6 0.1 1 131 131 46.0 131.0 39.7
C170_FR_SZ_95 1659 4955 22 63 0.3 21 62 0.5 – >23301932 53 66 22.0 52.0 27.0
C170_FR_SZ_96 1659 4955 22 64 0.3 22 63 0.4 – >10383703 67 82 22.0 53.0 28.6
C202_FS_RZ_44 1750 6199 12 18 0.3 21 27 0.5 – >7764186 29 54 12.1 18.0 11.9
C202_FS_SZ_104 1750 6201 30 35 0.3 29 34 0.5 – >4803992 70 94 19.0 24.0 15.3
C202_FS_SZ_121 1750 6181 20 22 0.3 21 23 0.5 0.1 4 22 24 20.0 22.0 11.4
C202_FS_SZ_122 1750 6179 19 33 0.3 19 33 0.6 0.1 1 33 33 19.0 33.0 10.2
C202_FS_SZ_74 1750 6355 34 150 0.7 34 150 0.7 – – – – 34.0 150.0 88.0
C202_FS_SZ_84 1750 6273 204 219 1.2 205 220 1.2 – – – – 199.0 214.0 599.3
C202_FS_SZ_95 1750 6184 11 13 0.3 13 15 0.5 – >6173760 35 50 6.0 7.0 5.4
C202_FS_SZ_97 1750 6250 26 33 0.4 24 34 0.6 – >5466033 99 125 21.0 28.0 17.1
C202_FW_RZ_57 1799 8685 30 213 1.0 30 213 1.3 0.4 1 213 213 30.0 213.0 81.3
C202_FW_SZ_100 1799 8738 22 26 0.4 29 30 0.8 – – – – 22.0 23.0 16.7
C202_FW_SZ_103 1799 10283 139 158 2.5 138 156 3.4 – – – – 133.0 148.0 424.5
C202_FW_SZ_118 1799 8811 47 130 0.8 47 130 0.9 – >11072627 130 132 46.0 129.0 110.5
C202_FW_SZ_123 1799 8686 21 37 0.5 21 37 0.7 0.2 4 36 38 20.0 36.0 13.2
C202_FW_SZ_124 1799 8684 19 33 0.4 19 33 0.8 0.1 1 33 33 19.0 33.0 11.2
C202_FW_SZ_61 1799 8745 16 18 0.4 16 18 0.7 – – – – 16.0 18.0 52.7
C202_FW_SZ_77 1799 8860 34 156 0.8 34 156 1.0 – – – – 34.0 156.0 135.9
C202_FW_SZ_87 1799 8946 243 379 3.1 244 398 3.6 – – – – 231.0 361.0 3519.9
C202_FW_SZ_96 1799 8849 205 210 1.5 215 220 1.8 – – – – 204.0 209.0 983.8
C202_FW_SZ_98 1799 8689 6 8 0.4 16 21 0.7 – – – – 6.0 7.0 6.5
C202_FW_UT_2814 2038 11352 22 28 0.6 15 19 1.1 – – – – 15.0 16.0 95.1
C202_FW_UT_2815 2038 11352 22 28 0.6 15 19 1.1 – – – – 15.0 16.0 96.9
C208_FA_RZ_43 1608 5297 8 9 0.2 6 8 0.4 – >87515 9 28 6.0 8.0 9.2
C208_FA_RZ_64 1608 5279 29 212 0.7 29 212 0.7 0.2 1 212 212 29.0 212.0 67.0
C208_FA_SZ_120 1608 5278 19 34 0.2 19 34 0.4 0.1 2 34 34 19.0 34.0 9.6
146 J Comb Optim (2009) 18: 124–150
Table 9 (Continued)
Instance zCore+ AMUSE+ CAMUS HS-C
Name V C V C t V C t t #MUS MUS Size V C t
Min Max
C208_FA_SZ_121 1608 5278 18 32 0.2 18 32 0.4 0.1 2 32 32 18.0 32.0 9.1
C208_FA_SZ_87 1608 5299 17 20 0.2 17 19 0.4 0.9 12884 18 27 17.0 18.0 15.1
C208_FA_UT_3254 1876 7334 40 68 0.5 41 70 0.8 0.7 17408 40 74 38.0 40.0 18.6
C208_FA_UT_3255 1876 7337 40 68 0.5 41 70 0.8 1.4 52736 40 74 38.0 40.0 18.7
C208_FC_RZ_65 1654 5591 12 14 0.2 11 12 0.4 – – – – 11.0 12.0 13.8
C208_FC_RZ_70 1654 5543 29 212 0.8 29 212 0.7 0.2 1 212 212 29.0 212.0 67.5
C208_FC_SZ_107 1654 5641 36 47 0.3 38 50 0.5 – – – – 31.0 44.0 26.1
C208_FC_SZ_127 1654 5542 19 34 0.2 19 34 0.5 0.1 1 34 34 19.0 34.0 10.2
C208_FC_SZ_128 1654 5542 18 32 0.3 18 32 0.5 0.1 1 32 32 18.0 32.0 9.6
niques such as zCore+ and AMUSE+. In order to test our algorithms in a differentcontext, we have generated instances which are much more difficult for exact algo-rithms. These five new instances were generated from random graph k-coloring prob-lems, using Joe Culberson’s converter (Culberson 2004). For each of these instances,we find a potential MUSV using the Insertion method. We then determine if thiscore is satisfiable using zChaff, and if the core is unsatisfiable, we have a proofthat the original instance is unsatisfiable. Table 11 summarizes the results of this ex-periment. The columns labeled “zChaff” and “Check.” give the computation timerequired by zChaff to solve the original instances and their extracted core, with acomputation time limit of 24 hours. We can observe that, out of the five tested in-stances, zChaff was only able to solve one instance within the imposed time limit,while it succeeded in solving three of the extracted cores. Furthermore, we notice thatzChaff required much less time to solve the cores. For example, while it failed tosolve instance DGHP50_5 within 24 hours, it was able to solve its core in just 7.74hours. Given that it took 0.87 hours to extract the core, the total time needed to provethis instance unsatisfiable was 8.61 hours. Notice that zCore+ and AMUSE+ wereunable to treat these instances within the 24 hours time limit.
3.4 Overall discussion of the results
In the previous section we have clearly observed that the proposed algorithms areslower than other algorithms such as zCore+ and AMUSE+ for extracting MUSCs.They have, however, many advantages when compared to the most popular algo-rithms. First of all, our algorithms can find not only MUSCs but also MUSVs. Hence,for example, while the MUSC version of Removal is similar to zMinimal, we haveshown that the MUSV version can first be applied to reduce the size of an instance,and a MUSC can then be extracted using the MUSC version of Removal. Theresulting algorithm, called REMOVAL has sometimes produced MUSCs with muchless variables and clauses than the other algorithms. The P+Insertion algorithmnot only follows the same approach (i.e., removing clauses from a MUSV to geta MUSC), but also uses a neighborhood weight heuristic in order to extract small
J Comb Optim (2009) 18: 124–150 147
Table 10 More computational results for Daimler Chrysler instances
Instance zCore+ AMUSE+ CAMUS HS-C
Name V C V C t V C t t #MUS MUS Size V C t
Min Max
C210_FS_RZ_23 1755 5778 22 35 0.3 23 41 0.6 – – – – 17.0 31.0 24.2
C210_FS_RZ_38 1755 5763 17 28 0.3 19 34 0.5 – >5365108 46 59 13.0 25.0 15.9
C210_FS_RZ_40 1755 5752 29 140 0.6 29 140 0.6 0.3 15 140 173 29.0 140.0 47.9
C210_FS_SZ_103 1755 5755 35 45 0.3 39 50 0.6 – – – – 35.0 45.0 31.9
C210_FS_SZ_107 1755 5762 11 15 0.3 11 15 0.5 – >2160988 48 65 11.0 15.0 6.9
C210_FS_SZ_123 1755 5921 49 176 0.7 49 176 0.8 – >10064216 177 234 49.0 176.0 83.9
C210_FS_SZ_129 1755 5753 20 33 0.3 20 33 0.5 0.1 1 33 33 20.0 33.0 10.2
C210_FS_SZ_130 1755 5753 19 31 0.3 19 31 0.5 0.1 1 31 31 19.0 31.0 9.6
C210_FS_SZ_55 1755 5781 28 45 0.3 27 46 0.6 – – – – 24.0 41.0 31.4
C210_FS_SZ_78 1755 5930 37 170 0.7 37 170 0.8 – – – – 37.0 170.0 116.5
C210_FW_RZ_30 1789 7426 22 39 0.4 27 48 0.8 – – – – 19.0 35.0 36.4
C210_FW_RZ_57 1789 7405 14 26 0.4 17 32 0.7 – >4597505 47 61 13.0 25.0 17.3
C210_FW_RZ_59 1789 7394 29 140 0.6 29 140 0.8 0.4 15 140 173 29.0 140.0 51.3
C210_FW_SZ_106 1789 7417 39 55 0.4 42 53 0.8 – – – – 38.0 49.0 37.2
C210_FW_SZ_111 1789 7404 11 15 0.4 11 15 0.6 – >6904528 39 51 11.0 15.0 7.8
C210_FW_SZ_128 1789 7412 14 23 0.4 13 23 0.7 – – – – 13.0 22.0 20.0
C210_FW_SZ_129 1789 7606 49 176 0.9 49 176 1.0 – >7528982 238 281 49.0 176.0 90.0
C210_FW_SZ_135 1789 7395 20 33 0.4 20 33 0.7 0.1 1 33 33 20.0 33.0 11.6
C210_FW_SZ_136 1789 7395 19 31 0.4 19 31 0.7 0.2 1 31 31 19.0 31.0 10.8
C210_FW_SZ_80 1789 7572 39 173 0.8 35 175 1.0 – – – – 37.0 171.0 155.3
C210_FW_SZ_90 1789 7994 216 279 3.1 228 291 3.9 – – – – 209.0 274.0 1225.7
C210_FW_SZ_91 1789 7721 214 277 2.6 216 279 3.1 – – – – 206.0 271.0 1093.4
C210_FW_UT_8630 2024 9721 29 38 0.6 25 38 1.1 – – – – 19.0 30 .0 39.7
C210_FW_UT_8634 2024 9719 28 32 0.5 28 35 1.0 – – – – 18.0 23.0 30.0
C220_FV_RZ_12 1728 4512 10 11 0.2 13 14 0.4 1.7 80272 11 35 10.0 11.0 5.6
C220_FV_RZ_13 1728 4509 9 10 0.2 13 14 0.3 0.3 6772 10 27 9.0 10.0 5.6
C220_FV_RZ_14 1728 4508 10 11 0.2 10 11 0.4 0.1 80 11 18 10.0 11.0 3.8
C220_FV_SZ_114 1728 4777 47 132 0.5 47 132 0.6 – >1356651 320 364 47.0 132.0 62.3
C220_FV_SZ_121 1728 4508 14 58 0.3 14 58 0.4 0.2 9 58 65 14.0 58 .0 17.3
C220_FV_SZ_39 1728 5263 199 205 2.2 211 217 2.8 – – – – 199.0 205.0 340.2
C220_FV_SZ_46 1728 4498 16 17 0.2 31 32 0.4 – >11089464 40 71 16.0 17.0 10.7
C220_FV_SZ_55 1728 5753 240 304 3.8 238 305 4.6 – >2822517 663 670 233.0 297.0 664.9
C220_FV_SZ_65 1728 4496 18 23 0.2 18 23 0.4 3.2 103442 23 40 18.3 23.0 9.7
MUSs. We have observed that such a technique make it possible, in many cases, toobtain MUSCs with significantly fewer clauses.
There are other important differences between the Insertion and the Re-moval algorithms. For example, the Insertion algorithm uses a tabu searchheuristic to solve the Max-WSAT problem, while Removal uses the (exact)zChaff algorithm to solve the SAT problem. In many cases, the tabu search al-gorithm was slower than zChaff, explaining why Insertion was often slower
148 J Comb Optim (2009) 18: 124–150
Table 11 Computational results for instances coming from graph coloring
Instance zChaff P+Insertion Check
Name V C t V C t t
DGHP50_1 350 3641 5.23h 259 2277 0.43h 2.48h
DGHP50_2 400 4618 >24h ∗336 ∗3450 1.43h >24h
DGHP50_3 400 4650 >24h 288 2796 0.70h 17.42h
DGHP50_4 400 4602 >24h ∗288 ∗2724 2.22h >24h
DGHP50_5 400 4706 >24h 280 2619 0.87h 7.74h
than Removal. However, we have built hard coloring instances that are very difficultfor zChaff, while still manageable by the tabu search heuristic. For these instances,the Insertion algorithm was the only one able to find MUSVs for proving theunsatisfiability of the considered instances. Another difference is that the Inser-tion algorithm builds MUSs in a constructive way. Therefore, when the tabu searchheuristic makes no mistake, the number of steps is smaller or equal to the numberof clauses in the MUSC. Hence, it is not surprising that the Insertion algorithmis sometimes faster than the Removal algorithm when the MUSCs have much lessclauses than the original instance.
The MUSC version of the HittingSet algorithm (i.e, HS-C) provides MUSCswith a minimum number of clauses (i.e., MCUSs). Unlike the Removal and the In-sertion algorithms, the number of steps of HS-C is exponential with respect to thenumber of clauses, in the worst case. Therefore, it is not surprising that the algorithmfailed to solve several instances, in particular some large DIMACS instances. How-ever, when compared to SMUS and CAMUS on the Daimler instances, HS-C was theunique algorithm able to solve all these instances generally in less than one hour. It ismoreover interesting to observe important differences between HS-C and the CAMUSalgorithm. As mentioned in Sect. 1, CAMUS first builds the set of all co-MSSs, andthen all MUSCs. When it is successful, this algorithm is much faster than our HS-Cmethod. However, this algorithm may fail (during phase 1) if the number of co-MSSsis too large, or even (during phase 2) if the number of MUSCs is too large. UnlikeCAMUS, our HS-C algorithm tries to discover a MCUS by generating only a limitednumber of (non necessarily minimal) co-MSSs (the sets denoted U in Fig. 3). Thismakes it possible to solve many instances that are out of reach for CAMUS.
4 Conclusion
We presented, in this paper, novel algorithms to find minimal unsatisfiable subsets(MUSs) of variables or clauses of a given SAT problem. We also described an algo-rithm that finds MUSs with a minimum number of clauses or variables. Furthermore,we showed how these algorithms can be used with exact or heuristic SAT solvers,depending on the difficulty of the SAT instances to solve. We saw that the extractionalgorithms guarantee unsatisfiability and minimality when using an exact solver, andthat a guarantee of minimality can be obtained, in some cases, while using a heuristic
J Comb Optim (2009) 18: 124–150 149
solver. Moreover, we presented additional techniques to speed-up the extraction of aMUS and to find MUSs containing less variables and clauses. Finally, we evaluatedour algorithms with some computational experiments on various instances, and com-pared our results with those obtained by state-of-the-art algorithms for the same task.These experiments have shown that our algorithms are not the fastest ones, but theygenerally produce MUSs with less variables and clauses. For very large instances onwhich exact algorithms (e.g., zChaff) cannot be applied to prove their unsatisfia-bility, our Insertion algorithm was the only one able to extract MUSs. Also, ourHS-C algorithm can determine MUSCs of minimum cardinality on instances whichare out of reach for algorithms like CAMUS. In summary, the proposed algorithmsrepresent an interesting alternative to existing algorithms for the extraction of MUSs.
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