The Probabilistic Minimum Coloring ProblemCécile Murat∗, Vangelis Th. Paschos∗

Résumé

Nous étudions le problème de coloration probabiliste (PCOLOR) sous une stra-tégie de modification consistant, pour une solution a priori C, de supprimer les som-mets qui son absents de C. Nous calculons la fonction objectif associée avec cettestratégie, donnons des bornes pour sa valeur et caractérisons la complexité algorith-mique du calcul de la solution a priori optimal. Nous démontrons que PCOLOR estNP-difficile et nous concevons an algorithme polynomial approché qui garantit unrapport non-trivial. Nous démontrons ensuite que, même dans les graphes bipartis, leproblème reste NP-difficile et que la 2-coloration unique d’un tel graphe est une ap-proximation à rapport constant. Enfin nous prouvons que PCOLOR est polynomialdans les compléments des graphes bipartis.

Mots-clefs : Approximation, Coloration, Complexité, Graphe, NP-complet

Abstract

We study the probabilistic coloring problem (PCOLOR) under a modificationstrategy consisting, given an a priori solution C, of removing the absent verticesfrom C. We compute the objective function associated with this strategy, we givebounds on its value, we characterize the complexity of computing it and the oneof computing the optimal solution associated with. We show that PCOLOR is NP-hard and design a polynomial time approximation algorithm achieving non-trivialapproximation ratio. We then show that probabilistic coloring remains NP-hard evenin bipartite graphs and that the unique 2-coloring in such graphs is a constant ra-tio approximation. We finally prove that PCOLOR is polynomial when dealing withcomplements of bipartite graphs.

Key words : Approximation, Coloring, Complexity, Graph, NP-complete

∗LAMSADE, CNRS UMR 7024, Université Paris-Dauphine, 75775 Paris Cedex 16, France {mu-rat,paschos}@lamsade.dauphine.fr

The Probabilistic Minimum Coloring Problem

1 Introduction

Consider a graph G(V,E) of order n. In minimum coloring problem, we wish to color Vwith as few colors as possible so that no two adjacent vertices receive the same color. Thechromatic number of a graph, denoted by χ(G), is the smallest number of colors that canfeasibly color its vertices. A graph G is called k-colorable if its vertices can be legallycolored by k colors, in other words if its chromatic number is at most k; it will be calledk-chromatic if k is its chromatic number. Minimum coloring was shown to be NP-hardin Karp’s original paper ([19]), and remains NP-complete even restricted to graphs ofconstant (independent on n) chromatic number at least 3 ([11]).

Since adjacent vertices are forbidden to be colored with the same color, a feasiblecoloring can be seen as a partition of V into vertex-sets such that, for each one of thesesets, no two of its vertices are mutually adjacent. Such sets are usually called independentsets. So, the optimal solution of minimum coloring is a minimum-cardinality partitioninto independent sets.

In this paper we deal with a probabilistic version of minimum coloring, denoted byPCOLOR in what follows. Consider any set V ′ ⊆ V and set G′ = G[V ′], the subgraphof G induced by V ′. In PCOLOR we are given:

• a graph G(V,E), and an n-vector Pr = (p1, . . . , pn) of vertex-probabilities; inother words, an instance of PCOLOR is a pair (G,Pr);

• a coloring C = (S1, . . . , Sk) for V ;

• a modification strategy M, i.e., an algorithm receiving C and G′ as inputs and mod-ifying C in order to produce a coloring C ′ for G′;

and the objective is to determine a coloring C∗ of G minimizing the quantity (commonlycalled functional) E(G,C, M) =

∑

V ′⊆V Pr[V ′]|C(V ′, M)| where C(V ′, M) is the solutioncomputed by M(C, V ′) (i.e., by M when executed with inputs the a priori solution C andthe subgraph of G induced by V ′) and Pr[V ′] =

∏

i∈V ′ pi

∏

i∈V \V ′(1 − pi) (there exist 2n

distinct sets V ′; therefore, explicit computation of E(G,C, M) is, a priori, not polynomial).The complexity of PCOLOR is the complexity of computing C∗. Using the terminology in-troduced in [6, 13], we will call solution C (in which the modification strategy is applied)an a priori solution; C∗ is then the optimal a priori solution.

The fact that strategy M intervenes in the formulation of the functional seems some-what unusual with respect to standard complexity theory where no algorithm intervenes inthe definition of the objective function of a problem. But M is absolutely not an algorithmfor PCOLOR, in the sense that it does not compute a coloring for G. It simply fits C (nomatter how it has been computed) to the subgraph of G induced by V ′, denoted by G[V ′]

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in what follows. This also implies that changing M one changes the definition of PCOLOR

itself. Strictly speaking, the PCOLOR-variants induced by the triples PCOLOR(M1) andPCOLOR(M2) (parameterized by two distinct modification strategies M1 and M1) are twodistinct probabilistic combinatorial optimization problems.

A priori optimization, i.e., searching for optimal a priori solutions of probabilisticcombinatorial optimization problems, has been studied in restricted versions of routingand network-design probabilistic minimization problems ([2, 4, 5, 6, 13, 14, 15, 16]), de-fined on complete graphs. Also, in [23] the minimum vertex covering problem in generaland in bipartite graphs is studied. Finally, a priori optimization has been used in [21, 22]to study probabilistic maximization problems, the longest path and the maximum inde-pendent set, respectively.

In what follows, we study PCOLOR(M) under the following simple but intuitive modifi-cation strategy M: given an a priori solution C, remove the absentvertices from C, i.e., take C ∩ V

′ as solution for G[V′]. Since M isfixed for the rest of the paper, we simplify notations using PCOLOR instead of PCOLOR(M).

We first motivate the study of this problem by two real-world applications, showingthat it is not simply a toy (even nice and pleasant) problem. Then, we are managed tocompute the functional, to give bounds on its value, to characterize the complexity ofcomputing it and the one of computing the optimal a priori solution associated with. Weshow that PCOLOR is NP-hard, so we try to face it with polynomial time approximationalgorithms achieving non-trivial approximation ratios. We then restrict ourselves to bi-partite graphs and show that PCOLOR is always NP-hard; we also prove that the unique2-coloring in bipartite graphs achieves approximation ratio 2.773 when used as a priorisolution. We finally show that PCOLOR is polynomial in bipartite complements of perfectmatchings, under identical vertex-probabilities, and in complements of bipartite graphs,without restriction on vertex-probabilities.

Let A be a polynomial time approximation algorithm for an NP-hard graph-prob-lem Π, let A(G) be the value of the solution provided by A on a graph G instanceof Π, and OPT(G) be the value of the optimal solution for G (following our notationfor PCOLOR, OPT(G) = E(G,C∗,M)). The approximation ratio ρA(G) of the algo-rithm A on graph G is defined as ρA(G) = A(G)/OPT(G). An approximation algorithmachieving ratio, at most, ρ on any instance of Π will be called ρ-approximation algorithm.

Since the modification strategy M is fixed for the rest of the paper we will simplifynotations by using E(G,C∗) instead of E(G,C∗, M) and C(V ′) instead of C(V ′, M).Moreover, we shall denote by pmax (resp., pmin), the maximum (resp., minimum) vertex-probability of V .

The Probabilistic Minimum Coloring Problem

2 Two real-world applications for probabilistic coloring

2.1 A scheduling application

Consider for a given University-fall a list of potential classes, students can follow: anystudent has to choose a subset of such classes. For any of them one knows the title,the teaching professor and the time slot assigned to it, each such slot being proposed bythe professor concerned. A class will be really given if the number of students havingchoosen it is above a given threshold. So, nobody knows a priori if a particular class willtake place before the closing of students registrations (we can reasonably assume that thechoice of any student is a function of the contents of the course and of the teacher). Onthe other hand, one can by statistically studying the behaviour of the students during thepast years assign probabilities on the fact that such or such class will really take place,the mandatory courses been assigned with probability 1. The problem for the Universityplanning services is how much rooms has to assign to the set of courses dealt.

This problem is typically an instance of PCOLOR if one considers courses as ver-tices and he/she links two such vertices if the corresponding classes cannot have place inthe same room (because they are planned with the same professor, or because they areassigned with overlapping time slots). This type of graph is known under the term incom-patibility graph. Here an independent set, i.e., a potential color, corresponds to a set of“compatible classes”, i.e., to classes that can be assigned with the same room. The num-ber of colors used in such a graph represents the total number of rooms assigned to theset of classes considered. The probabilities resulting from the statistical analysis on theformer students’ behaviour, are the presence probabilities for the vertices (i.e., the prob-abilities that the corresponding classes will really take place). Starting from an a priorisolution, i.e., a coloring of the incompatibility graph, the functional represents, in somesense, the average number of the necessary rooms for the courses planned.

2.2 A satellite shots planning application

Consider a planning aiding process for realizing satellite shots. We associate a vertexwith any shot requested and we link two vertices if they correspond to shots that cannotbe realized by the satellite on the same orbit. But a shot realized under, for example,strong cloud covering can be not usable for the purposes for which it has been requested.Using meteorological forecasting, one can assign to any shot requested a probability thatit will be usable. For a mean-term planning, one of the main problems is to decide ifon a given time slot (consequently, for a fixed number of orbits), a shot can or cannotbe realized. It has been shown ([10]) that this problem can be modelled as a minimumpartition into cliques of the vertices of the graph outlined just above. So, if one takes

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into account probabilities associated with meteorological forecasting, one has to solve aprobabilistic version of the problem mentioned. On the other hand, it is very well-knownthat the partition into cliques in a graph, amounts to a coloring problem in its complement.

3 The probabilistic coloring in general graphs

3.1 The complexity of probabilistic coloring

In this section, we first analytically express the functional for PCOLOR; then, based uponit, we show that it can be computed in polynomial time0 . Moreover, always based uponthe analytical expression obtained for the functional, we give a combinatorial characteri-zation of the optimal a priori solution.

Recall that given an a priori solution C = (S1, S2, . . . , Sk) of cardinality k and asubgraph G′ of G induced by a subset V ′ of V , the modification strategy adopted in thepaper consists of removing the vertices in C \ V ′. Denote by C(V ′) the coloring of G′

so-obtained and set k′ = |C(V ′)|. Denote also by 1F,the indicator function of a fact F .As it has already been noted, E(G,C) =

∑

V ′⊆V Pr[V ′]|C(V ′)|. Using the notations justabove,

E(G,C) =∑

V ′⊆V

Pr[V ′]|C(V ′)| =∑

V ′⊆V

Pr[V ′]k′ (1)

Consider the facts Fj: color Sj has at least a vertex and Fj: there is no vertex incolor Sj; then k′ can be written as k′ =

∑kj=1 1Fj

=∑k

j=1(1 − 1Fj) and (1) becomes:

E(G,C) =∑

V ′⊆V

Pr [V ′]

(

k∑

j=1

(

1 − 1Fj

)

)

=∑

V ′⊆V

Pr [V ′]

k∑

j=1

1 −∑

V ′⊆V

Pr [V ′]

k∑

j=1

1Sj∩V′=∅

=k∑

j=1

∑

V ′⊆V

Pr [V ′] −k∑

j=1

∑

V ′⊆V

Pr [V ′] 1Sj∩V′=∅ = k −k∑

j=1

∏

vi∈Sj

(1 − pi)

=k∑

j=1

1 −∏

vi∈Sj

(1 − pi)

(2)

It is easy to see that computation of E(G,C) can be performed in at most O(n2) steps,consequently, PCOLOR ∈ NP. On the other hand, from (2), we can easily characterize

0Recall that, as we have already mentioned in section 1, the functional is not a priori polynomiallycomputable for any problem and for any modification strategy.

The Probabilistic Minimum Coloring Problem

the optimal a priori solution C∗ for PCOLOR: if the value of an independent set Sj of G is1−∏

vi∈Sj(1−pi) then the optimal a priori coloring for G is the partition into independent

sets for which the sum of their values is the smallest over all such partitions. We finallynote that if we assume pi = 1, i = 1, . . . , n, then PCOLOR becomes the classical coloringproblem. This observation immediately deduces the NP-hardness of PCOLOR.

Theorem 1. PCOLOR is NP-hard.

We are here faced to a graph-problem completely different from the ones studiedin [22, 23]. There, when strategies as M were used, i.e., strategies consisting of droppingabsent vertices out of the a priori solution, the optimal a priori solutions were a maxi-mum weight independent set, or a minimum weight vertex-covering, of the input graphconsidering that its vertices receive their probabilities as weights; here, the weight of anindependent set is not an additive function. Let us mention that there exist also weightedversions of the minimum coloring. For example, one can consider that the weight of acolor is the maximum (or the minimum, or even, the average) weight of the vertices in theindependent set representing it, and the objective is to find a coloring minimizing the sumof the weights of the colors (see, for example, [8, 9] for a version of weighted coloring,where the weight of a color is the maximum over the weights of its vertices). The valuedcoloring dealt here is closer to the so-called chromatic sum problem ([3, 17, 24]) than tothe weighted coloring in [8, 9].

3.2 Bounds on E(G, C)

We give in this section upper and lower bounds on the value of E(G,C) valid for anygraph. Some of these bounds will be used later in section 3.3. Consider a coloring C =(S1, . . . , Sk). We first produce a framing for the term 1 −

∏

vi∈Sj(1 − pi) which will

be useful later. For simplicity, assume |Sj| = ` and arbitrarily denote vertices in Sj byv1, . . . , v`. Then, by induction in ` the following holds:

∑

i=1

pi −∑

i=1

∑

j=i+1

pipj 6 1 −∏

i=1

(1 − pi) 6∑

i=1

pi (3)

For the lefthand side one, observe first that it is true for ` = 1 and suppose it true for` = κ, i.e.,

κ∑

i=1

pi −κ∑

i=1

κ∑

j=i+1

pipj 6 1 −κ∏

i=1

(1 − pi) (4)

We will show the truth of the lefthand side of (3) for ` = κ + 1. Expression (4) implies∏κ

i=1(1−pi) 6 1−∑κ

i=1 pi+∑κ

i=1

∑κj=i+1 pipj . Multiply both terms of the last inequality

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by (1 − pκ+1); this leads to

κ+1∏

i=1

(1 − pi) 6

(

1 −κ∑

i=1

pi +κ∑

i=1

κ∑

j=i+1

pipj

)

(1 − pκ+1)

= 1 −κ∑

i=1

pi +κ∑

i=1

κ∑

j=i+1

pipj − pκ+1 + pκ+1

κ∑

i=1

pi − pκ+1

κ∑

i=1

κ∑

j=i+1

pipj

= 1 −κ+1∑

i=1

pi +

κ+1∑

i=1

κ∑

j=i+1

pipj − pκ+1

κ∑

i=1

κ∑

j=i+1

pipj

6 1 −κ+1∑

i=1

pi +κ+1∑

i=1

κ∑

j=i+1

pipj

which proves the lefthand side inequality.

For the righthand side, we prove by induction on ` that∏`

i=1(1 − pi) > 1 −∑`

i=1 pi.This is clearly true for ` = 1; suppose it also true for ` = κ, i.e.,

∏κi=1(1 − pi) >

1−∑κ

i=1 pi. Then, by multiplying both members of this inequality by (1− pκ+1), we getthat the product obtained is equal to 1 − pκ+1 −

∑κi=1 pi + pκ+1

∑κi=1 pi > 1 −

∑κ+1i=1 pi,

q.e.d.

Revisit (3) and take the sums of its members for m = 1 to k. The righthand sideinequality immediately gives E(G,C) 6

∑ni=1 pi. Moreover,

k∑

m=1

(

∑

i=1

pi −∑

i=1

∑

j=i+1

pipj

)

=n∑

i=1

pi −k∑

m=1

∑

i=1

∑

j=i+1

pipj >

n∑

i=1

pi −n∑

i=1

n∑

j=i+1

pipj

and the following (first) framing is immediately derived for the functional:

n∑

i=1

pi −n∑

i=1

n∑

j=i+1

pipj 6 E(G,C) 6

n∑

i=1

pi (5)

We now give additional lower and upper bounds for E(G,C) used also later. For theformer ones, note that

∏

vi∈Sj

(1 − pi) = exp

∑

vi∈Sj

log (1 − pi)

= exp

−∑

vi∈Sj

log

(

1

1 − pi

)

(6)

Consider function f(x) = x − log(1/(1 − x)) = x + log(1 − x); it is decreasingwith x ∈ [0, 1] and, moreover, f(0) = 0. Therefore, f(x) 6 0, in other words,

The Probabilistic Minimum Coloring Problem

pi 6 log(1/(1 − pi)). Using this inequality together with (6), we get:

E(G,C) =k∑

j=1

1 − exp

−∑

vi∈Sj

log

(

1

1 − pi

)

>

k∑

j=1

1 − exp

−∑

vi∈Sj

pi

(7)On the other hand,

∏

vi∈Sj(1−pi) 6 (1−pmin)

|Sj | 6 1−pmin. Consequently the followingbound is immediately get:

E(G,C) =k∑

j=1

1 −∏

vi∈Sj

(1 − pi)

>

k∑

j=1

pmin = kpmin (8)

The bounds for E(G,C) produced in (7) and (8) are immediately be combined into thefollowing:

E(G,C) > max

k∑

j=1

1 − exp

−∑

vi∈Sj

pi

, kpmin

(9)

For the upper bounds, the first one is immediately get by noticing that

E(G,C) =k∑

j=1

1 −∏

vi∈Sj

(1 − pi)

6 k (10)

For the second one, observe that 1−pi > 1−pmax; hence,∏

vi∈Sj(1−pi) > (1−pmax)

|Sj |.Let us prove that, for any ` > 0,

(1 − pmax)`> 1 − `pmax (11)

Inequality (11) is obviously true for ` = 1. Suppose now (11) true for ` 6 κ; in particular,for ` = κ we have: (1 − pmax)

κ > 1 − κpmax. At range κ + 1 we have:

(1 − pmax)κ+1

> (1 − κpmax) (1 − pmax) = 1− (κ + 1)pmax + κp2max > 1− (κ + 1)pmax

and (11) holds for any `. Using it for |Sj|, j = 1, . . . , k, we have∏

vi∈Sj(1 − pi) >

(1 − pmax)|Sj | > 1 − |Sj|pmax, that implies

1 −∏

vi∈Sj

(1 − pi) 6 |Sj|pmax (12)

Summing (12) for j = 1, . . . , k, we get E(G,C) =∑k

j=1(1 −∏

vi∈Sj(1 − pi)) 6

∑kj=1 |Sj |pmax = npmax. Combination of this bound with the one of (10) concludes

the following:

E(G,C) =k∑

j=1

1 −∏

vi∈Sj

(1 − pi)

6 min {k, npmax} (13)

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3.3 An approximation algorithm for PCOLOR in general graphs

We have already mentioned that, in the problem we deal with, the weight of an inde-pendent set is not an additive function. This makes that it becomes harder to be faced byapproximation algorithms than the probabilistic problems dealt in [22, 23]. In this section,we devise and analyze an approximation algorithm for general PCOLOR.

Let us, in a first time deal with fixed vertex-probabilities, i.e., with probabilities suchthat pmin > t, for some t. Then, the following lemma, that will be used later, holds.

Lemma 1. Assume a graph of order n, with vertex-probability vector Pr and sup-pose that pmin > t. Then, PCOLOR is approximable in polynomial time within ra-tio O(n log2 log n/(t log3 n)).

Proof. Assume pi > t. It is easy to see from (7) that if we denote by C∗ = (S∗1 , . . . , S

∗k∗)

an optimal a priori solution for PCOLOR, then

E (G,C∗) =k∗

∑

j=1

1 −∏

vi∈S∗

j

(1 − pi)

>

k∗

∑

j=1

(1 − exp{−t}) = t′(t)k∗ (14)

where t′(t) = t′ 6 1 only depends on t. On the other hand, set C = (S1, . . . , Sk), theapproximate coloring computed by a ρ-approximation coloring-algorithm A on G (i.e.,by do not taking probabilities into account). Then the functional E(G, C) for C, in other

words, the objective value of C for PCOLOR is, by (10): E(G, C) =∑k

j=1(1−∏

vi∈Sj(1−

pi)) 6 k. By hypothesis, k/χ(G) 6 ρ; on the other hand, C∗ being a feasible coloring,k∗ > χ(G). Therefore, k/k∗ 6 ρ and the approximation ratio of A for PCOLOR is,taking (14) into account, E(G, C)/E(G,C∗) 6 ρ/t′. Using for A the algorithm of [12],then E(G, C)/E(G,C∗) 6 O(n log2 log n/(t′ log3 n)). Since t′ depends only on t, theresult claimed follows.

Corollary 1. If pmin is a fixed constant, then PCOLOR is approximable in polynomialtime within ratio O(n log2 log n/ log3 n).

Remark 1. Assume that at least pmax is a fixed constant and denote it by p. Then, denot-ing, as previously, by C∗ = (S∗

1 , . . . , S∗k∗) an optimal a priori solution for PCOLOR, one

gets E(G,C∗) > 1 − exp{−pmax} = 1 − exp{−p}, which is a fixed constant since pis supposed fixed. If one applies the polynomial algorithm of [20] computing, in anygraph G of maximum degree ∆, a coloring of the vertices of G with at most ∆ colors,this algorithm, when used for PCOLOR, guarantees, using (13) an approximation ratioof O(∆).

The Probabilistic Minimum Coloring Problem

We are ready now to devise and to analyze an algorithm for PCOLOR in general graphs.Consider the graph G and two vertex probabilities p0 and p′, p0 < p′. The main idea ofthe algorithm proposed is to partition the vertices of G into three subsets: the first, Vsp

including the vertices with “small” probabilities, i.e., at most p0, the second, Vip, includ-ing the ones with “intermediate” probabilities, i.e., greater than p0 and at most p′, and thethird, Vlp, including the vertices with “large” probabilities, i.e., greater than p′. Then, itseparately colors the vertices of G[Vsp], G[Vip] and G[Vlp] by using a proper set of col-ors for any subgraph and it finally takes the union of the colors used as solution for G.For G[Vsp], and G[Vip], we can use, as we will see just below, any polynomial feasible-coloring algorithm; for G[Vlp] we will use the algorithm of [12]. For simplicity in theanalysis of the algorithm that follows, we fix p0 = 1/n. This, as we will see, has noimportant impact in the approximation ratio concluded; p′ will be fixed later. The follow-ing lemmata deal with the approximation ratios of the algorithm just sketched in G[Vsp],G[Vip] and G[Vlp], respectively. As previously, denote by C∗ = (S∗

1 , . . . , S∗k∗) an optimal

a priori solution and by C = (S1, . . . , Sk) the approximate coloring computed. In theproof of the three lemmata, just below C∗ and C will deal with G[Vsp], G[Vip] and G[Vlp],respectively.

Lemma 2. (The ratio in G[Vsp]) Any feasible polynomial time approximation algorithmfor PCOLOR achieves in G[Vsp] approximation ratio bounded above by 2.

Proof. Denote by nsp the order of G[Vsp]. From (5) we get the following:

E(

G [Vsp] , C)

6

nsp∑

i=1

pi (15)

E (G [Vsp] , C∗) >

nsp∑

i=1

pi −

nsp∑

i=1

nsp∑

j=i+1

pipj (16)

Combining (16) and (15) we get the following for the ratio E(G,C∗)/E(G, C):

E (G [Vsp] , C∗)

E(

G [Vsp] , C) > 1 −

nsp∑

i=1

nsp∑

j=i+1

pipj

nsp∑

i=1

pi

= 1 −

(

nsp∑

i=1

pi

)2

−nsp∑

i=1

p2i

2nsp∑

i=1

pi

> 1 −

nsp∑

i=1

pi

2+

nsp∑

i=1

p2i

2nsp∑

i=1

pi

> 1 −

nsp∑

i=1

pi

2(17)

Since pi’s are assumed smaller than 1/n and nsp 6 n, the righthand side of (17) is at least1 − nsp/2n > 1/2. Observe now that the approximation ratio of a coloring algorithm

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in G[Vsp] is E(G[Vsp], C)/E(G[Vsp], C∗) which is smaller than, or equal to 2, and the

proof of lemma 2 is complete.

Lemma 3. (The ratio in G[Vip]) Any feasible polynomial time approximation algorithmfor PCOLOR achieves in G[Vip] approximation ratio bounded above by O(np′).

Proof. We deal here with a subgraph of G for which, for any vertex vi, pi > p0 = 1/n.Obviously,

∏

vi∈S∗

j

(1 − pi) 6

(

1 −1

n

)|S∗

j |

1 −∏

vi∈S∗

j

(1 − pi) > 1 −

(

1 −1

n

)|S∗

j |>

∣

∣S∗j

∣

∣

n−

∣

∣S∗j

∣

∣

(∣

∣S∗j

∣

∣− 1)

2n2(18)

where the last inequality in (18) is an easy application of the lefthand side of (3) withpi = 1/n for any vertex vi. Furthermore,

∣

∣S∗j

∣

∣

n−

∣

∣S∗j

∣

∣

(∣

∣S∗j

∣

∣− 1)

2n2=

∣

∣S∗j

∣

∣

n

(

1 −

∣

∣S∗j

∣

∣− 1

2n

)

>

∣

∣S∗j

∣

∣

n

n + 1

2n>

∣

∣S∗j

∣

∣

2n(19)

Summing inequality (19) for j = 1, . . . , k∗, we get E(G[Vip], C∗) > nip/2n, where nip

is the order of G[Vip]. On the other hand, using (13) we immediately get E(G[Vip], C) 6

nipp′. Consequently, using the bounds for E(G[Vip], C

∗) and E(G[Vip], C) just provided,E(G[Vip], C)/E(G[Vip], C

∗) 6 2np′ = O(np′), and the proof of lemma 3 is complete.

Finally remark that lemma 1 induces the following lemma dealing with approximationratio of [12] in G[Vlp].

Lemma 4. (The ratio in G[Vlp]) The algorithm of [12] achieves in G[Vlp] approximationratio bounded above by O(nlp log2 log nlp/p

′ log3 nlp) 6 O(n log2 log n/p′ log3 n) whenused to solve PCOLOR.

Theorem 2. PCOLOR can be approximated within ratio O(n log log n/ log3/2 n) in poly-nomial time.

Proof. In what follows, denote by C∗ an optimal a priori coloring for PCOLOR in G,by C∗[Vsp], C∗[V ip] and C∗[Vlp] the solutions induced by C∗ in G[Vsp], G[Vip] and G[Vlp],respectively, and by C∗

sp, C∗ip and C∗

lp, Csp, Cip and Clp the optimal and approximated apriori solutions in G[Vsp], G[Vip] and G[Vlp], respectively. We will prove that, for any x ∈

The Probabilistic Minimum Coloring Problem

{sp, ip, lp}, E(G,C∗) > E(G[Vx], C∗[Vx]) > E(G[Vx], C

∗x). Remark first that C∗[Vx] is a

particular feasible solution for G[Vx]; hence, E(G[Vx], C∗[Vx]) > E(G[Vx], C

∗x). In order

to prove the first inequality, fix an x and consider a color, say S∗j of C∗. If S∗j ∩ Vx = ∅,

then its contribution in C∗ is positive while in C∗[Vx] this contribution is null. If S∗j ⊂ Vx,

then its contribution is the same in both C∗ and C∗[Vx]. Suppose finally that none ofthe above cases does not hold, i.e., that a subset of S∗

j belongs to Vx. Set |S∗j | = `

and, without loss of generality, denote by p1, p2, . . . , p`, the vertex-probabilities in S∗j ;

suppose, furthermore, that {v1, . . . vi} ∈ S∗j ∩ Vx, while {vi+1, . . . , v`} ∈ S∗

j \ Vx. Then,

the contribution of S∗j in C∗[Vx] is 1−

∏im=1(1− pm) 6 1−

∏`m=1(1− pm) which is its

contribution in C∗. Iterating this argument for all the colors in C∗[Vx], the claim follows.Recall finally, that the algorithm sketched before theorem’s statement, colors the verticesof any G[Vx], x ∈ {sp, ip, lp} with a distinct set of colors and the a priori solution Cfinally provided is the union of these sets. Consequently, E(G,C) = E(G[Vsp], Csp) +

E(G[Vip], Cip) + E(G[Vip], Clp). So, using the fact that E(G,C∗) is at least as large asany of E(G[Vx], C

∗x), x ∈ {sp, ip, lp}, shown above, one immediately deduces that the

overall ratio of the algorithm in G is at most the sum of the ratios proved by lemmata 2, 3and 4, i.e., at most O(2 + np′ + (n log2 log n/p′ log3 n). Remark that the ratio claimed inlemma 3 is increasing with p′, while the one in lemma 4 is decreasing with p′. Equality ofexpressions np′ and n log2 log n/p′ log3 n holds for p′ = log log n/ log3/2 n. In this casethe value of the ratio obtained is O(n log log n/ log3/2 n). The proof of the theorem isnow complete.

4 The restriction of probabilistic coloring in bipartitegraphs

4.1 The complexity of PCOLOR

In what follows we denote by B(V, U,E) a bipartite graph with bipartition V and Uand edge-set E. We first do the following emphasized preliminary observation: in anybipartite graph, the bipartition (bicoloring) of its vertices is unique. Another observationis that the unique 2-coloring is not always the best a priori solution PCOLOR in a bipartitegraph B as it is shown in figure 1. There, the functional of the 3-coloring consisting oftaking v1, v2, v4 and u2 in the first color, u1 and u3 in the second color and v3 in the thirdcolor, is equal to 1.3896 and better than the one induced by the 2-coloring (V, U), equalto 1.8364.

We prove that PCOLOR is NP-hard even in bipartite graphs. For doing this we firstneed to introduce a variant of PCOLOR and to prove an initial completeness result that willserve us as a basis. We consider the following problem denoting by PCOLOR(B,Pr, 3).

Annales du LAMSADE n◦ 1

PSfrag replacements

0.9 0.9 0.2

0.8

0.8

0.10.1

v1 v2 v3 v4

u1 u2 u3

V

U

Figure 1: A bipartite graph B(V, U,E) where the 2-coloring is not the best-functional apriori solution.

It is a variant of PCOLOR where, given a probabilistic bipartite graph B, we are look-ing for the best 3-coloring, i.e., the three coloring for which the functional (2) associ-ated is the best over the ones of any other 3-coloring of B. The decision version ofPCOLOR(B,Pr, 3), denoted by PCOLOR(B,Pr, 3, K) is the one where we look for a3-coloring of functional’s value at most K.

Proposition 1. PCOLOR(B,Pr, 3, K) is NP-complete.

Proof. PCOLOR(B,Pr, 3, K) is obviously in NP. The completeness will be proved byreduction from the following problem, called 3-PRECOLORING: “given a bipartite graphB(V, U,E) with |V ∪ U | > 3 and three vertices v1, v2, v3, does there exist a 3-coloring(S1, S2, S3) of B such that vi ∈ Si for i = 1, 2, 3?”. 3-PRECOLORING was shown tobe NP-complete in [7]. Consider an instance B ′(V, U ′, E ′, v1, v2, v3) of 3-PRECOLORING

and remark that we can assume that v1, v2, v3 belong all either to V or to U ′; in theopposite case it is easy to see that 3-PRECOLORING is polynomial. Suppose that v1, v2, v3

are in V . We transform B ′(V, U ′, E ′, v1, v2, v3) into an instance of PCOLOR(B,Pr, 3, K)in the following way:

• add in U ′ three new vertices u1, u2, u3 and set U = U ′ ∪ {u1, u2, u3}; add in E ′

the edge-set E ′′ = {viuj : i, j = 1, 2, 3, i 6= j} and take E = E ′ ∪ E ′′; setB = B(V, U,E);

• the probability vector Pr is as follows: p(u1) = p(v1) = ε, p(u2) = p(v2) = ε2,p(u3) = p(v3) = ε3, for ε 6 1/10, p(vi) = 0, vi ∈ (V ∪ U) \ {vi, ui : i = 1, 2, 3};

• set K = 2ε + ε2 + 2ε3 − ε4 − ε6.

Obviously, the transformation of B ′ into B can be performed in polynomial time. More-over, we claim that (B,Pr, 3, K) has a 3-coloring with functional at most 2ε+ ε2 +2ε3 −

The Probabilistic Minimum Coloring Problem

PSfrag replacementsv1 v2 v3

u1 u2 u3

Figure 2: The graph M3,3 = B[{vi, ui : i = 1, 2, 3}].

PSfrag replacements

v1 v2 v3

u1 u2 u3

Figure 3: The 3-coloring C∗ restricted to M3,3; E(B,C∗) = 2ε + ε2 + 2ε3 − ε4 − ε6.

Annales du LAMSADE n◦ 1

ε4 − ε6 iff we can 3-color B ′(V, U ′, E ′, v1, v2, v3) by assigning any of v1, v2, v3 with adistinct color.

It is easy to see that the contribution of any vertex in (V ∪ U) \ {vi, ui : i = 1, 2, 3}in any coloring of B is null. Denote by M3,3 the graph B[{vi, ui : i = 1, 2, 3}] (thisgraph is a kind of bipartite complement of a perfect matching on 3 edges, in our case onedges viui, i = 1, 2, 3) and observe that the (non-zero) value of any coloring of B is valueof some coloring of M3,3. In figure 2 the graphM3,3 = B[{vi, ui : i = 1, 2, 3}] is shown.Observe also that the value of K, introduced in the third item just above, corresponds to a3-coloring C∗ of B taking {vi, ui}, i = 1, 2, 3 in the same color, say Si; this coloring hasfunctional equal to

∑3i=1(1− (1− εi)2) = 2ε+ ε2 +2ε3 − ε4 − ε6. In figure 3, the optimal

functional-value 3-coloring C∗ restricted to M3,3 is presented. By a simple inspection ofany other 3-coloring C of B (there exist 7 distinct functional-value such colorings), onecan easily see that the functional of C is, for ε 6 1/10, greater than the functional Kof C∗. In figure 4 the 6 distinct-value 3-colorings, other than C∗, of M3,3 together withthe values of the functionals associated are illustrated.

So, if a 3-coloring C∗ of B is polynomially computed with functional K = 2ε +ε2 + 2ε3 − ε4 − ε6, then C∗ restricted to M3,3 is of the form of figure 3 (recall that thecontribution of the vertices of (V ∪ U) \ {vi, ui : i = 1, 2, 3} in any coloring of Bis 0). Consequently, C∗ 3-colors the vertices of B ′ by assigning a distinct color to eachof v1, v2, v3.

Conversely, if a 3-coloring assigning a distinct color, say S1, S2 and S3 to eachof v1, v2, v3, respectively, is computed for B ′, then (S1 ∪ {u1}, S2 ∪ {u2}, S3 ∪ {u3})is a coloring for B with functional K = 2ε + ε2 + 2ε3 − ε4 − ε6.

Remark 2. The functional of the (unique) 2-coloring of B[{vi, ui : i = 1, 2, 3}] (figure 5)has value 2(ε + ε2 − ε4 − ε5 + ε6) > 2ε + ε2 + 2ε3 − ε4 − ε6 for ε 6 1/10.

We now consider problem PCOLOR(B,Pr, k), where we look for the best k-coloring forany k ∈ {4, . . . , n} and its decision version PCOLOR(B,Pr, k,K). We will establish thatPCOLOR(B,Pr, k,K) is NP-complete for any such k.

Consider a bipartite graph Mk,k, the bipartite complement of a perfect matching with kedges (i.e., a bipartite graph B(V, U,E) with |V | = |U | = k and with E = E(Bk,k) \{viui, vi ∈ V, ui ∈ U, i = 1, . . . , k}), where by Bk,k we denote the complete bipartitegraph with |V | = |U | = k. Call a color horizontal if it is a proper subset either of V , orof U ; a coloring will be called horizontal if it is composed only by horizontal colors. Onthe other hand, call a color vertical if it contains vertices from both V and U ; a coloringof Mk,k will be called vertical if all its colors are vertical, otherwise it will be callednon-vertical. The following properties hold for the colorings of Mk,k:

The Probabilistic Minimum Coloring Problem

PSfrag replacements

v1 v2 v3

u1 u2 u3

(a) E(B, C) = 2ε + 2ε2 − ε4 −

ε5 + ε6

PSfrag replacements

v1 v2 v3

u1 u2 u3

(b) E(B, C) = 2ε+2ε2+ε3−ε4−

2ε5 + ε6

PSfrag replacements

v1 v2 v3

u1 u2 u3

(c) E(B, C) = 2ε + 2ε2 + ε3 −

2ε4 − ε5 + ε6

PSfrag replacements

v1 v2 v3

u1 u2 u3

(d) E(B, C) = 2ε+ε2+2ε3−2ε5

PSfrag replacements

v1 v2 v3

u1 u2 u3

(e) E(B, C) = 2ε+2ε2 +2ε3 −

3ε4

PSfrag replacements

v1 v2 v3

u1 u2 u3

(f) E(B, C) = 2ε + 2ε2 − ε6

Figure 4: The 6 distinct-value non-optimal 3-colorings of B[{vi, ui : i = 1, 2, 3}] withthe values of the functionals associated.

Annales du LAMSADE n◦ 1

PSfrag replacements

v1 v2 v3

u1 u2 u3

Figure 5: The 2-coloring of B[{vi, ui : i = 1, 2, 3}] with functional of value 2(ε + ε2 −ε4 − ε5 + ε6).

1. any vertical color of Mk,k is exclusively of the form {vi, ui}, i = 1, . . . , k; thisis easily deduced from the particular form of Mk,k implying that independent sets{vi, ui}, i = 1, . . . , k are all maximal for the inclusion;

2. the non-vertical colors of any non-vertical coloring of Mk,k are horizontal, i.e., thereis no coloring of Mk,k with other than horizontal or vertical colors;

3. for any i = 1, . . . , k, if vi and ui belong to two different colors, these colors arehorizontal; this is concluded by the fact that vi excludes any vertex of U (otherthan ui), while ui excludes any vertex of V (other than vi).

Proposition 2. Consider Mk,k and assume that there exists a vertex-probability systemPr with p(vi) = p(ui) = pi, i = 1, . . . , k, such that: (i) for any i, 3 6 i < k, the func-tional of a vertical coloring of any subgraph Mi,i of Mk,k is smaller than the functional ofthe 2-coloring of Mi,i and (ii) for any induced subgraph B ′ of Mk,k, the functional-valueof any horizontal coloring is greater than the one of the 2-coloring of B ′. Consider ak-coloring C of Mk,k with value equal to

∑ki=1(1 − (1 − pi)

2). Then this coloring isvertical, i.e., of the form {{vi, ui} : i = 1, . . . , k} and the functional associated with it, isthe smallest over the functional of any feasible coloring of Mk,k.

Proof. Set C = (S1, . . . , Sk), and remove vertices vk and uk from Mk,k together withtheir incident edges, in order to obtain graph Mk−1,k−1. Then, the following three casescan appear: (a) Mk−1,k−1 remains colored with k colors; (b) Mk−1,k−1 is colored withk− 1 colors, i.e., one color is removed from C; (c) Mk−1,k−1 is colored with k− 2 colors,i.e., two colors are removed from C.

The Probabilistic Minimum Coloring Problem

Study of case (a): Mk−1,k−1 remains colored with k colors

By property 3, vk and uk belong to two distinct horizontal colors, say S1 (⊂ V ) and S2

(⊂ U ), respectively. Set S ′1 = S1 \ {vk}, S ′

2 = S2 \ {uk}; for simplicity, set∏

S′

1=

∏

vj∈S′

1(1 − pj),

∏

S′

2=∏

uj∈S′

2(1 − pj) and for, i = 3, . . . k, set

∏

Si=∏

x`∈Si(1 − p`).

Denote by E(Mk,k, C) the value of the initial k-coloring C in Mk,k; then,

E(

Mk,k, C)

=

1 − (1 − pk)∏

S′

1

+

1 − (1 − pk)∏

S′

2

+k∑

i=3

(

1 −∏

Si

)

(20)

Assume now that C = (S1, . . . , Sk) has m horizontal and k − m vertical colors. Assumealso that, up to a reordering of the colors, the m first ones are horizontal and the k − mlast ones are vertical; in other words, the vertical colors in C are Si+1 = {vi, ui}, i =m, . . . , k − 1 (both vk and uk belong to horizontal colors). Remark, finally that, underthis assumption, both graphs, the one induced by S1 ∪ . . . ∪ Sm and the one induced bySm+1 ∪ . . . ∪ Sk are both bipartite complements of a perfect matching on vertex-sets:{vi, ui : i = 1, . . . ,m − 1, k}, the former, and {vi, ui : i = m, . . . , k − 1}, the latter.

Set C ′ = (S ′1, S

′2, S3, . . . , Sk); C ′ is identical to C except for independent sets S ′

1

and S ′2 which they miss in vk and uk with respect to S1 and S2. Consider Mk−1,k−1; based

upon the remarks just above, the subgraphs of Mk−1,k−1 induced by S ′1 ∪S ′

2 ∪S3 . . .∪Sm

and by Sm+1∪ . . .∪Sk, respectively, are both bipartite complements of a perfect matchingon vertex-sets: {vi, ui : i = 1, . . . ,m − 1}, the former, and {vi, ui : i = m, . . . , k − 1},the latter. Then, the functional associated with the k-coloring C ′ of Mk−1,k−1 is:

E(

Mk−1,k−1, C′)

=

1 −∏

S′

1

+

1 −∏

S′

2

+m∑

i=3

(

1 −∏

Si

)

+k∑

i=m+1

(

1 − (1 − pi−1)2) (21)

The three first terms in the righthand side of (21) correspond to the value of an horizontalcoloring in the subgraph B ′ of Mk,k induced by S ′

1∪S ′2∪S3∪ . . .∪Sm. By assumption (ii)

in the statement of the proposition, the value of this coloring is greater than the value of the2-coloring of B ′, i.e., greater than 2−2

∏m−1i=1 (1−pi). In other words, m−(

∏

S′

1+∏

S′

2)−

∑mi=3

∏

Si> 2 − 2

∏m−1i=1 (1 − pi), implying

m − 2 −

∏

S′

1

+∏

S′

2

−m∑

i=3

∏

Si

> −2m−1∏

i=1

(1 − pi) (22)

Annales du LAMSADE n◦ 1

Remark that, using (21), (20) can be written as:

E(

Mk,k, C)

= 2 − (1 − pk)∏

S′

1

− (1 − pk)∏

S′

2

+m∑

i=3

(

1 −∏

Si

)

+k∑

i=m+1

(

1 − (1 − pi−1)2)

= 2 + m − 2 −

∏

S′

1

+∏

S′

2

−m∑

i=3

∏

Si

+pk

∏

S′

1

+∏

S′

2

+k∑

i=m+1

(

1 − (1 − pi−1)2)

> 2 − 2m−1∏

i=1

(1 − pi) + pk

∏

S′

1

+∏

S′

2

+k∑

i=m+1

(

1 − (1 − pi−1)2)(23)

where, the inequality in (23) holds thanks to (22). We now show that

2 − 2m−1∏

i=1

(1 − pi) + pk

∏

S′

1

+∏

S′

2

+k∑

i=m+1

(

1 − (1 − pi−1)2)

>

2 − 2

(

m−1∏

i=1

(1 − pi)

)

(1 − pk) +k∑

i=m+1

(

1 − (1 − pi−1)2) (24)

The difference between lefthand and righthand sides of (24) is −2∏m−1

i=1 (1 − pi) +pk(∏

S′

1+∏

S′

2) + 2

∏m−1i=1 (1 − pi)(1 − pk) = pk(

∏

S′

1+∏

S′

2) − 2pk

∏m−1i=1 (1 − pi) =

pk(∏

S′

1+∏

S′

2−2∏m−1

i=1 (1− pi)) > 0 since∏

S′

1and

∏

S′

2are subproducts of

∏m−1i=1 (1−

pi).

Remark that the first two terms in the expression of righthand side of (24) is the valueof the 2-coloring in the graph induced by the subgraph Mm,m of Mk,k induced by ver-tices v1, u1, . . . , vm−1, um−1, vk, uk. By assumption (i) in the statement of the proposition,this value is greater than the value of a vertical coloring in Mm,m. Consequently, theassumption of case (a) is in contradiction with the value of the coloring C assumed; inother words case (a) cannot occur under the hypotheses of the proposition. The proof ofcase (a) is now complete.

The Probabilistic Minimum Coloring Problem

Study of case (b): Mk−1,k−1 is colored with k − 1 colors

One color is removed from C with the removal of {vk, uk}. Denote by C ′ =(S1, . . . , Sk−1) the coloring so-obtained. Two subcases may appear here: (b.1) both vk

and uk belong to Sk and (b.2) vk and uk belong to two distinct colors.

Study of subcase (b.1): both vk and uk belong to Sk

Since {vk, uk} belong to the same color Sk, by property 1 stated above no other vertexcan simultaneously belong to it. In this case, the value of C ′ is

∑k−1i=1 (1 − (1 − pi)

2) andthe proof of subcase (b.1) is complete.

Study of subcase (b.2): vk and uk belong to two distinct colors

Assume that one among vk, uk, say vk, belongs to a color in C ′, say S1; then uk is acolor by itself, i.e., Sk = {uk}. By property 3, S1 is horizontal; set S ′

1 = S1 \ {vk} andC ′′ = (S ′

1, . . . , Sk−1) and use the product notations of case (a). Assume as previouslythat S1, . . . , Sm are horizontal and that Sm+1, . . . , Sk−1 are vertical. Then the value of C ′′

for Mk−1,k−1 is E(Mk−1,k−1, C′′) = m −

∏

S′

1−∑m

i=2

∏

Si+∑k−1

i=m+1(1 − (1 − pi)2).

Observe that m −∏

S′

1−∑m

i=2

∏

Siis the value of a feasible coloring in the subgraph

of Mk,k induced by S ′1 ∪S2 ∪ . . .∪Sm and thus, by assumption (ii) in the statement of the

proposition, it is greater than 2 − 2∏m

i=1(1 − pi). We so have E(Mk−1,k−1, C′′) = m −

∏

S′

1−∑m

i=2

∏

Si+∑k−1

i=m+1(1−(1−pi)2) > 2−2

∏mi=1(1−pi)+

∑k−1i=m+1(1−(1−pi)

2).In other words, m −

∏

S′

1−∑m

i=2

∏

Si> 2 − 2

∏mi=1(1 − pi), or

m − 1 −∏

S′

1

−m∑

i=2

∏

Si

> 1 − 2

m∏

i=1

(1 − pi) (25)

On the other hand, the value of C assumed for Mk,k is

E(

Mk,k, C)

= 1 −∏

S′

1

(1 − pk) + m − 1 −m∑

i=2

∏

Si

+k−1∑

i=m+1

(

1 − (1 − pi)2)+ 1 − (1 − pk)

= 1 + pk

∏

S′

1

+

m − 1 −∏

S′

1

−m∑

i=2

∏

Si

Annales du LAMSADE n◦ 1

+k−1∑

i=m+1

(

1 − (1 − pi)2)+ pk (26)

Using (25) for the term in square brackets of (26), we get: E(Mk,k, C) > 2− 2∏m

i=1(1−

pi) + pk

∏

S′

1+∑k−1

i=m+1(1 − (1 − pi)2) + pk. We show that

2 − 2m∏

i=1

(1 − pi) + pk

∏

S′

1

+k−1∑

i=m+1

(

1 − (1 − pi)2)+ pk >

2 − 2m∏

i=1

(1 − pi) (1 − pk) +k−1∑

i=m+1

(

1 − (1 − pi)2) (27)

The difference between lefthand and righthand sides of (27) is, with some easy algebra,pk(∏

S′

1+1 − 2

∏mi=1(1 − pi)) > 0 since

∏

S′

1is a subproduct of

∏mi=1(1 − pi) and 1 >

∏mi=1(1 − pi).

Since the first two terms in righthand side of (27) is the value of the 2-coloring of thesubgraph of Mk,k induced by S1∪ . . .∪Sm∪{uk}, it is, by assumption (i) of proposition’sstatement, greater than, or equal to, the value of a vertical coloring in this subgraph.Consequently, the assumption of subcase (b.2) is in contradiction with the value of Cclaimed in proposition’s statement; in other words, subcase (b.2) does never occur. Thestudy of subcase (b.2) is now complete.

Study of case (c): Mk−1,k−1 is colored with k − 2 colors

Here, vk and uk are two distinct colors by themselves, say Sk−1 and Sk. By a similarreasoning as previously (renumbering, if necessary, the colors), we get: E(Mk,k, C) >∑k−2

i=1 (1 − (1 − pi)2) + 2pk >

∑ki=1(1 − (1 − pi)

2), contradicting so the value of Cassumed in the statement of the proposition. Once more, case (c) cannot occur and itsproof is complete.

In all we have proved that, under the assumptions made, only subcase (b.1) can hold.Moreover, from the proofs of the cases (a), (b) and (c), one can immediately deducethat the vertical coloring of Mk,k is the one with the smallest functional over any otherfeasible coloring of Mk,k. An easy backwards induction shows finally that the claims ofthe proposition remain valid for any k > 3 and this completes its proof.

We now show that a vertex-probability system satisfying assumptions (i) and (ii) inthe statement of proposition 2 really exists. We first start from the latter assumption andwe show that it holds, in fact, for any vertex-probability system and for any bipartitegraph B(V, U,E).

The Probabilistic Minimum Coloring Problem

Lemma 5. Consider a bipartite graph B(V, U,E) of order n and denote by Pr =(p1, p2, . . . , pn) the vector of its vertex-probabilities. Consider also a horizontal k-coloring C = (S1, . . . Sk) of B. Then, E(B,C) > E(B, (V, U)).

Proof. Let S1, . . . , S` be the colors of V and S`+1, . . . , Sk be the ones of U . We willuse the product notations of the proof of proposition 2. We prove, by induction on ` that` −

∑`j=1

∏

Sj> 1 −

∏

vi∈U (1 − pi).

Assume ` = 2, i.e., V = S1 ∪ S2. We will prove that 2 −∏

S1−∏

S2>

1 −∏

vi∈S1∪S2(1 − pi). Then, after some very easy algebra, 1 −

∏

vi∈S1(1 − pi) + 1 −

∏

vi∈V \S1(1−pi)−1+

∏

vi∈V (1−pi) = (1−∏

vi∈S1(1−pi))(1−

∏

vi∈V \S1(1−pi)) > 0;

hence the claim is true for ` = 2.

Suppose it true for any ` from 2 to m, i.e., ` −∑`

j=1

∏

Sj> 1 −

∏

vi∈∪`j=1

Sj(1 − pi).

Consider now ` = m + 1, V = ∪m+1j=1 Sj and quantity m + 1 −

∑m+1j=1

∏

Sj= m −

∑mj=1

∏

Sj+1 −

∏

Sm+1. The first two terms of the sum are, by the induction hypothesis,

greater than 1−∏

vi∈∪mj=1

Sj(1−pi); so, m−

∑mj=1

∏

Sj+1−

∏

Sm+1> 1−

∏

vi∈∪mj=1

Sj(1−

pi) + 1 −∏

vi∈Sm+1(1 − pi) > 1 −

∏

vi∈∪m+1j=1

Sj(1 − pi), where the last inequality holds

thanks to the induction basis (considering S1 = ∪mj=1Sj and S2 = Sm+1).

Obviously the same proof holds for colors S`+1, . . . , Sk and U . Consequently,E(B,C) = k −

∑ki=1

∏

Si> 2 −

∏

vi∈V (1 − pi) −∏

vi∈U (1 − pi) = E(B, (V, U))and the proof of the lemma is complete.

The following lemma deals with assumption (i) of proposition 2, that there exists aprobability system for the vertices of a bipartite graph Mk,k such that, for any i, 3 6 i < k,the functional of a vertical coloring of any subgraph Mi,i of Mk,k is smaller than thefunctional of the 2-coloring of Mi,i.

Lemma 6. Consider a bipartite graph Mn,n, set V = {v1, . . . , vn} and U = {u1, . . . , un}both sets ranged in decreasing vertex probability. Set p(vi) = p(ui) = εi, for ε 6 1/3.Then this vertex-probability system verifies assumption (i) of proposition 2.

Proof. We fix a k 6 n and we show that for any isomorphicMk,k of Mn,n, the functionalof a vertical coloring of Mk,k is smaller than the functional of the 2-coloring Mk,k. Orderthe vertices of Mk,k in decreasing order of probabilities and, without loss of generality,set Vk = {vk1

, . . . , vkk} and Uk = {uk1

, . . . , ukk}. We will compute an ε such that, if

we set pi = p(vi) = p(ui) = εi, i = 1, . . . n, and if we assume a vertical coloringCk = {{vki

, uki} : i = 1, . . . , k} for Mk,k, then:

k∑

i=1

(

1 − (1 − pki)2)

6 2 − 2k∏

i=1

(1 − pki) (28)

Annales du LAMSADE n◦ 1

For the lefthand side of (28) we have:

k∑

i=1

(

1 − (1 − pki)2) = 2

k∑

i=1

pki−

k∑

i=1

p2ki

(29)

For the righthand side of (28), we get, using the first inequality of (3):

2 − 2k∏

i=1

(1 − pki) > 2

k∑

i=1

pki− 2

k∑

i=1

k∑

j=i+1

pkipkj

(30)

Using (30), in order to prove (28) it suffices to compute pkisuch that 2

∑ki=1 pki

−∑k

i=1 p2ki

6 2∑k

i=1 pki− 2

∑ki=1

∑kj=i+1 pki

pkj, or

k∑

i=1

p2ki

> 2k∑

i=1

pki

k∑

j=i+1

pkj(31)

Fix ki ∈ {1, . . . , k}; suppose that vertex vki∈ Vk corresponds to vertex v` ∈ V (` > ki)

and recall that vertices are ordered in decreasing probability-order. We want to com-pute pki

(of the form εx for some integer x ∈ 1, . . . , n), in such a way that (31) is satisfied,i.e., pki

> 2∑k

j=i+1 pkj. Obviously, 2

∑kj=i+1 pkj

6 2∑n

j=`+1 pj; therefore, we want tocompute ε such that pki

= ε` > 2∑n

j=`+1 εj . The righthand side of this inequality is ageometric series with ratio ε. Its value is (ε`+1−εn+1)/(1−ε); so, we compute ε such thatε` > 2(ε`+1−εn+1)/(1−ε), or 1 > 2(ε−εn−`+1)/(1−ε). Function ` 7→ (ε−εn−`+1)/(1−ε)is decreasing with `; so, 2(ε − εn−`+1)/(1 − ε) 6 2(ε − εn)/(1 − ε). So, we compute εsatisfying 1 > 2(ε − εn)/(1 − ε), i.e., 1 > 3ε − 2εn which is true for ε 6 1/3, q.e.d.

Lemmata 5 and 6 guarantee that the assumptions of proposition 2 are realistic. We arewell prepared now use them in order to face the main complexity result of this section,namely that PCOLOR(B,Pr, k) is NP-hard, for any k > 3.

Theorem 3. PCOLOR(B,Pr, k,K) is NP-complete.

Proof. Our problem is obviously in NP. On the other hand in proposition 1 we have re-ally proved that PCOLOR(B,Pr, 3, K) is NP-complete even for bipartite graphs B ′ hav-ing the following four additional characteristics: (a) only six vertices vi, ui, i = 1, 2, 3of B′ have non-zero probabilities; (b) the subgraph of B ′ induced by these six verticesis a M3,3; (c) pi = p(vi) = p(ui) = εi, i = 1, 2, 3, for a suitable ε, for exampleε < 1/10. We reduce PCOLOR(B ′,Pr, 3, K ′), where B′ fits characteristics (a) to (c)and K ′ =

∑3i=1(1 − (1 − pi)

2), to PCOLOR(B,Pr, k,K). We consider an instance ofPCOLOR(B′,Pr, 3, K ′) and construct B as follows: we put B ′ together with a Mk−3,k−3;we link the vertices of V (B ′) with the ones of U(Mk−3,k−3) in such a way that the graph

The Probabilistic Minimum Coloring Problem

induced by V (B ′) ∪ U(Mk−3,k−3) is a complete bipartite graph; we do so with U(B ′)and V (Mk−3,k−3). Setting V (Mk−3,k−3) = {v4, . . . vk} and U(Mk−3,k−3) = {u4, . . . uk},we set pi = p(vi) = p(ui) = εi, i = 4, . . . , k. Finally, we set K =

∑ki=1(1 − (1 − pi)

2).By what has been discussed previously, around proposition 2 and property 1 (a verticalcolor on vi and ui cannot have vertices other than these two ones) and by the fact that thecontribution of any other vertex of B ′ is the functional of any coloring is 0, one can imme-diately deduce that B has a k-coloring with functional at most K =

∑ki=1(1− (1− pi)

2),if and only if B′ has a 3-coloring with functional at most K ′ =

∑3i=1(1 − (1 − pi)

2),q.e.d.

4.2 On the approximation of PCOLOR in bipartite graphs

Consider a bipartite graph B(V, U,E) of order n and the 2-coloring C = (V, U). Fixan optimal-functional coloring C∗ = (S∗

1 , . . . , S∗k∗) of B and an ε ∈ (0, 2). Assume that

the first k∗ colors are such that, for any S∗j ∈ (S∗

1 , . . . S∗k∗

),∑

vi∈S∗

jpi 6 ε, while for the

k∗ − k∗ remaining ones, the sum of the probabilities of the vertices in any color is greaterthan ε. Set C∗ = (S∗

1 , . . . S∗k∗

) and denote by n the order of the subgraph of B inducedby ∪S∗

j ∈C∗S∗j . The following inequality, proved by an easy reduction on i will be used

later:

mini=1,...,n

{

ai

bi

}

6

n∑

i=1

ai

n∑

i=1

bi

6 maxi=1,...,n

{

ai

bi

}

(32)

Theorem 4. In any bipartite graph, its unique 2-coloring achieves approximation ratiobounded above by 2.773 for PCOLOR.

Proof. We study the ratio E(B,C∗)/E(B,C), i.e., the inverse of the approximation ratioof the solution C. Set |S∗

j | = `∗j ; Using (3), (7) and (13), we obtain

E (B,C∗)

E(B,C)>

∑

S∗

j ∈C∗

(

∑

vi∈S∗

j

pi −∑

vi∈S∗

j

`∗j∑

k=i+1

pipk

)

+∑

S∗

j ∈C∗\C∗

(

1 − exp

{

−∑

vi∈S∗

j

pi

})

min

{

n∑

i=1pi, 2

}

>

∑

S∗

j ∈C∗

(

∑

vi∈S∗

j

pi −∑

vi∈S∗

j

`∗j∑

k=i+1

pipk

)

+∑

S∗

j ∈C∗\C∗

(

1 − exp

{

−∑

vi∈S∗

j

pi

})

n∑

i=1pi + 2

Annales du LAMSADE n◦ 1

(32)>

min

n∑

i=1pi −

∑

S∗

j ∈C∗

∑

vi∈S∗

j

`∗j∑

k=i+1

pipk

n∑

i=1pi

,

∑

S∗

j ∈C∗\C∗

(

1 − exp

{

−∑

vi∈S∗

j

pi

})

2

(33)

Get the first term in the min-expression of (33); we have:

n∑

i=1

pi −∑

S∗

j ∈C∗

∑

vi∈S∗

j

`∗j∑

k=i+1

pipk

n∑

i=1

pi

= 1 −

∑

S∗

j ∈C∗

∑

vi∈S∗

j

`∗j∑

k=i+1

pipk

n∑

i=1

pi

(34)

On the other hand,

∑

vi∈S∗

j

`∗j∑

k=i+1

pipk =

(

∑

vi∈S∗

j

pi

)2

−∑

vi∈S∗

j

p2i

26

(

∑

i=1

pi

)2

2(35)

and, combining (34) and (35), we get:

1 −

∑

S∗

j ∈C∗

∑

vi∈S∗

j

`∗j∑

k=i+1

pipk

n∑

i=1

pi

> 1 −

∑

S∗

j ∈C∗

(

∑

vi∈S∗

j

pi

)2

2n∑

i=1

pi

(36)

Using (32) and taking into account that∑n

i=1 pi =∑

S∗

j ∈C∗

∑

vi∈S∗

jpi, we get from (36):

1 −

∑

S∗

j ∈C∗

(

∑

vi∈S∗

j

pi

)2

2n∑

i=1

pi

> 1 −

(

∑

vi∈S∗

0

pi

)2

2∑

vi∈S∗

0

pi

= 1 −

∑

vi∈S∗

0

pi

2> 1 −

ε

2(37)

The Probabilistic Minimum Coloring Problem

(the last inequality in (37) holds thanks the assumption on the probabilities-sum in themembers of C∗), where we have denoted by S∗

0 the color of C∗ realizing the maximumfor the quantity (

∑

vi∈S∗

jpi)

2/∑

vi∈S∗

jpi.

On the other hand, for the second term in the min-expression of (33) holds:

∑

S∗

j ∈C∗\C∗

(

1 − exp

{

−∑

vi∈S∗

j

pi

})

2>

1 − exp

{

−∑

vi∈S∗

j

pi

}

2>

1 − exp{−ε}

2(38)

because of the assumption on the probabilities-sum in the color-collection C∗ \ C∗.

In all, using (37) and (38): E(B,C∗)/E(B,C) > min{1 − ε/2, (1 − exp{−ε})/2};so, the approximation ratio of the solution C = (V, U) for PCOLOR in any bipartitegraph B(V, U,E) is bounded above by max{2/(2 − ε), 2/(1 − exp{−ε})}. Equality forthese terms implies ε ≈ 1.278 and then, the value of both them are smaller than 2.773.

4.3 The graphs Mn,n under identical vertex-probabilities

Let us focus ourselves in the case of identical vertex-probabilities and consider a Mn,n.Using the notations of section 4.1, any coloring here is either a vertical one, or an horizon-tal one or, finally, a mixed one with some vertical and some horizontal independent sets.Recall that lemma 5 holds for any vertex-probability system. Consequently, a 2-coloringof Mn,n always dominates any other horizontal coloring. On the other hand, since proba-bilities are all identical, the values of any mixed coloring with a precise number of verticalcolors and a 2-coloring for the graph induced by the rest of the vertices are all identical too(depending on the number of the vertical colors). Consequently, an algorithm consistingof: (a) evaluating the functional associated with the 2-coloring of Mn,n; (b) for i = 1 to nevaluating the functional associated with a mixed coloring consisting of i vertical colorsand a 2-coloring on the subgraph of Mn,n induced by its non-colored vertices; (c) evalu-ating the value of a vertical coloring; (d) retaining the solution associated with the best ofthe functionals computed in steps (a) to (c); constitutes an optimal polynomial algorithmfor PCOLOR in Mn,n, for the case where vertex-probabilities are all identical.

Proposition 3. PCOLOR is polynomial in Mn,n, in the case where vertex-probabilitiesare all equal.

4.4 The complements of bipartite graphs

Given a bipartite graph B(V, U,E) its complement B(V, U, E) is a loopless consisting oftwo cliques, one on V and one on U , plus the set of edges E ′ = {vivj /∈ E : vi ∈ V, uj ∈

Annales du LAMSADE n◦ 1

U}; in other words the edges of E are the edges of the cliques K|V | and K|U | and the edgesbetween V and U missing from E. These graphs have the property that any independentset is of cardinality at most 2. In other words, any coloring there, is a collection ofindependent sets of size 2 and of singletons. The following lemma characterizes thefunctional’s value of a such a coloring.

Lemma 7. Let B be the complement of bipartite graph B, let n be the order of B and B,let C be a coloring of B and let S = {{vik , vjk

} : k = 1, . . . , |S|} be the collection of

independent sets of size 2 in C. Then E(B, C) =∑n

i=1 pi −∑|S|

k=1 pikpjk.

Proof. The value of a color {vik , vjk} ∈ C will be 1−(1−pik)(1−pjk

) = pik+pjk−pikpjk

;on the other hand, the value of a singleton {vi} ∈ C will be pi. Consequently, it is easyto see that the functional of C will be E(B, C) =

∑ni=1 pi −

∑|S|k=1 pikpjk

as claimed.

It is easy to see from lemma 7 that the first term of the functional is constant;so, E(B, C) is minimized when its second term is maximized. Consider the bipartitegraph B′(V, U,E(B′)) with E(B′) = (V × U) \ E ′ and assign to any edge vivj ∈ E(B′)

weight pipj . Then, collection S becomes a matching of B ′ and the term∑|S|

k=1 pikpjk

the total weight of this matching. Recall finally that a maximum weight matching canbe polynomially computed in any graph ([25]). Then consider the following algorithm:given B transform it into B ′ and weight any of its edges with the product of the probabil-ities of its endpoints; compute a maximum weight matching M in B ′; color endpoints ofany edge of M with an unused color (the same for both endpoints); color the remainingvertices of B′ with an unused color by such vertex. From what has been discussed thecoloring so produced is optimal and the following result concludes the section.

Theorem 5. PCOLOR is polynomial in complements of bipartite graphs.

5 Concluding remarks

There exists a list of interesting open problems dealing with the results of this paper. Forexample, the complexity of PCOLOR remains open, notably for natural classes of bipar-tite graphs as chains, or trees, or even for classes of graphs “close” to bipartite ones, forexample, the split graphs. In another order of ideas, an interesting approximation strategyfor solving hard minimization problem is the so-called “master-slave” approximation. Itconsists of solving a minimization problem (the master one) by repeatedly solving a maxi-mization one (the slave problem) (for more details on this technique, cf., [1, 18, 26]). Thiskind of technique has a very natural application in the case of minimum coloring wherethe slave problem is the maximum independent set. It consists of iteratively computingan independent set in the graph, of coloring its vertices with the same unused color, of

The Probabilistic Minimum Coloring Problem

removing it from the graph and of repeating these stages in the subsequent surviving sub-graphs until all vertices are colored. The slave independent set problem for PCOLOR is theone of determining the independent set S∗ maximizing quantity |S|/(1−

∏

vi∈S(1− pi))over any independent set of the input graph. Obviously, this problem is NP-hard in gen-eral graphs since for pi = 1 for any vertex of the input graph we recover the classicalmaximum independent set problem. However, approximation of it in general graphs andcomplexity and, eventually, approximation results in graph-families as the ones dealt inthis paper seem us interesting to be studied.

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