Introduction to the min cost homomorphism problem for undirected and directed graphs Gregory Gutin...

Introduction to the min cost homomorphism problem

for undirected and directed graphs

Gregory Gutin

Royal Holloway, U. London, UK

and U. Haifa, Israel

Homomorphisms For a pair of graphs G and H, a mapping

h:V(G) → V(H) is called a homomorphism if xy ε E(G) implies h(x)h(y) ε E(H) (also called H-coloring).

uv

xy

w z 1 2 3

GH

The Homomorphism Problem

Fix a graph H. H-HOM: For an input graph G, check whether there is a homomorphism of G to H.

Theorem (Hell & Nešetřil, 1990) Let H be an unditected graph. H-HOM is polynomial

time solvable if H is bipartite or has a loop. If H is not bipartite and it has no loop, then H-HOM is NP-complete.

Theorem (Bang-Jensen, Hell & MacGillivray, 1988) Let H be a semicomplete graph. H-HOM is polynomial

time solvable if H has at most one cycle. If H has at least two cycles, then H-HOM is NP-complete.

The List Homomorphism Problem

Fix a graph H. H-ListHOM: For an input graph G and a list L(v) for each v ε V(G), check if there is a homomorphism f of G to H s.t. f(v) ε L(v).

Theorem (Feder, Hell & Huang, 1999) Let H be an undirected loopless graph. H-ListHOM is

polynomial-time solvable if H is bipartite and the complement of a circular-arc graph. Otherwise, H-ListHOM is NP-complete.

Theorem (Gutin,Rafiey,Yeo, 2006) If H is a semicomplete digraph with at most one cycle, H-ListHOM is polynomial-time solvable. If H is a SD with at least two cycles, then H-ListHOM is NP-complete.

The Min Cost Homomorphism Problem

Introduced in Gutin, Rafiey, Yeo and Tso, 2006. Fix H. MinHOM(H): Given a graph G and a cost ci(u) of mapping u to i for each u ε V(G), i ε V(H), find if there is a homomorphism of G to H and if it does, then find a homomorphism f of G to H of minimum cost.

cost(f)= ΣuεV(G) cf(u)(u)

Min Cost vs ListHOM

H-ListHOM: G; L(v), v ε V(G)

Special MinHOM(H): ci(v)=0 if i ε L(v) and ci(v)=1, otherwise. Э H-coloring of cost 0?

Motivation: LORA

• Level of Repair Analysis (LORA): procedure for defence logistics, optimal provision of repair and maintenance facilities to minimize overall life-cycle costs

• Complex system with thousands of assemblies, sub-assemblies, components, etc.

• Has λ ≥2 levels of indenture and with r ≥ 2 repair decisions

• LORA can be reduced to MinHOM(H) for some bipartite graphs H (Gutin, Rafiey, Yeo, Tso, ‘06)

LORA

• Introduced and studied by Barros (1998) and Barros and Riley (2001) who designed branch-and-bound heuristics for LORA

• We showed that LORA is polynomial-time solvable for some practical cases

Important Polynomial Case of MinHOM(H) and LORA

• Let HBR=(Z1,Z2;T) be a bipartite graph with partite sets Z1={D,C,L} (subsystem repair options) and Z2 = {d,c,ℓ} (module repair options) and with T={Dd,Cd,Cc,Ld,Lc,Lℓ}.

L d

cC

D ℓ

Other Applications

• General Optimum Cost Chromatic Partition: H=Kp (many applications)

• Special Cases:

• Optimum Cost Chromatic Partition: ci(u)=f(i)≥0

• Minsum colorings:, ci(u)=i

Easy Polynomial Cases of MinHOM(H): H is a di-Ck

Easy Polynomial Cases of MinHOM(H): H is an extended L

Replacing each vertex of H by an independent set of vertices, we get an extended H.

If MinHOM(L) is polytime solvable and H is an extended L, then MinHOM(H) is polytime solvable.

E.g. MinHOM(ext-di-Ck)

xz

Y

u

x

y

z1

z2

u1

u2

Easy NP-hard Case

Let H be a connected undirected graph in which there are vertices with and without loops. Then MinHOM(H) is NP-hard. Indeed:

(1) H has an edge ij such that ii is a loop and jj is not. Set cj(x)=0 and ci(x)=1 for each x in G.

(2) Let J be a maximum independent set of G. A cheapest H-coloring assigns j to each x in J and i to each x not in J.

(3) MaxIndepSet ≤ MinHOM(H)(4) The maximum independent set is NP-hard.

Dichotomy for directed Ck with possible loops

Theorem (Gutin and Kim, submitted)

Let H be a di-Ck (k≥3) with at least one loop.Then MinHOM(H) is NP-hard.Proof: Let kk be a loop in H, G input digraph

of order n. To obtain D replace every x in V(G) by the path x1 x2 … xk-1 and every arc xy by xk-1 y1. Costs: ci(xi)=0, cj(xi )=(k-1)n+1, ck(xi )=1. Observe that h(xi )=k is an H-coloring of D of cost (k-1)n .

Proof continuation

Let f be a minimum cost H-coloring of D. Then for each x in G we have: f(xi )=i for all i or f(xi )=k for all i . Let f(x1)= f(y1 )=1 and xy an arc of G. Then xk-1y1 is an arc in D, a contradiction since f(xk-1)=k-1. Thus, I={ x ε V(G): f(x1)=1} is an independent set in G and cost(f)=(k-1)(n-|I|).

Conversely, if I is indep. in G set f(xi )=i if

x in G and f(xi )=k, otherwise; cost(f)=(k-1)(n-|I|).

Dichotomy

Theorem (Gutin and Kim, submitted)

Let H be a di-Ck (k≥2) with possible loops. If di-Ck has no loops or k=2 and there are two loops, then MinHOM(H) is polytime solvable. Otherwise, MinHOM(H) is NP-hard.

Min-Max Ordering for Digraphs

A digraph H=(V,A), an ordering v1,…,vp and is Min-Max if vivj ε A and vrvs ε A imply vavb

ε A for both a = min{i,r}, b = min{j,s} and a = max{i,r}, b = max{j,s}.

MinHOM(H) and Min-Max ordering

Theorem (Gutin, Rafiey, Yeo, 2006)

If a digraph H has a Min-Max ordering of V(H), then MinHOM(H) is polytime solvable.

Let TTp be the transitive tournament on vertices 1,2,…,p (ij arc iff i<j).

Corollary MinHOM(H) is polytime solvable if H=TTp or TTp- {1p}.

Dichotomy for SMDs

Theorem (Gutin,Rafiey,Yeo,submitted) Let H be a semicomplete k-partite digraph, k≥3. Then MinHOM(H) is polytime solvable if H is an extension of TTk or TTk+1-{(1,k+1)} or di-C3 . Otherwise, MinHOM(H) is NP-hard.

Theorem (Gutin,Rafiey,Yeo,2006) Let H be a semicomplete digraph. Then MinHOM(H) is polytime solvable if H is TTk or di-C3 . Otherwise, MinHOM(H) is NP-hard.

Min-Max Orderings for Bipartite Graphs

• A bipartite graph H=(U,W;E), orderings u1,…,up and w1,…,wq of U and W are Min-Max orderings if uiwj ε E and urws ε E imply uawb ε E for both a = min{i,r}, b = min{j,s} and a = max{i,r}, b = max{j,s}

• implies • Theorem (Spinrad, Brandstadt, Stewart,

1987) A bipartite graph H has Min-Max orderings iff H is a proper interval bigraph.

Interval Bigraphs

• G=(R,L;E) is an interval bigraph if there are families {I(u): u ε R} and {J(v): v ε L} of intervals such that uv ε E iff I(u) intersects J(v)

• An interval bigraph G=(R,L;E) is proper iff no interval in either family contains another interval in the family

Illustration (from LORA) HBR has Min-Max orderings; HBR is an interval bigraph

L d

cC

D ℓ

L

D

Cℓ

c

d

L ℓ

c

d

C

DHBR Min-Max orderings

Polynomial Cases

• Corollary (Gutin,Hell,Rafiey,Yeo, 2007)

(a) If a bipartite graph H has Min-Max orderings, then MinHOM(H) is polytime solvable; (b) If H is a proper interval bigraph, then MinHOM(H) is polytime solvable.

NP-hardness

• Key Remark: If MinHOM(H’) is NP-hard and H’ is an induced subgraph of H, then MinHOM(H) is NP-hard as well.

Forbidden Subgraphs

• Theorem (Hell & Huang, 2004) A bipartite graph is not a proper interval bigraph iff it has an induced

subgraph Cn , n≥6, or a bipartite claw, or a bipartite net, or a bipartite tent.

Dichotomy

• Feder, Hell & Huang, 1999: Cn -ListHOM (n≥6) is NP-hard.

• MinHOM(H) is NP-hard if H is a bipartite claw, net, or tent (reduction from max independent set in 3-partite graphs with fixed partite sets).

• Theorem (Gutin,Hell,Rafiey,Yeo,2007) Let H be an undirected graph. If every component of H is a proper interval bigraph or a reflexive interval graph, then MinHOM(H) is polytime solvable. Otherwise, MinHOM(H) is NP-hard.

Digraph with Possible Loops• L is a digraph on vertices 1,2,…,k.

Replacing i by S1 we get L[S1, S2 ,…, Sk].

• An undirected graph US(L) is obtained from L by deleting all arcs xy for which yx is not an arc and replacing all remaining arcs by edges.

• R :

Dichotomy for Semicomplete Digraphs with Possible Loops

Theorem (Kim & Gutin, submitted) Let H is a semicomplete digraph wpl. Let H= TTk[S1, S2 ,…, Sk] where each Si is either a single vertex without a loop, or a reflexive semicomplete digraph which does not contain R as an induced subdigraph and for which US(Si ) is a connected proper interval graph. Then, MinHOM(H) is polytime solvable. Otherwise, MinHOM(H) is NP-hard.

k-Min-Max Ordering

• A collection V1,…,Vk of subsets of a set V is called

a k-partition of V if V=V1 U … U Vk, and Vi ∩ Vj = ø provided i ≠ j.

• Let H=(V,A) be a loopless digraph and let k ≥ 2 be an integer; H has a k-Min-Max ordering if there is k-partition of V into V1,…,Vk and there is an ordering v1(i),…, vm(i)(i) of Vi for each i such that

(a) Every arc of H is an arc from Vi to Vi+1 for some i

(b) v1(i),…, vm(i)(i) v1(i+1),…, vm(i+1)(i+1) is a Min-Max ordering of the subdigraph of H induced by V=Vi U Vi+1 for each i.

k-Min-Max Ordering Theorem

Theorem (Gutin, Rafiey, Yeo, submitted) If

a digraph H has a k-Min-Max ordering for some k, then MinHOM(H) is polytime solvable.

Proof: A reduction to the min cut problem.

Dichotomy for SBDs

Theorem (Gutin, Rafiey, Yeo, submitted) Let H be a semicomplete digraph. If H is an

extension of di-C4 or H has a 2-Min-Max ordering, then MinHOM(H) is polytime solvable. Otherwise, MinHOM(H) is NP-hard.

Corollary (Gutin, Rafiey, Yeo, submitted) Let H be a bipartite tournament. If H is an

extension of di-C4 or H is acyclic, then MinHOM(H) is polytime solvable. Otherwise, MinHOM(H) is NP-hard.

Further Research

• P: Dichotomy for other classes of digraphs

• P: Dichotomy for acyclic multipartite tournaments with possible loops?

• Q: Existence of dichotomy for all digraphs?

• For ListHOM, Bulatov proved the existence of dichotomy (no characterization)

Thank you!

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