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!

• Questions?

• Comments?

• Remarks?

• Suggestions?

• Criticism?