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Introduction to the min cost homomorphism problem for undirected and directed graphs Gregory Gutin Royal Holloway, U. London, UK and U. Haifa, Israel

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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). u v x y w z 123 G H

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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.

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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.

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The Min Cost Homomorphism Problem Introduced in Gutin, Rafiey, Yeo and Tso, Fix H. MinHOM(H): Given a graph G and a cost c i (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)= Σ uεV(G) c f(u) (u)

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Min Cost vs ListHOM H-ListHOM: G; L(v), v ε V(G) Special MinHOM(H): c i (v)=0 if i ε L(v) and c i (v)=1, otherwise. Э H-coloring of cost 0?

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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)

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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

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Important Polynomial Case of MinHOM(H) and LORA Let H BR =(Z 1,Z 2 ;T) be a bipartite graph with partite sets Z 1 ={D,C,L} (subsystem repair options) and Z 2 = {d,c,ℓ} (module repair options) and with T={Dd,Cd,Cc,Ld,Lc,Lℓ}. Ld c C D ℓ

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Other Applications General Optimum Cost Chromatic Partition: H=K p (many applications) Special Cases: Optimum Cost Chromatic Partition: c i (u)=f(i)≥0 Minsum colorings:, c i (u)=i

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

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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-C k ) x z Y u x y z1z1 z2z2 u1u1 u2u2

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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 c j (x)=0 and c i (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.

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Dichotomy for directed C k with possible loops Theorem (Gutin and Kim, submitted) Let H be a di-C k (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 x 1 x 2 … x k-1 and every arc xy by x k-1 y 1. Costs: c i (x i )=0, c j (x i )=(k- 1)n+1, c k (x i )=1. Observe that h(x i )=k is an H-coloring of D of cost (k-1)n.

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Proof continuation Let f be a minimum cost H-coloring of D. Then for each x in G we have: f(x i )=i for all i or f(x i )=k for all i. Let f(x 1 )= f(y 1 )=1 and xy an arc of G. Then x k-1 y 1 is an arc in D, a contradiction since f(x k-1 )=k-1. Thus, I={ x ε V(G): f(x 1 )=1} is an independent set in G and cost(f)=(k-1)(n-|I|). Conversely, if I is indep. in G set f(x i )=i if x in G and f(x i )=k, otherwise; cost(f)=(k-1)(n-|I|).

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

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Min-Max Ordering for Digraphs A digraph H=(V,A), an ordering v 1,…,v p and is Min-Max if v i v j ε A and v r v s ε A imply v a v b ε A for both a = min{i,r}, b = min{j,s} and a = max{i,r}, b = max{j,s}.

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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 TT p be the transitive tournament on vertices 1,2,…,p (ij arc iff i

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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 TT k or TT k+1 -{(1,k+1)} or di-C 3. 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 TT k or di-C 3. Otherwise, MinHOM(H) is NP-hard.

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Min-Max Orderings for Bipartite Graphs A bipartite graph H=(U,W;E), orderings u 1,…,u p and w 1,…,w q of U and W are Min- Max orderings if u i w j ε E and u r w s ε E imply u a w b 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.

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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

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Illustration (from LORA) H BR has Min-Max orderings; H BR is an interval bigraph Ld c C D ℓ L D C ℓ c d Lℓ c d C D H BR Min-Max orderings

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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.

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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.

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Forbidden Subgraphs Theorem (Hell & Huang, 2004) A bipartite graph is not a proper interval bigraph iff it has an induced subgraph C n, n≥6, or a bipartite claw, or a bipartite net, or a bipartite tent.

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Dichotomy Feder, Hell & Huang, 1999: C n -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.

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Digraph with Possible Loops L is a digraph on vertices 1,2,…,k. Replacing i by S 1 we get L[S 1, S 2,…, S k ]. 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 :

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Dichotomy for Semicomplete Digraphs with Possible Loops Theorem (Kim & Gutin, submitted) Let H is a semicomplete digraph wpl. Let H= TT k [S 1, S 2,…, S k ] where each S i 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(S i ) is a connected proper interval graph. Then, MinHOM(H) is polytime solvable. Otherwise, MinHOM(H) is NP-hard.

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k-Min-Max Ordering A collection V 1,…,V k of subsets of a set V is called a k-partition of V if V=V 1 U … U V k, and V i ∩ V j = 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 V 1,…,V k and there is an ordering v 1 (i),…, v m(i) (i) of V i for each i such that (a) Every arc of H is an arc from V i to V i+1 for some i (b) v 1 (i),…, v m(i) (i) v 1 (i+1),…, v m(i+1) (i+1) is a Min-Max ordering of the subdigraph of H induced by V=V i U V i+1 for each i.

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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.

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Dichotomy for SBDs Theorem (Gutin, Rafiey, Yeo, submitted) Let H be a semicomplete digraph. If H is an extension of di-C 4 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-C 4 or H is acyclic, then MinHOM(H) is polytime solvable. Otherwise, MinHOM(H) is NP-hard.

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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)

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Thank you! Questions? Comments? Remarks? Suggestions? Criticism?

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