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

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

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

  2. Homomorphisms For a pair of graphsGandH, a mappingh:V(G) → V(H)is called ahomomorphismifxy ε E(G)impliesh(x)h(y) ε E(H) (also called H-coloring). v u w 1 2 3 z G H y x

  3. 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 His bipartite or has a loop. If H is not bipartite and it has no loop, thenH-HOMis NP-complete. Theorem (Bang-Jensen, Hell & MacGillivray, 1988) Let H be a semicomplete graph. H-HOM is polynomial time solvable if Hhas at most one cycle. If H has at least two cycles, thenH-HOMis NP-complete.

  4. 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 His bipartite and the complement of a circular-arc graph. Otherwise, H-ListHOMis NP-complete. Theorem (Gutin,Rafiey,Yeo, 2006) If H is a semicomplete digraph with at most one cycle, H-ListHOMis polynomial-time solvable. If H is a SD with at least two cycles, then H-ListHOMis NP-complete.

  5. The Min Cost Homomorphism Problem Introduced in Gutin, Rafiey, Yeo and Tso, 2006. Fix H. MinHOM(H): Given a graphGand 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 ofGtoHand if it does, then find ahomomorphism f of G to H of minimum cost. cost(f)= ΣuεV(G) cf(u)(u)

  6. 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. ЭH-coloring of cost 0?

  7. 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≥2levels 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)

  8. 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

  9. Important Polynomial Case of MinHOM(H) and LORA • LetHBR=(Z1,Z2;T)be a bipartite graph with partite setsZ1={D,C,L}(subsystem repair options) andZ2 ={d,c,ℓ}(module repair options) and with T={Dd,Cd,Cc,Ld,Lc,Lℓ}. L d C c D ℓ

  10. 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

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

  12. x z u Y x z1 u1 y u2 z2 Easy Polynomial Cases of MinHOM(H): H is an extended L Replacing each vertex ofHby an independent set of vertices, we getan extended H. If MinHOM(L)is polytime solvable andHis an extendedL, then MinHOM(H)is polytime solvable. E.g. MinHOM(ext-di-Ck)

  13. 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: • 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. • 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. • MaxIndepSet ≤ MinHOM(H) • The maximum independent set is NP-hard.

  14. 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-1and every arc xy by xk-1 y1. Costs: ci(xi)=0, cj(xi )=(k-1)n+1, ck(xi )=1. Observethat h(xi )=k is an H-coloring of D of cost (k-1)n .

  15. 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|).

  16. 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.

  17. Min-Max Ordering for Digraphs A digraph H=(V,A), an ordering v1,…,vpandisMin-Max if vivj εA and vrvs ε A imply vavb εA for both a = min{i,r}, b = min{j,s} and a = max{i,r}, b = max{j,s}.

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

  19. 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.

  20. Min-Max Orderings for Bipartite Graphs • A bipartite graph H=(U,W;E), orderings u1,…,upandw1,…,wq of U and W areMin-Max orderings if uiwj εE and urws ε E imply uawb εE 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 graphHhas Min-Max orderings iffHis a proper interval bigraph.

  21. 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

  22. Illustration (from LORA) HBRhas Min-Max orderings; HBR is an interval bigraph L ℓ L d ℓ L C c C c d C c D D d D HBR ℓ Min-Max orderings

  23. Polynomial Cases • Corollary (Gutin,Hell,Rafiey,Yeo, 2007) (a) If a bipartite graph H has Min-Max orderings, thenMinHOM(H)is polytimesolvable; (b) IfH is a proper interval bigraph, then MinHOM(H)is polytime solvable.

  24. 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.

  25. 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.

  26. 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) LetHbe an undirected graph.If every component of H is a proper interval bigraph or a reflexive interval graph, thenMinHOM(H)is polytime solvable. Otherwise,MinHOM(H)is NP-hard.

  27. 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 :

  28. 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.

  29. 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 = øprovided 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 • Every arc of H is an arc from Vi to Vi+1 for some i • 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.

  30. 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.

  31. 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.

  32. 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)

  33. Thank you! • Questions? • Comments? • Remarks? • Suggestions? • Criticism?

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