Presentation is loading. Please wait.

Presentation is loading. Please wait.

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

Similar presentations


Presentation on theme: "Introduction to the min cost homomorphism problem for undirected and directed graphs Gregory Gutin Royal Holloway, U. London, UK and U. Haifa, Israel."— Presentation transcript:

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

2 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

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 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.

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 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.

5 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)

6 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?

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 ≥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)

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 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

10 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

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

12 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

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: (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.

14 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.

15 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|).

16 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.

17 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}.

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

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 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.

20 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.

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) 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

23 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.

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 C n, n≥6, or a bipartite claw, or a bipartite net, or a bipartite tent.

26 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.

27 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 :

28 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.

29 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.

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-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.

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?


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

Similar presentations


Ads by Google