Gregory Gutin Royal Holloway U London UK and
- Slides: 33
Gregory Gutin Royal Holloway, U. London, UK and U. Haifa, Israel Introduction to the min cost homomorphism problem for undirected and directed graphs
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). v u w G x z 1 2 H y 3
The Homomorphism Problem Fix a graph H. H-HOM: For an input graph G, check whethere 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 & Mac. Gillivray, 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-List. HOM: 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-List. HOM is polynomial-time solvable if H is bipartite and the complement of a circular-arc graph. Otherwise, HList. HOM is NP-complete. Theorem (Gutin, Rafiey, Yeo, 2006) If H is a semicomplete digraph with at most one cycle, H-List. HOM is polynomialtime solvable. If H is a SD with at least two cycles, then H-List. HOM is NP-complete.
The Min Cost Homomorphism Problem Introduced in Gutin, Rafiey, Yeo and Tso, 2006. Fix H. Min. HOM(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(f)= ΣuεV(G) cf(u)(u)
Min Cost vs List. HOM H-List. HOM: G; L(v), v ε V(G) Special Min. HOM(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 lifecycle 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 Min. HOM(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 Min. HOM(H) and LORA • Let HBR=(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, Cc, Ld, Lc, Lℓ}. L d C c 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 Min. HOM(H): H is a di-Ck
Easy Polynomial Cases of Min. HOM(H): H is an extended L Replacing each vertex of H by an independent set of vertices, we get an extended H. If Min. HOM(L) is polytime solvable and H is an extended L, then Min. HOM(H) is polytime solvable. E. g. Min. HOM(ext-di-Ck) x z u Y x z 1 u 1 y u 2 z 2
Easy NP-hard Case Let H be a connected undirected graph in which there are vertices with and without loops. Then Min. HOM(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) Max. Indep. Set ≤ Min. HOM(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 Min. HOM(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 … xk-1 and every arc xy by xk-1 y 1. 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(x 1)= f(y 1 )=1 and xy an arc of G. Then xk-1 y 1 is an arc in D, a contradiction since f(xk 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(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 Min. HOM(H) is polytime solvable. Otherwise, Min. HOM(H) is NPhard.
Min-Max Ordering for Digraphs A digraph H=(V, A), an ordering v 1, …, 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}.
Min. HOM(H) and Min-Max ordering Theorem (Gutin, Rafiey, Yeo, 2006) If a digraph H has a Min-Max ordering of V(H), then Min. HOM(H) is polytime solvable. Let TTp be the transitive tournament on vertices 1, 2, …, p (ij arc iff i<j). Corollary Min. HOM(H) is polytime solvable if H=TTp or TTp- {1 p}.
Dichotomy for SMDs Theorem (Gutin, Rafiey, Yeo, submitted) Let H be a semicomplete k-partite digraph, k≥ 3. Then Min. HOM(H) is polytime solvable if H is an extension of TTk or TTk+1 -{(1, k+1)} or di-C 3. Otherwise, Min. HOM(H) is NP-hard. Theorem (Gutin, Rafiey, Yeo, 2006) Let H be a semicomplete digraph. Then Min. HOM(H) is polytime solvable if H is TTk or di-C 3. Otherwise, Min. HOM(H) is NP-hard.
Min-Max Orderings for Bipartite Graphs • A bipartite graph H=(U, W; E), orderings u 1, …, up and w 1, …, 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 C D c HBR ℓ L C D ℓ L ℓ C c c d D d Min-Max orderings
Polynomial Cases • Corollary (Gutin, Hell, Rafiey, Yeo, 2007) (a) If a bipartite graph H has Min-Max orderings, then Min. HOM(H) is polytime solvable; (b) If H is a proper interval bigraph, then Min. HOM(H) is polytime solvable.
NP-hardness • Key Remark: If Min. HOM(H’) is NP-hard and H’ is an induced subgraph of H, then Min. HOM(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 -List. HOM (n≥ 6) is NP-hard. • Min. HOM(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 Min. HOM(H) is polytime solvable. Otherwise, Min. HOM(H) is NP-hard.
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 , …, 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[S 1, S 2 , …, 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, Min. HOM(H) is polytime solvable. Otherwise, Min. HOM(H) is NP-hard.
k-Min-Max Ordering • A collection V 1, …, Vk of subsets of a set V is called a k-partition of V if V=V 1 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 V 1, …, Vk and there is an ordering v 1(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) v 1(i), …, vm(i)(i) v 1(i+1), …, vm(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 (Gutin, Rafiey, Yeo, submitted) If a digraph H has a k-Min-Max ordering for some k, then Min. HOM(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-C 4 or H has a 2 -Min-Max ordering, then Min. HOM(H) is polytime solvable. Otherwise, Min. HOM(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 Min. HOM(H) is polytime solvable. Otherwise, Min. HOM(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 List. HOM, Bulatov proved the existence of dichotomy (no characterization)
Thank you! • • • Questions? Comments? Remarks? Suggestions? Criticism?
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