The increasing availability of network data is creating a great potential for knowledge discovery from graph data. In many applications, feature vectors are given in addition to graph data, where nodes represent entities, edges relationships between entities, and feature vectors associated with the nodes represent properties of entities. Often features and edges contain complementary information. In such scenarios the simultaneous use of both data types promises more meaningful and accurate results. Along these lines, we introduce the novel problem of mining cohesive patterns from graphs with feature vectors, which combines the concepts of dense subgraphs and subspace clusters into a very expressive problem definition. A cohesive pattern is a dense and connected subgraph that has homogeneous values in a large enough feature subspace. We argue that this problem definition is natural in identifying small communities in social networks and functional modules in Protein-Protein interaction networks. We present the algorithm CoPaM (Cohesive Pattern Miner), which exploits various pruning strategies to efficiently find all maximal cohesive patterns. Our theoretical analysis proves the correctness of CoPaM, and our experimental evaluation demonstrates its effectiveness and efficiency.

For graphs G and H, a mapping f : V (G)→V (H) is a homomorphism of G to H if uv ∈ E(G) implies f (u)f (v) ∈ E(H). If, moreover, each vertex u ∈ V (G) is associated with costs c i (u), i ∈ V (H), then the cost of the homomorphism f is u∈V (G) c f (u) (u). For each fixed graph H, we have the minimum cost homomorphism problem, written as MinHOM(H). The problem is to decide, for an input graph G with costs c i (u), u ∈ V (G), i ∈ V (H), whether there exists a homomorphism of G to H and, if one exists, to find one of minimum cost. Minimum cost homomorphism problems encompass (or are related to) many well studied optimization problems. We prove a dichotomy of the minimum cost homomorphism problems for graphs H, with loops allowed. When each connected component of H is either a reflexive proper interval graph or an irreflexive proper interval bigraph, the problem MinHOM(H) is polynomial time solvable. In all other cases the problem MinHOM(H) is NP-hard. This solves an open problem from an earlier paper.

No abstract

For digraphs D and H, a mapping f : V (D)→V (H) is a homomorphism of D to H if uv ∈ A(D) implies f (u)f (v) ∈ A(H). Let H be a fixed directed or undirected graph. The homomorphism problem for H asks whether a directed or undirected graph input digraph D admits a homomorphism to H. The list homomorphism problem for H is a generalization of the homomorphism problem for H, where every vertex x ∈ V (D) is assigned a set L x of possible colors (vertices of H).The following optimization version of these decision problems was introduced in [12], where it was motivated by a real-world problem in defence logistics. Suppose we are given a pair of digraphs D, H and a positive cost c i (u) for each u ∈ V (D) and i ∈ V (H). The cost of a homomorphism f of D to H is u∈V (D) c f (u) (u). For a fixed digraph H, the minimum cost homomorphism problem for H, MinHOMP(H), is stated as follows: For an input digraph D and costs c i (u) for each u ∈ V (D) and i ∈ V (H), verify whether there is a homomorphism of D to H and, if it exists, find such a homomorphism of minimum cost.We obtain dichotomy classifications of the computational complexity of the list homomorphism problem and MinHOMP(H), when H is a semicomplete digraph (a digraph in which every two vertices have at least one arc between them). Our dichotomy for the list homomorphism problem coincides with the one obtained by Bang-Jensen, Hell and MacGillivray in 1988 for the homomorphism problem when H is a semicomplete digraph: both problems are polynomial solvable if H has at most one cycle; otherwise, both problems are NP-complete. The dichotomy for MinHOMP(H) is different: the problem is polynomial time solvable if H is acyclic or H is a cycle of length 2 or 3; otherwise, the problem is NP-hard.

Level of repair analysis (LORA) is a prescribed procedure for defense logistics support planning. For a complex engineering system containing perhaps thousands of assemblies, sub-assemblies, components, etc. organized into several levels of indenture and with a number of possible repair decisions, LORA seeks to determine an optimal provision of repair and maintenance facilities to minimize overall life-cycle costs. For a LORA problem with two levels of indenture with three possible repair decisions, which is of interest in UK and US military and which we call LORA-BR, Barros [The optimisation of repair decisions using life-cycle cost parameters. IMA J. Management Math. 9 (1998) 403-413] and Barros and Riley [A combinatorial approach to level of repair analysis, European J. Oper. Res. 129 (2001) 242-251] developed certain branch-and-bound heuristics. The surprising result of this paper is that LORA-BR is, in fact, polynomial-time solvable. To obtain this result, we formulate the general LORA problem as an optimization homomorphism problem on bipartite graphs, and reduce a generalization of LORA-BR, LORA-M, to the maximum weight independent set problem on a bipartite graph. We prove that the general LORA problem is NP-hard by using an important result on list homomorphisms of graphs. We introduce the minimum cost graph homomorphism problem, provide partial results and pose an open problem. Finally, we show that our result for LORA-BR can be applied to prove that an extension of the maximum weight independent set problem on bipartite graphs is polynomial time solvable.