Introduction to algorithms
Finding critical independent sets and critical vertex subsets are polynomial problems
SIAM Journal on Discrete Mathematics
On Finding Critical Independent and Vertex Sets
SIAM Journal on Discrete Mathematics
Improved Algorithms for Bipartite Network Flow
SIAM Journal on Computing
Bipartite graphs and their applications
Bipartite graphs and their applications
A combinatorial algorithm for weighted stable sets in bipartite graphs
Discrete Applied Mathematics - Special issue: 2nd cologne/twente workshop on graphs and combinatorial optimization (CTW 2003)
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Level of Repair Analysis (LORA) is a prescribed procedure for defence 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 (1998) and Barros and Riley (2001) 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 and provide partial results. 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.