Resilience and optimization of identifiable bipartite graphs

A bipartite graph G = (L,R;E) with at least one edge is said to be identifiable if for every vertex v ∈ L, the subgraph induced by its non-neighbors has a matching of cardinality |L|− 1. This definition arises in the context of low-rank matrix factorization and is motivated by signal processing applications. In this paper, we study the resilience of identifiability with respect to edge additions, edge deletions and edge modifications. These can all be seen as measures of evaluating how strongly a bipartite graph possesses the identifiability property. On the one hand, we show that computing the resilience of this nonmonotone property can be done in polynomial time for edge additions or edge modifications. On the other hand, for edge deletions this is an NP-complete problem. Our polynomial results are based on polynomial algorithms for computing the surplus of a bipartite graph G and finding a tight set in G, which ∗Corresponding author, +52 (81)14920363 Email addresses: [email protected] (Epameinondas Fritzilas), [email protected] (Martin Milanič), [email protected] (Jérôme Monnot), [email protected] (Yasmin A. Rios-Solis*) Preprint submitted to Discrete Applied Mathematics December 10, 2011 might be of independent interest. We also deal with some complexity results for the optimization problem related to the isolation of a smallest set J ⊆ L that, together with all vertices with neighbors only in J , induces an identifiable subgraph. We obtain an APXhardness result for the problem and identify some polynomially solvable cases.