Theoretical and computational advances for network diversion

The network-diversion problem (ND) is defined on a directed or undirected graph G = (V , E ) having nonnegative edge weights, a source vertex s, a sink vertex t , and a “diversion edge” e ′. This problem, with intelligence-gathering and war-fighting applications, seeks a minimum-weight, minimal s-t cut EC ⊆ E in G such that e ′ ∈ EC . We present (a) a new NP-completeness proof for ND on directed graphs, (b) the first polynomialtime solution algorithm for a special graph topology, (c) an improved mixed-integer programming formulation (MIP), and (d) useful valid inequalities for that MIP. The proof strengthens known results by showing, for instance, that ND is strongly NP-complete on a directed graph even when e ′ is incident from s or into t , but not both, and even when G is acyclic; a corollary shows the NP-completeness of a vertex-deletion version of ND on undirected graphs. The polynomial-time algorithm solves ND on s-t planar graphs. Compared to a MIP from the literature, the new MIP, coupled with valid inequalities, reduces the average duality gap by 10–50% on certain classes of test problems. It can also reduce solution times by an order of magnitude. We successfully solve unweighted problems with roughly 90,000 vertices and 360,000 edges and weighted problems with roughly 10,000 vertices and 40,000 edges. © 2013 Wiley Periodicals, Inc. NETWORKS, Vol. 000(00), 000–00