The All-Pairs Vitality-Maximization (VIMAX) Problem

TL;DR

This paper introduces the all-pairs vitality maximization (VIMAX) problem, aiming to identify vertex removals that maximize flow through a critical node, with formulations, complexity analysis, and heuristic solutions demonstrated on real and simulated networks.

Contribution

It formulates the VIMAX problem as a mixed integer program, proves its NP-hardness, and compares optimization and heuristic methods for maximizing vertex vitality.

Findings

MIP formulation of VIMAX problem

NP-hardness of VIMAX established

Heuristic methods improve vitality of key vertices

Abstract

Traditional network interdiction problems focus on removing vertices or edges from a network so as to disconnect or lengthen paths in the network; network diversion problems seek to remove vertices or edges to reroute flow through a designated critical vertex or edge. We introduce the all-pairs vitality maximization problem (VIMAX), in which vertex deletion attempts to maximize the amount of flow passing through a critical vertex, measured as the all-pairs vitality of the vertex. The assumption in this problem is that in a network for which the structure is known but the physical locations of vertices may not be known (e.g. a social network), locating a person or asset of interest might require the ability to detect a sufficient amount of flow (e.g., communications or financial transactions) passing through the corresponding vertex in the network. We formulate VIMAX as a mixed integer program, and show that it is NP-Hard. We compare the performance of the MIP and a simulated annealing heuristic on both real and simulated data sets and highlight the potential increase in vitality of key vertices that can be attained by subset removal. We also present graph theoretic results that can be used to narrow the set of vertices to consider for removal.