Maximizing Convergence Time in Network Averaging Dynamics Subject to Edge Removal

TL;DR

This paper studies the problem of maximizing consensus convergence time in networks by removing edges, showing its computational hardness, and proposing approximation algorithms with practical performance demonstrated through numerical experiments.

Contribution

It establishes the NP-hardness of the consensus interdiction problem and related effective resistance interdiction, and introduces polynomial-time approximation algorithms for these problems.

Findings

CIP is NP-hard even for simple bipartite graphs.

A polynomial-time $mn$-approximation algorithm for ERIP is developed.

An iterative approximation algorithm for CIP performs well in numerical tests.

Abstract

We consider the consensus interdiction problem (CIP), in which the goal is to maximize the convergence time of consensus averaging dynamics subject to removing a limited number of network edges. We first show that CIP can be cast as an effective resistance interdiction problem (ERIP), in which the goal is to remove a limited number of network edges to maximize the effective resistance between a source node and a sink node. We show that ERIP is strongly NP-hard, even for bipartite graphs of diameter three with fixed source/sink edges, and establish the same hardness result for the CIP. We then show that both ERIP and CIP cannot be approximated up to a (nearly) polynomial factor assuming exponential time hypothesis. Subsequently, we devise a polynomial-time $mn$-approximation algorithm for the ERIP that only depends on the number of nodes $n$ and the number of edges $m$, but is independent of the size of edge resistances. Finally, using a quadratic program formulation for the CIP, we devise an iterative approximation algorithm to find a first-order stationary solution for the CIP and evaluate its good performance through numerical results.