Constraint Generation Algorithm for the Minimum Connectivity Inference Problem

TL;DR

This paper introduces a constraint generation algorithm for the Minimum Connectivity Inference problem, demonstrating improved efficiency over existing flow-based MILP methods and exploring enumeration approaches.

Contribution

It proposes a novel constraint generation approach for solving the problem more efficiently than traditional flow-based MILP formulations.

Findings

The constraint generation algorithm outperforms previous MILP methods on random instances.

The approach is faster and potentially applicable to other connectivity-related optimization problems.

An enumeration algorithm for the problem is also presented.

Abstract

Given a hypergraph $H$, the Minimum Connectivity Inference problem asks for a graph on the same vertex set as $H$ with the minimum number of edges such that the subgraph induced by every hyperedge of $H$ is connected. This problem has received a lot of attention these recent years, both from a theoretical and practical perspective, leading to several implemented approximation, greedy and heuristic algorithms. Concerning exact algorithms, only Mixed Integer Linear Programming (MILP) formulations have been experimented, all representing connectivity constraints by the means of graph flows. In this work, we investigate the efficiency of a constraint generation algorithm, where we iteratively add cut constraints to a simple ILP until a feasible (and optimal) solution is found. It turns out that our method is faster than the previous best flow-based MILP algorithm on random generated instances, which suggests that a constraint generation approach might be also useful for other optimization problems dealing with connectivity constraints. At last, we present the results of an enumeration algorithm for the problem.