Fractional Decomposition Tree Algorithm: A tool for studying the integrality gap of Integer Programs

TL;DR

The paper introduces the Fractional Decomposition Tree algorithm, which efficiently finds feasible solutions for binary integer programs and aids in analyzing their integrality gaps, with promising experimental results.

Contribution

It presents a novel polynomial-time algorithm, FDT, for solving IPs with bounded integrality gaps and provides tools for studying the integrality gap of IP formulations.

Findings

FDT guarantees feasible solutions when the integrality gap is bounded.

FDT outperforms previous algorithms on certain hard-to-decompose solutions.

FDT compares favorably with the feasibility pump in vertex cover instances.

Abstract

We present a new algorithm, Fractional Decomposition Tree (FDT) for finding a feasible solution for an integer program (IP) where all variables are binary. FDT runs in polynomial time and is guaranteed to find a feasible integer solution provided the integrality gap is bounded. The algorithm gives a construction for Carr and Vempala's theorem that any feasible solution to the IP's linear-programming relaxation, when scaled by the instance integrality gap, dominates a convex combination of feasible solutions. FDT is also a tool for studying the integrality gap of IP formulations. We demonstrate that with experiments studying the integrality gap of two problems: optimally augmenting a tree to a 2-edge-connected graph and finding a minimum-cost 2-edge-connected multi-subgraph (2EC). We also give a simplified algorithm, Dom2IP, that more quickly determines if an instance has an unbounded integrality gap. We show that FDT's speed and approximation quality compare well to that of feasibility pump on moderate-sized instances of the vertex cover problem. For a particular set of hard-to-decompose fractional 2EC solutions, FDT always gave a better integer solution than the best previous approximation algorithm (Christofides).