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

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 of an instance’s polyhedron, independent of objective function, 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. The upper bound on the integrality gap of an FDT solution can be exponentially large. However our experiments demonstrate that FDT can be effective in practice. We study 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, DomToIP, that finds a feasible solution to an IP instance, or concludes that it has unbounded integrality gap. We show that FDT’s speed and approximation quality compare well to that of the original feasibility pump heuristic 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-of-Many Christofides Algorithm (BOMC).