The Cunningham-Geelen Method in Practice: Branch-Decompositions and Integer Programming

Branch-decompositions and Integer Programming S. Margulies Department of Computational and Applied Math, Rice University, Houston, Texas, {[email protected]} J. Ma Department of Management Science and Engineering, Stanford University, Palo Alto, California, {[email protected]} I.V. Hicks Department of Computational and Applied Math, Rice University, Houston, Texas, {[email protected]} Cunningham and Geelen [7] describe an algorithm for solving the integer program max{cT x : Ax = b, x ≥ 0, x ∈ Zn}, where A ∈ Zm×n ≥0 , b ∈ Z, and c ∈ Z, which utilizes a branchdecomposition of the matrix A and techniques from dynamic programming. In this paper, we report on the first implementation of the CG algorithm, and compare our results with the commercial integer programming software Gurobi [3]. Using branch-decomposition trees produced by the heuristics developed by Ma et. al [12], and optimal trees produced by the algorithm designed by Hicks [10], we test both a memory-intensive and low-memory version of the CG algorithm on problem instances such as graph 3-coloring, set partition, market split and knapsack. We isolate a class of set partition instances where the CG algorithm runs twice as fast as Gurobi, and demonstrate that certain infeasible market split and knapsacks instances with width ≤ 6 range from running twice as fast as Gurobi, to running in a matter of minutes versus a matter of hours.