A Wide Branching Strategy for the Graph Coloring Problem

Abstract

Branch-and-price algorithms for the graph coloring problem use an exponentially-sized independent set based integer programming formulation to produce usually tight lower bounds to enable more aggressive pruning in the branch-and-bound tree. One major problem inherent to any branch-and-price scheme for graph coloring is that in order to avoid destroying the pricing problem structure during column generation, difficult-to-implement branching rules that modify the underlying graph must be used. This paper proposes an alternate branching strategy that does not change the graph in order to solve the pricing problem, but rather modifies the search tree to require fewer calls to difficult instances of the pricing problem. This approach, called wide branching, generates many subproblems at each node in the branch-and-price tree, which significantly reduces the length of any path through the search tree. In contrast, traditional deep branching only creates two subproblems per node, assigning a variable to either zero or one. A delayed branching procedure is introduced that prevents the branching factor at any particular node from growing too large in this scheme. Finally, computational results are presented that show the wide branching strategy to be competitive with state-of-the-art graph coloring solvers.

An Improved DSATUR‐Based Branch‐and‐Bound Algorithm for the Vertex Coloring Problem

“…Held et al also work on this formulation tackling numerical difficulties in the context of column generation, deriving a way of computing numerically safe bounds. Finally, Morrison et al work on new branching rules that preserve the graph structure at each node of the branching tree.…”

Solving the Pricing Problem in a Branch-and-Price Algorithm for Graph Coloring Using Zero-Suppressed Binary Decision Diagrams

“…It was observed that modifying the initial pool size can dramatically improve the running time of B&P+ZDD; for example, running the initialization procedure for 6100 seconds (the default initialization time limit in Malaguti et al (2011) and Morrison et al (2014a)) allows B&P+ZDD to solve DSJC125.5 in 31 seconds. Similarly, running the initialization procedure for only 3 seconds allows B&P+ZDD to solve queen9_9 in 2.3 seconds.…”

“…For example, some branching rules modify the problem structure at each subproblem in the search tree (e.g., the graph coloring rule of Mehrotra and Trick, 1996); others branch on original (non-reformulated) problem variables, or problem constraints (Vanderbeck, 2011). A related scheme by Morrison et al (2014a) uses a modified branching scheme called wide branching, which does not wholly eliminate calls to the constrained pricing problem, but restructures the search tree in an attempt to reduce the number of such calls.…”

Cyclic best first search: Using contours to guide branch‐and‐bound algorithms

“…The CBFS strategy is a generalization of other common search strategies, and can be thought of as a different hybridization of the BFS and DFS strategies. Originally called distributed best‐first search by Kao et al , CBFS has been used effectively in a number of different settings, including a variety of different scheduling problems and two different algorithms for the graph coloring problem .…”