An improved, easily computable combinatorial lower bound for weighted graph bipartitioning

Abstract

There has recently been much progress on exact algorithms for the (un)weighted graph (bi)partitioning problem using branch-and-bound and related methods. In this note we present and improve an easily computable, purely combinatorial lower bound for the weighted bipartitioning problem. The bound is computable in $O(n\log n+m)$ time steps for weighted graphs with $n$ vertices and $m$ edges. In the branch-and-bound setting, the bound for each new subproblem can be updated in $O(n+(m/n)\log n)$ time steps amortized over a series of $n$ branching steps; a rarely triggered tightening of the bound requires search on the graph of unassigned vertices and can take from $O(n+m)$ to $O(nm+n^2\log n)$ steps depending on implementation and possible bound quality. Representing a subproblem uses $O(n)$ space. Although the bound is weak, we believe that it can be advantageous in a parallel setting to be able to generate many subproblems fast, possibly out-weighting the advantages of tighter, but much more expensive (algebraic, spectral, flow) lower bounds. We use a recent priority task-scheduling framework for giving a parallel implementation, and show the relative improvements in bound quality and solution speed by the different contributions of the lower bound. A detailed comparison with standardized input graphs to other lower bounds and frameworks is pending. Detailed investigations of branching and subproblem selection rules are likewise not the focus here, but various options are discussed.