A (probably) optimal algorithm for Bisection on bounded-treewidth graphs

TL;DR

This paper presents a new algorithm for the maximum/minimum bisection problems on bounded-treewidth graphs that is asymptotically optimal and matches the best possible complexity bounds under certain complexity assumptions.

Contribution

The authors develop an $O(2^t(tn)^2)$-time algorithm for bisection problems on bounded-treewidth graphs, achieving asymptotic optimality under the Strong Exponential Time Hypothesis.

Findings

The new algorithm is asymptotically tight to existing lower bounds.

Exponential dependency on treewidth is proven to be optimal under SETH.

Discussion on tractability of bisection problems for special graph classes.

Abstract

The maximum/minimum bisection problems are, given an edge-weighted graph, to find a bipartition of the vertex set into two sets whose sizes differ by at most one, such that the total weight of edges between the two sets is maximized/minimized. Although these two problems are known to be NP-hard, there is an efficient algorithm for bounded-treewidth graphs. In particular, Jansen et al. (SIAM J. Comput. 2005) gave an $O(2^tn^3)$-time algorithm when given a tree decomposition of width $t$ of the input graph, where $n$ is the number of vertices of the input graph. Eiben et al. (ESA 2019) improved the dependency of $n$ in the running time by giving an $O(8^tt^5n^2\log n)$-time algorithm. Moreover, they showed that there is no $O(n^{2-\varepsilon})$-time algorithm for trees under some reasonable complexity assumption. In this paper, we show an $O(2^t(tn)^2)$-time algorithm for both problems, which is asymptotically tight to their conditional lower bound. We also show that the exponential dependency of the treewidth is asymptotically optimal under the Strong Exponential Time Hypothesis. Finally, we discuss the (in)tractability of both problems with respect to special graph classes.