# An optimal algorithm for bisection for bounded-treewidth graph

Tesshu Hanaka, Yasuaki Kobayashi, Taiga Sone

Research output: Chapter in Book/Report/Conference proceedingConference contribution

1 Citation (Scopus)

## 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 running time to $$O(8^tt^5n^2\log n)$$. 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. Moreover, we discuss the (in)tractability of both problems with respect to special graph classes.

Original language English Frontiers in Algorithmics - 14th International Workshop, FAW 2020, Proceedings Minming Li Springer Science and Business Media Deutschland GmbH 25-36 12 9783030599003 https://doi.org/10.1007/978-3-030-59901-0_3 Published - 2020 Yes 14th International Workshop on Frontiers in Algorithmics, FAW 2020 - Haikou, ChinaDuration: Oct 19 2020 → Oct 21 2020

### Publication series

Name Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) 12340 LNCS 0302-9743 1611-3349

### Conference

Conference 14th International Workshop on Frontiers in Algorithmics, FAW 2020 China Haikou 10/19/20 → 10/21/20

## All Science Journal Classification (ASJC) codes

• Theoretical Computer Science
• Computer Science(all)

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