TY - GEN

T1 - An optimal algorithm for bisection for bounded-treewidth graph

AU - Hanaka, Tesshu

AU - Kobayashi, Yasuaki

AU - Sone, Taiga

N1 - Funding Information:
This work is partially supported by JSPS KAKENHI Grant JP17H01788 and JST CREST JPMJCR1401.
Funding Information:
This work is partially supported by JSPS KAKENHI Grant Numbers JP19K21537, JP17H01788 and JST CREST JPMJCR1401.
Publisher Copyright:
© Springer Nature Switzerland AG 2020.

PY - 2020

Y1 - 2020

N2 - 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.

AB - 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.

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U2 - 10.1007/978-3-030-59901-0_3

DO - 10.1007/978-3-030-59901-0_3

M3 - Conference contribution

AN - SCOPUS:85092175843

SN - 9783030599003

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 25

EP - 36

BT - Frontiers in Algorithmics - 14th International Workshop, FAW 2020, Proceedings

A2 - Li, Minming

PB - Springer Science and Business Media Deutschland GmbH

T2 - 14th International Workshop on Frontiers in Algorithmics, FAW 2020

Y2 - 19 October 2020 through 21 October 2020

ER -