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 -