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Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2021.92
URN: urn:nbn:de:0030-drops-141616
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/14161/
Lampis, Michael
Minimum Stable Cut and Treewidth
Abstract
A stable or locally-optimal cut of a graph is a cut whose weight cannot be increased by changing the side of a single vertex. Equivalently, a cut is stable if all vertices have the (weighted) majority of their neighbors on the other side. Finding a stable cut is a prototypical PLS-complete problem that has been studied in the context of local search and of algorithmic game theory.
In this paper we study Min Stable Cut, the problem of finding a stable cut of minimum weight, which is closely related to the Price of Anarchy of the Max Cut game. Since this problem is NP-hard, we study its complexity on graphs of low treewidth, low degree, or both. We begin by showing that the problem remains weakly NP-hard on severely restricted trees, so bounding treewidth alone cannot make it tractable. We match this hardness with a pseudo-polynomial DP algorithm solving the problem in time (Δ⋅ W)^{O(tw)}n^{O(1)}, where tw is the treewidth, Δ the maximum degree, and W the maximum weight. On the other hand, bounding Δ is also not enough, as the problem is NP-hard for unweighted graphs of bounded degree. We therefore parameterize Min Stable Cut by both tw and Δ and obtain an FPT algorithm running in time 2^{O(Δtw)}(n+log W)^{O(1)}. Our main result for the weighted problem is to provide a reduction showing that both aforementioned algorithms are essentially optimal, even if we replace treewidth by pathwidth: if there exists an algorithm running in (nW)^{o(pw)} or 2^{o(Δpw)}(n+log W)^{O(1)}, then the ETH is false. Complementing this, we show that we can, however, obtain an FPT approximation scheme parameterized by treewidth, if we consider almost-stable solutions, that is, solutions where no single vertex can unilaterally increase the weight of its incident cut edges by more than a factor of (1+ε).
Motivated by these mostly negative results, we consider Unweighted Min Stable Cut. Here our results already imply a much faster exact algorithm running in time Δ^{O(tw)}n^{O(1)}. We show that this is also probably essentially optimal: an algorithm running in n^{o(pw)} would contradict the ETH.
BibTeX - Entry
@InProceedings{lampis:LIPIcs.ICALP.2021.92,
author = {Lampis, Michael},
title = {{Minimum Stable Cut and Treewidth}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {92:1--92:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-195-5},
ISSN = {1868-8969},
year = {2021},
volume = {198},
editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/14161},
URN = {urn:nbn:de:0030-drops-141616},
doi = {10.4230/LIPIcs.ICALP.2021.92},
annote = {Keywords: Treewidth, Local Max-Cut, Nash Stability}
}
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Keywords: |
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Treewidth, Local Max-Cut, Nash Stability |
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Collection: |
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48th International Colloquium on Automata, Languages, and Programming (ICALP 2021) |
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Issue Date: |
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2021 |
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Date of publication: |
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02.07.2021 |
