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Algorithms 2018, 11(7), 98; https://doi.org/10.3390/a11070098

Width, Depth, and Space: Tradeoffs between Branching and Dynamic Programming

1
AROBOT Innovation, Taiwan
2
Royal Holloway, University of London, TW20 0EX, UK
3
Department of Computer Science, RWTH Aachen University, 52062 Aachen, Germany
*
Author to whom correspondence should be addressed.
Received: 31 March 2018 / Revised: 27 June 2018 / Accepted: 28 June 2018 / Published: 1 July 2018
(This article belongs to the Special Issue Algorithms for Hard Problems: Approximation and Parameterization)
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Abstract

Treedepth is a well-established width measure which has recently seen a resurgence of interest. Since graphs of bounded treedepth are more restricted than graphs of bounded tree- or pathwidth, we are interested in the algorithmic utility of this additional structure. On the negative side, we show with a novel approach that the space consumption of any (single-pass) dynamic programming algorithm on treedepth decompositions of depth d cannot be bounded by (2ϵ)d·logO(1)n for Vertex Cover, (3ϵ)d·logO(1)n for 3-Coloring and (3ϵ)d·logO(1)n for Dominating Set for any ϵ>0. This formalizes the common intuition that dynamic programming algorithms on graph decompositions necessarily consume a lot of space and complements known results of the time-complexity of problems restricted to low-treewidth classes. We then show that treedepth lends itself to the design of branching algorithms. Specifically, we design two novel algorithms for Dominating Set on graphs of treedepth d: A pure branching algorithm that runs in time dO(d2)·n and uses space O(d3logd+dlogn) and a hybrid of branching and dynamic programming that achieves a running time of O(3dlogd·n) while using O(2ddlogd+dlogn) space. View Full-Text
Keywords: treewidth; treedepth; dynamic programming; branching algorithm; space lower bound treewidth; treedepth; dynamic programming; branching algorithm; space lower bound
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).
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Chen, L.-H.; Reidl, F.; Rossmanith, P.; Sánchez Villaamil, F. Width, Depth, and Space: Tradeoffs between Branching and Dynamic Programming. Algorithms 2018, 11, 98.

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