Approximate counting of K-paths: Deterministic and in polynomial space

Andreas Björklund, Daniel Lokshtanov, Saket Saurabh, Meirav Zehavi

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

4 Scopus citations

Abstract

A few years ago, Alon et al. [ISMB 2008] gave a simple randomized O((2e)km∊2)-time exponential-space algorithm to approximately compute the number of paths on k vertices in a graph G up to a multiplicative error of 1 ± ∊. Shortly afterwards, Alon and Gutner [IWPEC 2009, TALG 2010] gave a deterministic exponential-space algorithm with running time (2e)k+O(log3 k)m log n whenever ∊1 = kO(1). Recently, Brand et al. [STOC 2018] provided a speed-up at the cost of reintroducing randomization. Specifically, they gave a randomized O(4km∊2)-time exponential-space algorithm. In this article, we revisit the algorithm by Alon and Gutner. We modify the foundation of their work, and with a novel twist, obtain the following results. We present a deterministic 4k+O(√k(log2 k+log2 ∊−1))m log n-time polynomial-space algorithm. This matches the running time of the best known deterministic polynomial-space algorithm for deciding whether a given graph G has a path on k vertices. Additionally, we present a randomized 4k+O(log k(log k+log ∊1))m log n-time polynomial-space algorithm. While Brand et al. make non-trivial use of exterior algebra, our algorithm is very simple; we only make elementary use of the probabilistic method. Thus, the algorithm by Brand et al. runs in time 4k+o(k)m whenever ∊1 = 2o(k), while our deterministic and randomized algorithms run in time 4k+o(k)m log n whenever ∊1 = 2o(k 4 ) and 1 ∊1 = 2o(log k k ), respectively. Prior to our work, no 2O(k)nO(1)-time polynomial-space algorithm was known. Additionally, our approach is embeddable in the classic framework of divide-and-color, hence it immediately extends to approximate counting of graphs of bounded treewidth; in comparison, Brand et al. note that their approach is limited to graphs of bounded pathwidth.

Original language English 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019 Christel Baier, Ioannis Chatzigiannakis, Paola Flocchini, Stefano Leonardi Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing 9783959771092 https://doi.org/10.4230/LIPIcs.ICALP.2019.24 Published - 1 Jul 2019 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019 - Patras, GreeceDuration: 9 Jul 2019 → 12 Jul 2019

Publication series

Name Leibniz International Proceedings in Informatics, LIPIcs 132 1868-8969

Conference

Conference 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019 Greece Patras 9/07/19 → 12/07/19

Keywords

• Approximate counting
• K-Path
• Parameterized complexity

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