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Exact Weight Subgraphs and the k-Sum Conjecture

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Automata, Languages, and Programming (ICALP 2013)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7965))

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Abstract

We consider the Exact-Weight-H problem of finding a (not necessarily induced) subgraph H of weight 0 in an edge-weighted graph G. We show that for every H, the complexity of this problem is strongly related to that of the infamous k-sum problem. In particular, we show that under the k-sum Conjecture, we can achieve tight upper and lower bounds for the Exact-Weight-H problem for various subgraphs H such as matching, star, path, and cycle.

One interesting consequence is that improving on the O(n 3) upper bound for Exact-Weight-4-path or Exact-Weight-5-path will imply improved algorithms for 3-sum, 5-sum, All-Pairs Shortest Paths and other fundamental problems. This is in sharp contrast to the minimum-weight and (unweighted) detection versions, which can be solved easily in time O(n 2). We also show that a faster algorithm for any of the following three problems would yield faster algorithms for the others: 3-sum, Exact-Weight-3-matching, and Exact-Weight-3-star.

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Authors and Affiliations

  1. Computer Science Department, Stanford University, USA

    Amir Abboud & Kevin Lewi

Authors
  1. Amir Abboud
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  2. Kevin Lewi
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Editors and Affiliations

  1. Department of Informatics, University of Bergen, Postboks 7803, 5020, Bergen, Norway

    Fedor V. Fomin

  2. Faculty of Computing, University of Latvia, Raina bulv. 19, 1586, Riga, Latvia

    Rūsiņš Freivalds

  3. Department of Computer Science, University of Oxford, Wolfson Building, Parks Road, OX1 3QD, Oxford, UK

    Marta Kwiatkowska

  4. Faculty of Mathematics and Computer Science, Weizmann Institute of Science, POB 26, 76100, Rehovot, Israel

    David Peleg

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Abboud, A., Lewi, K. (2013). Exact Weight Subgraphs and the k-Sum Conjecture. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds) Automata, Languages, and Programming. ICALP 2013. Lecture Notes in Computer Science, vol 7965. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39206-1_1

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  • DOI: https://doi.org/10.1007/978-3-642-39206-1_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-39205-4

  • Online ISBN: 978-3-642-39206-1

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