License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
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DOI: 10.4230/LIPIcs.ISAAC.2021.28
URN: urn:nbn:de:0030-drops-154612
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Nikoletseas, Sotiris ; Raptopoulos, Christoforos ; Spirakis, Paul

MAX CUT in Weighted Random Intersection Graphs and Discrepancy of Sparse Random Set Systems

LIPIcs-ISAAC-2021-28.pdf (0.9 MB)


Let V be a set of n vertices, M a set of m labels, and let 𝐑 be an m × n matrix of independent Bernoulli random variables with probability of success p; columns of 𝐑 are incidence vectors of label sets assigned to vertices. A random instance G(V, E, 𝐑^T 𝐑) of the weighted random intersection graph model is constructed by drawing an edge with weight equal to the number of common labels (namely [𝐑^T 𝐑]_{v,u}) between any two vertices u, v for which this weight is strictly larger than 0. In this paper we study the average case analysis of Weighted Max Cut, assuming the input is a weighted random intersection graph, i.e. given G(V, E, 𝐑^T 𝐑) we wish to find a partition of V into two sets so that the total weight of the edges having exactly one endpoint in each set is maximized.
In particular, we initially prove that the weight of a maximum cut of G(V, E, 𝐑^T 𝐑) is concentrated around its expected value, and then show that, when the number of labels is much smaller than the number of vertices (in particular, m = n^α, α < 1), a random partition of the vertices achieves asymptotically optimal cut weight with high probability. Furthermore, in the case n = m and constant average degree (i.e. p = Θ(1)/n), we show that with high probability, a majority type randomized algorithm outputs a cut with weight that is larger than the weight of a random cut by a multiplicative constant strictly larger than 1. Then, we formally prove a connection between the computational problem of finding a (weighted) maximum cut in G(V, E, 𝐑^T 𝐑) and the problem of finding a 2-coloring that achieves minimum discrepancy for a set system Σ with incidence matrix 𝐑 (i.e. minimum imbalance over all sets in Σ). We exploit this connection by proposing a (weak) bipartization algorithm for the case m = n, p = Θ(1)/n that, when it terminates, its output can be used to find a 2-coloring with minimum discrepancy in a set system with incidence matrix 𝐑. In fact, with high probability, the latter 2-coloring corresponds to a bipartition with maximum cut-weight in G(V, E, 𝐑^T 𝐑). Finally, we prove that our (weak) bipartization algorithm terminates in polynomial time, with high probability, at least when p = c/n, c < 1.

BibTeX - Entry

  author =	{Nikoletseas, Sotiris and Raptopoulos, Christoforos and Spirakis, Paul},
  title =	{{MAX CUT in Weighted Random Intersection Graphs and Discrepancy of Sparse Random Set Systems}},
  booktitle =	{32nd International Symposium on Algorithms and Computation (ISAAC 2021)},
  pages =	{28:1--28:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-214-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{212},
  editor =	{Ahn, Hee-Kap and Sadakane, Kunihiko},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-154612},
  doi =		{10.4230/LIPIcs.ISAAC.2021.28},
  annote =	{Keywords: Random Intersection Graphs, Maximum Cut, Discrepancy}

Keywords: Random Intersection Graphs, Maximum Cut, Discrepancy
Collection: 32nd International Symposium on Algorithms and Computation (ISAAC 2021)
Issue Date: 2021
Date of publication: 30.11.2021

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