2 Search Results for "Bamas, Étienne"

Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2021.22
The Submodular Santa Claus Problem in the Restricted Assignment Case

Authors: Etienne Bamas, Paritosh Garg, and Lars Rohwedder

Published in: LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

Abstract
The submodular Santa Claus problem was introduced in a seminal work by Goemans, Harvey, Iwata, and Mirrokni (SODA'09) as an application of their structural result. In the mentioned problem n unsplittable resources have to be assigned to m players, each with a monotone submodular utility function f_i. The goal is to maximize min_i f_i(S_i) where S₁,...,S_m is a partition of the resources. The result by Goemans et al. implies a polynomial time O(n^{1/2 +ε})-approximation algorithm. Since then progress on this problem was limited to the linear case, that is, all f_i are linear functions. In particular, a line of research has shown that there is a polynomial time constant approximation algorithm for linear valuation functions in the restricted assignment case. This is the special case where each player is given a set of desired resources Γ_i and the individual valuation functions are defined as f_i(S) = f(S ∩ Γ_i) for a global linear function f. This can also be interpreted as maximizing min_i f(S_i) with additional assignment restrictions, i.e., resources can only be assigned to certain players. In this paper we make comparable progress for the submodular variant: If f is a monotone submodular function, we can in polynomial time compute an O(log log(n))-approximate solution.
Cite as

Etienne Bamas, Paritosh Garg, and Lars Rohwedder. The Submodular Santa Claus Problem in the Restricted Assignment Case. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 22:1-22:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bamas_et_al:LIPIcs.ICALP.2021.22,
  author =	{Bamas, Etienne and Garg, Paritosh and Rohwedder, Lars},
  title =	{{The Submodular Santa Claus Problem in the Restricted Assignment Case}},
  booktitle =	{48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
  pages =	{22:1--22:18},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.22},
  URN =		{urn:nbn:de:0030-drops-140912},
  doi =		{10.4230/LIPIcs.ICALP.2021.22},
  annote =	{Keywords: Scheduling, submodularity, approximation algorithm, hypergraph matching}
}
Document
DOI: 10.4230/LIPIcs.STACS.2019.10
Distributed Coloring of Graphs with an Optimal Number of Colors

Authors: Étienne Bamas and Louis Esperet

Published in: LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

Abstract
This paper studies sufficient conditions to obtain efficient distributed algorithms coloring graphs optimally (i.e. with the minimum number of colors) in the LOCAL model of computation. Most of the work on distributed vertex coloring so far has focused on coloring graphs of maximum degree Delta with at most Delta+1 colors (or Delta colors when some simple obstructions are forbidden). When Delta is sufficiently large and c >= Delta-k_Delta+1, for some integer k_Delta ~~ sqrt{Delta}-2, we give a distributed algorithm that given a c-colorable graph G of maximum degree Delta, finds a c-coloring of G in min{O((log Delta)^{13/12}log n), 2^{O(log Delta+sqrt{log log n})}} rounds, with high probability. The lower bound Delta-k_Delta+1 is best possible in the sense that for infinitely many values of Delta, we prove that when chi(G) <= Delta-k_Delta, finding an optimal coloring of G requires Omega(n) rounds. Our proof is a light adaptation of a remarkable result of Molloy and Reed, who proved that for Delta large enough, for any c >= Delta-k_Delta deciding whether chi(G) <= c is in P, while Embden-Weinert et al. proved that for c <= Delta-k_Delta-1, the same problem is NP-complete. Note that the sequential and distributed thresholds differ by one. Our first result covers the case where the chromatic number of the graph ranges between Delta-sqrt{Delta} and Delta+1. Our second result covers a larger range, but gives a weaker bound on the number of colors: For any sufficiently large Delta, and Omega(log Delta) <= k <= Delta/100, we prove that every graph of maximum degree Delta and clique number at most Delta-k can be efficiently colored with at most Delta-epsilon k colors, for some absolute constant epsilon >0, with a randomized algorithm running in O(log n/log log n) rounds with high probability.
Cite as

Étienne Bamas and Louis Esperet. Distributed Coloring of Graphs with an Optimal Number of Colors. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 10:1-10:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

Copy BibTex To Clipboard

@InProceedings{bamas_et_al:LIPIcs.STACS.2019.10,
  author =	{Bamas, \'{E}tienne and Esperet, Louis},
  title =	{{Distributed Coloring of Graphs with an Optimal Number of Colors}},
  booktitle =	{36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)},
  pages =	{10:1--10:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-100-9},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{126},
  editor =	{Niedermeier, Rolf and Paul, Christophe},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.10},
  URN =		{urn:nbn:de:0030-drops-102496},
  doi =		{10.4230/LIPIcs.STACS.2019.10},
  annote =	{Keywords: Graph coloring, distributed algorithm, maximum degree}
}
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