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DOI: 10.4230/LIPIcs.MFCS.2018.24

Hardness Results for Consensus-Halving

Authors: Aris Filos-Ratsikas, Søren Kristoffer Stiil Frederiksen, Paul W. Goldberg, and Jie Zhang

Published in: LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

Abstract

The Consensus-halving problem is the problem of dividing an object into two portions, such that each of n agents has equal valuation for the two portions. We study the epsilon-approximate version, which allows each agent to have an epsilon discrepancy on the values of the portions. It was recently proven in [Filos-Ratsikas and Goldberg, 2018] that the problem of computing an epsilon-approximate Consensus-halving solution (for n agents and n cuts) is PPA-complete when epsilon is inverse-exponential. In this paper, we prove that when epsilon is constant, the problem is PPAD-hard and the problem remains PPAD-hard when we allow a constant number of additional cuts. Additionally, we prove that deciding whether a solution with n-1 cuts exists for the problem is NP-hard.

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Aris Filos-Ratsikas, Søren Kristoffer Stiil Frederiksen, Paul W. Goldberg, and Jie Zhang. Hardness Results for Consensus-Halving. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 24:1-24:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{filosratsikas_et_al:LIPIcs.MFCS.2018.24,
  author =	{Filos-Ratsikas, Aris and Frederiksen, S{\o}ren Kristoffer Stiil and Goldberg, Paul W. and Zhang, Jie},
  title =	{{Hardness Results for Consensus-Halving}},
  booktitle =	{43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)},
  pages =	{24:1--24:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-086-6},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{117},
  editor =	{Potapov, Igor and Spirakis, Paul 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.MFCS.2018.24},
  URN =		{urn:nbn:de:0030-drops-96069},
  doi =		{10.4230/LIPIcs.MFCS.2018.24},
  annote =	{Keywords: PPAD, PPA, consensus halving, generalized-circuit, reduction}
}

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DOI: 10.4230/LIPIcs.MFCS.2017.80

Walrasian Pricing in Multi-Unit Auctions

Authors: Simina Brânzei, Aris Filos-Ratsikas, Peter Bro Miltersen, and Yulong Zeng

Published in: LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

Abstract

Multi-unit auctions are a paradigmatic model, where a seller brings multiple units of a good, while several buyers bring monetary endowments. It is well known that Walrasian equilibria do not always exist in this model, however compelling relaxations such as Walrasian envy-free pricing do. In this paper we design an optimal envy-free mechanism for multi-unit auctions with budgets. When the market is even mildly competitive, the approximation ratios of this mechanism are small constants for both the revenue and welfare objectives, and in fact for welfare the approximation converges to 1 as the market becomes fully competitive. We also give an impossibility theorem, showing that truthfulness requires discarding resources, and in particular, is incompatible with (Pareto) efficiency.

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Simina Brânzei, Aris Filos-Ratsikas, Peter Bro Miltersen, and Yulong Zeng. Walrasian Pricing in Multi-Unit Auctions. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 80:1-80:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{branzei_et_al:LIPIcs.MFCS.2017.80,
  author =	{Br\^{a}nzei, Simina and Filos-Ratsikas, Aris and Miltersen, Peter Bro and Zeng, Yulong},
  title =	{{Walrasian Pricing in Multi-Unit Auctions}},
  booktitle =	{42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)},
  pages =	{80:1--80:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-046-0},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{83},
  editor =	{Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.80},
  URN =		{urn:nbn:de:0030-drops-81197},
  doi =		{10.4230/LIPIcs.MFCS.2017.80},
  annote =	{Keywords: mechanism design, multi-unit auctions, Walrasian pricing, market share}
}

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Documents authored by Filos-Ratsikas, Aris

Hardness Results for Consensus-Halving

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Walrasian Pricing in Multi-Unit Auctions

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