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

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.

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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Invited Talk

**Published in:** LIPIcs, Volume 25, 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)

We survey recent applications of real algebraic and semi-algebraic geometry in (computational) game theory.

Peter Bro Miltersen. Semi-algebraic geometry in computational game theory - a consumer's perspective (Invited Talk). In 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 25, pp. 11-12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{miltersen:LIPIcs.STACS.2014.11, author = {Miltersen, Peter Bro}, title = {{Semi-algebraic geometry in computational game theory - a consumer's perspective}}, booktitle = {31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)}, pages = {11--12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-65-1}, ISSN = {1868-8969}, year = {2014}, volume = {25}, editor = {Mayr, Ernst W. and Portier, Natacha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2014.11}, URN = {urn:nbn:de:0030-drops-44994}, doi = {10.4230/LIPIcs.STACS.2014.11}, annote = {Keywords: Real Algebraic Geometry, Computational Game Theory} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 10171, Equilibrium Computation (2010)

From April 25 to April 30, 2010, the Dagstuhl Seminar 10171 ``Equilibrium Computation'' was held in Schloss Dagstuhl~--~Leibniz Center for Informatics.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available.

Edith Elkind, Nimrod Megiddo, Peter Bro Miltersen, Bernhard von Stengel, and Vijay V. Vazirani. 10171 Abstracts Collection – Equilibrium Computation. In Equilibrium Computation. Dagstuhl Seminar Proceedings, Volume 10171, pp. 1-18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{elkind_et_al:DagSemProc.10171.1, author = {Elkind, Edith and Megiddo, Nimrod and Miltersen, Peter Bro and von Stengel, Bernhard and Vazirani, Vijay V.}, title = {{10171 Abstracts Collection – Equilibrium Computation}}, booktitle = {Equilibrium Computation}, pages = {1--18}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2010}, volume = {10171}, editor = {Edith Elkind and Nimrod Megiddo and Peter Bro Miltersen and Vijay V. Vazirani and Bernahrd von Stengel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.10171.1}, URN = {urn:nbn:de:0030-drops-26738}, doi = {10.4230/DagSemProc.10171.1}, annote = {Keywords: Equilibrium computation, algorithmic game theory} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 8381, Computational Complexity of Discrete Problems (2008)

From the 14th of September to the 19th of September, the Dagstuhl Seminar
08381 ``Computational Complexity of Discrete Problems'' was held in Schloss Dagstuhl - Leibniz Center for Informatics.
During the seminar, several participants presented their current
research, and ongoing work as well as open problems were discussed.
Abstracts of the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this report. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available.

Peter Bro Miltersen, Rüdiger Reischuk, Georg Schnitger, and Dieter van Melkebeek. 08381 Abstracts Collection – Computational Complexity of Discrete Problems. In Computational Complexity of Discrete Problems. Dagstuhl Seminar Proceedings, Volume 8381, pp. 1-18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{miltersen_et_al:DagSemProc.08381.1, author = {Miltersen, Peter Bro and Reischuk, R\"{u}diger and Schnitger, Georg and van Melkebeek, Dieter}, title = {{08381 Abstracts Collection – Computational Complexity of Discrete Problems}}, booktitle = {Computational Complexity of Discrete Problems}, pages = {1--18}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2008}, volume = {8381}, editor = {Peter Bro Miltersen and R\"{u}diger Reischuk and Georg Schnitger and Dieter van Melkebeek}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.08381.1}, URN = {none}, doi = {10.4230/DagSemProc.08381.1}, annote = {Keywords: Computational complexity, discrete problems, Turing machines, circuits, proof complexity, pseudorandomness, derandomization, cryptography, computational learning, communication complexity, query complexity, hardness of approximation} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 8381, Computational Complexity of Discrete Problems (2008)

Estimating the computational complexity of discrete problems constitutes one of the central and classical topics in the theory of computation. Mathematicians and computer scientists have long tried to classify natural families of Boolean relations according to fundamental complexity measures like time and space, both in the uniform and in the nonuniform setting. Several models of computation have been developed in order to capture the various capabilities of digital computing devices, including parallelism, randomness, and quantum interference.

Peter Bro Miltersen, Rüdiger Reischuk, Georg Schnitger, and Dieter van Melkebeek. 08381 Executive Summary – Computational Complexity of Discrete Problems. In Computational Complexity of Discrete Problems. Dagstuhl Seminar Proceedings, Volume 8381, pp. 1-7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{miltersen_et_al:DagSemProc.08381.2, author = {Miltersen, Peter Bro and Reischuk, R\"{u}diger and Schnitger, Georg and van Melkebeek, Dieter}, title = {{08381 Executive Summary – Computational Complexity of Discrete Problems}}, booktitle = {Computational Complexity of Discrete Problems}, pages = {1--7}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2008}, volume = {8381}, editor = {Peter Bro Miltersen and R\"{u}diger Reischuk and Georg Schnitger and Dieter van Melkebeek}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.08381.2}, URN = {urn:nbn:de:0030-drops-17789}, doi = {10.4230/DagSemProc.08381.2}, annote = {Keywords: Computational complexity, discrete problems, Turing machines, circuits, proof complexity, pseudorandomness, derandomization, cryptography, computational learning, communication complexity, query complexity, hardness of approximation} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 6111, Complexity of Boolean Functions (2006)

We study two quite different approaches to understanding the complexity of fundamental problems in numerical analysis. We show that both hinge on the question of understanding the complexity of the following problem, which we call PosSlp: Given a division-free straight-line program producing an integer N, decide whether N>0. We show that OrdSlp lies in the counting hierarchy, and combining our results with work of Tiwari, we show that the Euclidean Traveling Salesman Problem lies in the counting hierarchy – the previous best upper bound for this important problem (in terms of classical complexity classes) being PSPACE.

Eric Allender, Peter Bürgisser, Johan Kjeldgaard-Pedersen, and Peter Bro Miltersen. On the Complexity of Numerical Analysis. In Complexity of Boolean Functions. Dagstuhl Seminar Proceedings, Volume 6111, pp. 1-9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{allender_et_al:DagSemProc.06111.12, author = {Allender, Eric and B\"{u}rgisser, Peter and Kjeldgaard-Pedersen, Johan and Miltersen, Peter Bro}, title = {{On the Complexity of Numerical Analysis}}, booktitle = {Complexity of Boolean Functions}, pages = {1--9}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2006}, volume = {6111}, editor = {Matthias Krause and Pavel Pudl\'{a}k and R\"{u}diger Reischuk and Dieter van Melkebeek}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.06111.12}, URN = {urn:nbn:de:0030-drops-6130}, doi = {10.4230/DagSemProc.06111.12}, annote = {Keywords: Blum-Shub-Smale Model, Euclidean Traveling Salesman Problem, Counting Hierarchy} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 6111, Complexity of Boolean Functions (2006)

In the cell probe model with word size 1 (the bit probe model), a
static data structure problem is given by a map
$f: {0,1}^n imes {0,1}^m
ightarrow {0,1}$,
where ${0,1}^n$ is a set of possible data to be stored,
${0,1}^m$ is a set of possible queries (for natural problems, we
have $m ll n$) and $f(x,y)$ is
the answer to question $y$ about data $x$.
A solution is given by a
representation $phi: {0,1}^n
ightarrow {0,1}^s$ and a query algorithm
$q$ so that $q(phi(x), y) = f(x,y)$. The time $t$ of the query algorithm
is the number of bits it reads in $phi(x)$.
In this paper, we consider the case of {em succinct} representations
where $s = n + r$ for some {em redundancy} $r ll n$.
For
a boolean version of the problem of polynomial
evaluation with preprocessing of coefficients, we show a lower bound on
the redundancy-query time tradeoff of the form
[ (r+1) t geq Omega(n/log n).]
In particular, for very small
redundancies $r$, we get an almost optimal lower bound stating that the
query algorithm has to inspect almost the entire data structure
(up to a logarithmic factor).
We show similar lower bounds for problems satisfying a certain
combinatorial property of a coding theoretic flavor.
Previously, no $omega(m)$ lower bounds were known on $t$
in the general model for explicit functions, even for very small
redundancies.
By restricting our attention to {em systematic} or {em index}
structures $phi$ satisfying $phi(x) = x cdot phi^*(x)$ for some
map $phi^*$ (where $cdot$ denotes concatenation) we show
similar lower bounds on the redundancy-query time tradeoff
for the natural data structuring problems of Prefix Sum
and Substring Search.

Anna Gál and Peter Bro Miltersen. The Cell Probe Complexity of Succinct Data Structures. In Complexity of Boolean Functions. Dagstuhl Seminar Proceedings, Volume 6111, pp. 1-13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{gal_et_al:DagSemProc.06111.17, author = {G\'{a}l, Anna and Miltersen, Peter Bro}, title = {{The Cell Probe Complexity of Succinct Data Structures}}, booktitle = {Complexity of Boolean Functions}, pages = {1--13}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2006}, volume = {6111}, editor = {Matthias Krause and Pavel Pudl\'{a}k and R\"{u}diger Reischuk and Dieter van Melkebeek}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.06111.17}, URN = {urn:nbn:de:0030-drops-6065}, doi = {10.4230/DagSemProc.06111.17}, annote = {Keywords: Cell probe model, data structures, lower bounds, time-space tradeoffs} }

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