Dagstuhl Seminar Proceedings, Volume 4421

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04421 Abstracts Collection – Algebraic Methods in Computational Complexity

Authors: Harry Buhrman, Lance Fortnow, and Thomas Thierauf

Abstract

From 10.10.04 to 15.10.04, the Dagstuhl Seminar 04421 ``Algebraic Methods in Computational Complexity'' was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. 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.

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Harry Buhrman, Lance Fortnow, and Thomas Thierauf. 04421 Abstracts Collection – Algebraic Methods in Computational Complexity. In Algebraic Methods in Computational Complexity. Dagstuhl Seminar Proceedings, Volume 4421, pp. 1-14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{buhrman_et_al:DagSemProc.04421.1,
  author =	{Buhrman, Harry and Fortnow, Lance and Thierauf, Thomas},
  title =	{{04421 Abstracts Collection – Algebraic Methods in Computational Complexity}},
  booktitle =	{Algebraic Methods in Computational Complexity},
  pages =	{1--14},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2005},
  volume =	{4421},
  editor =	{Harry Buhrman and Lance Fortnow and Thomas Thierauf},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.04421.1},
  URN =		{urn:nbn:de:0030-drops-1162},
  doi =		{10.4230/DagSemProc.04421.1},
  annote =	{Keywords: Computational complexity , algebraic methods , quantum computations , lower bounds}
}

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DOI: 10.4230/DagSemProc.04421.2

Exact-Four-Colorability, Exact Domatic Number Problems, and the Boolean Hierarchy

Authors: Jörg Rothe

Abstract

This talk surveys some of the work that was inspired by Wagner's general technique to prove completeness in the levels of the boolean hierarchy over NP. In particular, we show that it is DP-complete to decide whether or not a given graph can be colored with exactly four colors. DP is the second level of the boolean hierarchy. This result solves a question raised by Wagner in his 1987 TCS paper; its proof uses a clever reduction by Guruswami and Khanna. Similar results on various versions of the exact domatic number problem are also discussed. The result on Exact-Four-Colorability appeared in IPL, 2003. The results on exact domatic number problems, obtained jointly with Tobias Riege, are to appear in TOCS.

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Jörg Rothe. Exact-Four-Colorability, Exact Domatic Number Problems, and the Boolean Hierarchy. In Algebraic Methods in Computational Complexity. Dagstuhl Seminar Proceedings, Volume 4421, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{rothe:DagSemProc.04421.2,
  author =	{Rothe, J\"{o}rg},
  title =	{{Exact-Four-Colorability, Exact Domatic Number Problems, and the Boolean Hierarchy}},
  booktitle =	{Algebraic Methods in Computational Complexity},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2005},
  volume =	{4421},
  editor =	{Harry Buhrman and Lance Fortnow and Thomas Thierauf},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.04421.2},
  URN =		{urn:nbn:de:0030-drops-1059},
  doi =		{10.4230/DagSemProc.04421.2},
  annote =	{Keywords: Exact Colorability , exact domatic number , boolean hierarchy completeness}
}

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DOI: 10.4230/DagSemProc.04421.3

Finding Isolated Cliques by Queries – An Approach to Fault Diagnosis with Many Faults

Authors: William Gasarch and Frank Stephan

Abstract

A well-studied problem in fault diagnosis is to identify the set of all good processors in a given set $\{p_1,p_2,\ldots,p_n\}$ of processors via asking some processors $p_i$ to test whether processor $p_j$ is good or faulty. Mathematically, the set $C$ of the indices of good processors forms an isolated clique in the graph with the edges $E = \{(i,j):$ if you ask $p_i$ to test $p_j$ then $p_i$ states that ``$p_j$ is good''$\}$; where $C$ is an isolated clique iff it holds for every $i \in C$ and $j \neq i$ that $(i,j) \in E$ iff $j \in C$. In the present work, the classical setting of fault diagnosis is modified by no longer requiring that $C$ contains at least $\frac{n+1}{2}$ of the $n$ nodes of the graph. Instead, one is given a lower bound $a$ on the size of $C$ and the number $n$ of nodes and one has to find a list of up to $n/a$ candidates containing all isolated cliques of size $a$ or more where the number of queries whether a given edge is in $E$ is as small as possible. It is shown that the number of queries necessary differs at most by $n$ for the case of directed and undirected graphs. Furthermore, for directed graphs the lower bound $n^2/(2a-2)-3n$ and the upper bound $2n^2/a$ are established. For some constant values of $a$, better bounds are given. In the case of parallel queries, the number of rounds is at least $n/(a-1)-6$ and at most $O(\log(a)n/a)$.

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William Gasarch and Frank Stephan. Finding Isolated Cliques by Queries – An Approach to Fault Diagnosis with Many Faults. In Algebraic Methods in Computational Complexity. Dagstuhl Seminar Proceedings, Volume 4421, pp. 1-16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{gasarch_et_al:DagSemProc.04421.3,
  author =	{Gasarch, William and Stephan, Frank},
  title =	{{Finding Isolated Cliques by Queries – An Approach to Fault Diagnosis with Many Faults}},
  booktitle =	{Algebraic Methods in Computational Complexity},
  pages =	{1--16},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2005},
  volume =	{4421},
  editor =	{Harry Buhrman and Lance Fortnow and Thomas Thierauf},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.04421.3},
  URN =		{urn:nbn:de:0030-drops-1066},
  doi =		{10.4230/DagSemProc.04421.3},
  annote =	{Keywords: Isolated Cliques , Query-Complexity , Fault Diagnosis}
}

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DOI: 10.4230/DagSemProc.04421.4

Preferences and Domination

Authors: Judy Goldsmith

Abstract

CP-nets are a succinct formalism for specifying preferences over a multi-featured domain. A CP-net consists of a directed graph, with nodes representing the features of the domain, and edges indicating conditional preferences. An instance in the domain is an assignment of values to the features. An instance alpha is preferred to an instance beta if there are a sequence of "improving flips" from alpha to beta, where an improving flip changes the value of one feature to a more-preferred value, based on the values of the parents of that feature. We say alpha dominates beta if such a sequence exists. We show that recognizing dominance is PSPACE hard for cyclic CP-nets.

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Judy Goldsmith. Preferences and Domination. In Algebraic Methods in Computational Complexity. Dagstuhl Seminar Proceedings, Volume 4421, pp. 1-10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{goldsmith:DagSemProc.04421.4,
  author =	{Goldsmith, Judy},
  title =	{{Preferences and Domination}},
  booktitle =	{Algebraic Methods in Computational Complexity},
  pages =	{1--10},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2005},
  volume =	{4421},
  editor =	{Harry Buhrman and Lance Fortnow and Thomas Thierauf},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.04421.4},
  URN =		{urn:nbn:de:0030-drops-1038},
  doi =		{10.4230/DagSemProc.04421.4},
  annote =	{Keywords: Preferences , CP-nets , PSPACE-complete , reductions}
}

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DOI: 10.4230/DagSemProc.04421.5

Randomized QuickSort and the Entropy of the Random Source

Authors: Beatrice List, Markus Maucher, Uwe Schöning, and Rainer Schuler

Abstract

The worst-case complexity of an implementation of Quicksort depends on the random number generator that is used to select the pivot elements. In this paper we estimate the expected number of comparisons of Quicksort as a function in the entropy of the random source. We give upper and lower bounds and show that the expected number of comparisons increases from $n\log n$ to $n^2$, if the entropy of the random source is bounded. As examples we show explicit bounds for distributions with bounded min-entropy and the geometrical distribution.

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Beatrice List, Markus Maucher, Uwe Schöning, and Rainer Schuler. Randomized QuickSort and the Entropy of the Random Source. In Algebraic Methods in Computational Complexity. Dagstuhl Seminar Proceedings, Volume 4421, pp. 1-15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{list_et_al:DagSemProc.04421.5,
  author =	{List, Beatrice and Maucher, Markus and Sch\"{o}ning, Uwe and Schuler, Rainer},
  title =	{{Randomized QuickSort and the Entropy of the Random Source}},
  booktitle =	{Algebraic Methods in Computational Complexity},
  pages =	{1--15},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2005},
  volume =	{4421},
  editor =	{Harry Buhrman and Lance Fortnow and Thomas Thierauf},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.04421.5},
  URN =		{urn:nbn:de:0030-drops-1043},
  doi =		{10.4230/DagSemProc.04421.5},
  annote =	{Keywords: Randomized Algorithms , QuickSort , Entropy}
}

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DOI: 10.4230/DagSemProc.04421.6

The communication complexity of the Exact-N Problem revisited

Authors: William Gasarch, James Glenn, and Andre Utis

Abstract

If Alice has x, y, Bob has x, z and Carol has y, z can they determine if x+y+z=N? They can if (say) Alice broadcasts x to Bob and Carol; can they do better? Chandra, Furst, and Lipton studied this problem and showed sublinear upper bounds. They also had matching (up to an additive constant) lower bounds. We give an exposition of their result with some attention to what happens for particular values of N.

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William Gasarch, James Glenn, and Andre Utis. The communication complexity of the Exact-N Problem revisited. In Algebraic Methods in Computational Complexity. Dagstuhl Seminar Proceedings, Volume 4421, pp. 1-14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{gasarch_et_al:DagSemProc.04421.6,
  author =	{Gasarch, William and Glenn, James and Utis, Andre},
  title =	{{The communication complexity of the Exact-N Problem revisited}},
  booktitle =	{Algebraic Methods in Computational Complexity},
  pages =	{1--14},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2005},
  volume =	{4421},
  editor =	{Harry Buhrman and Lance Fortnow and Thomas Thierauf},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.04421.6},
  URN =		{urn:nbn:de:0030-drops-1026},
  doi =		{10.4230/DagSemProc.04421.6},
  annote =	{Keywords: Communication Complexity , Exact-N problem , Arithmetic Sequences}
}

Dagstuhl Seminar Proceedings, Volume 4421

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04421 Abstracts Collection – Algebraic Methods in Computational Complexity

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Exact-Four-Colorability, Exact Domatic Number Problems, and the Boolean Hierarchy

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Finding Isolated Cliques by Queries – An Approach to Fault Diagnosis with Many Faults

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Preferences and Domination

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Randomized QuickSort and the Entropy of the Random Source

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The communication complexity of the Exact-N Problem revisited

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