4 Search Results for "Saks, Michael E."

Document
DOI: 10.4230/LIPIcs.SEA.2023.13
Integer Programming Formulations and Cutting Plane Algorithms for the Maximum Selective Tree Problem

Authors: Ömer Burak Onar, Tınaz Ekim, and Z. Caner Taşkın

Published in: LIPIcs, Volume 265, 21st International Symposium on Experimental Algorithms (SEA 2023)

Abstract
This paper considers the Maximum Selective Tree Problem (MSelTP) as a generalization of the Maximum Induced Tree problem. Given an undirected graph with a partition of its vertex set into clusters, MSelTP aims to choose the maximum number of vertices such that at most one vertex per cluster is selected and the graph induced by the selected vertices is a tree. To the best of our knowledge, MSelTP has not been studied before although several related optimization problems have been investigated in the literature. We propose two mixed integer programming formulations for MSelTP; one based on connectivity constraints, the other based on cycle elimination constraints. In addition, we develop two exact cutting plane procedures to solve the problem to optimality. On graphs with up to 25 clusters, up to 250 vertices, and varying densities, we conduct computational experiments to compare the results of two solution procedures with solving a compact integer programming formulation of MSelTP. Our experiments indicate that the algorithm CPAXnY outperforms the other procedures overall except for graphs with low density and large cluster size, and that the algorithm CPAX yields better results in terms of the average time of instances optimally solved and the overall average time.
Cite as

Ömer Burak Onar, Tınaz Ekim, and Z. Caner Taşkın. Integer Programming Formulations and Cutting Plane Algorithms for the Maximum Selective Tree Problem. In 21st International Symposium on Experimental Algorithms (SEA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 265, pp. 13:1-13:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{onar_et_al:LIPIcs.SEA.2023.13,
  author =	{Onar, \"{O}mer Burak and Ekim, T{\i}naz and Ta\c{s}k{\i}n, Z. Caner},
  title =	{{Integer Programming Formulations and Cutting Plane Algorithms for the Maximum Selective Tree Problem}},
  booktitle =	{21st International Symposium on Experimental Algorithms (SEA 2023)},
  pages =	{13:1--13:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-279-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{265},
  editor =	{Georgiadis, Loukas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2023.13},
  URN =		{urn:nbn:de:0030-drops-183634},
  doi =		{10.4230/LIPIcs.SEA.2023.13},
  annote =	{Keywords: maximum induced tree, selective tree, cutting plane, separation algorithm, mixed integer programming}
}
Document
DOI: 10.4230/LIPIcs.CCC.2020.22
NP-Hardness of Circuit Minimization for Multi-Output Functions

Authors: Rahul Ilango, Bruno Loff, and Igor C. Oliveira

Published in: LIPIcs, Volume 169, 35th Computational Complexity Conference (CCC 2020)

Abstract
Can we design efficient algorithms for finding fast algorithms? This question is captured by various circuit minimization problems, and algorithms for the corresponding tasks have significant practical applications. Following the work of Cook and Levin in the early 1970s, a central question is whether minimizing the circuit size of an explicitly given function is NP-complete. While this is known to hold in restricted models such as DNFs, making progress with respect to more expressive classes of circuits has been elusive. In this work, we establish the first NP-hardness result for circuit minimization of total functions in the setting of general (unrestricted) Boolean circuits. More precisely, we show that computing the minimum circuit size of a given multi-output Boolean function f : {0,1}^n → {0,1}^m is NP-hard under many-one polynomial-time randomized reductions. Our argument builds on a simpler NP-hardness proof for the circuit minimization problem for (single-output) Boolean functions under an extended set of generators. Complementing these results, we investigate the computational hardness of minimizing communication. We establish that several variants of this problem are NP-hard under deterministic reductions. In particular, unless 𝖯 = 𝖭𝖯, no polynomial-time computable function can approximate the deterministic two-party communication complexity of a partial Boolean function up to a polynomial. This has consequences for the class of structural results that one might hope to show about the communication complexity of partial functions.
Cite as

Rahul Ilango, Bruno Loff, and Igor C. Oliveira. NP-Hardness of Circuit Minimization for Multi-Output Functions. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 22:1-22:36, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{ilango_et_al:LIPIcs.CCC.2020.22,
  author =	{Ilango, Rahul and Loff, Bruno and Oliveira, Igor C.},
  title =	{{NP-Hardness of Circuit Minimization for Multi-Output Functions}},
  booktitle =	{35th Computational Complexity Conference (CCC 2020)},
  pages =	{22:1--22:36},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-156-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{169},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.22},
  URN =		{urn:nbn:de:0030-drops-125744},
  doi =		{10.4230/LIPIcs.CCC.2020.22},
  annote =	{Keywords: MCSP, circuit minimization, communication complexity, Boolean circuit}
}
Document
DOI: 10.4230/LIPIcs.STACS.2018.23
Lower Bounds for Combinatorial Algorithms for Boolean Matrix Multiplication

Authors: Debarati Das, Michal Koucký, and Michael Saks

Published in: LIPIcs, Volume 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)

Abstract
In this paper we propose models of combinatorial algorithms for the Boolean Matrix Multiplication (BMM), and prove lower bounds on computing BMM in these models. First, we give a relatively relaxed combinatorial model which is an extension of the model by Angluin (1976), and we prove that the time required by any algorithm for the BMM is at least Omega(n^3 / 2^{O( sqrt{ log n })}). Subsequently, we propose a more general model capable of simulating the "Four Russian Algorithm". We prove a lower bound of Omega(n^{7/3} / 2^{O(sqrt{ log n })}) for the BMM under this model. We use a special class of graphs, called (r,t)-graphs, originally discovered by Rusza and Szemeredi (1978), along with randomization, to construct matrices that are hard instances for our combinatorial models.
Cite as

Debarati Das, Michal Koucký, and Michael Saks. Lower Bounds for Combinatorial Algorithms for Boolean Matrix Multiplication. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 23:1-23:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{das_et_al:LIPIcs.STACS.2018.23,
  author =	{Das, Debarati and Kouck\'{y}, Michal and Saks, Michael},
  title =	{{Lower Bounds for Combinatorial Algorithms for Boolean Matrix Multiplication}},
  booktitle =	{35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
  pages =	{23:1--23:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-062-0},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{96},
  editor =	{Niedermeier, Rolf and Vall\'{e}e, Brigitte},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2018.23},
  URN =		{urn:nbn:de:0030-drops-85050},
  doi =		{10.4230/LIPIcs.STACS.2018.23},
  annote =	{Keywords: Lower bounds, Combinatorial algorithm, Boolean matrix multiplication}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2014.596
The Power of Super-logarithmic Number of Players

Authors: Arkadev Chattopadhyay and Michael E. Saks

Published in: LIPIcs, Volume 28, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)

Abstract
In the `Number-on-Forehead' (NOF) model of multiparty communication, the input is a k times m boolean matrix A (where k is the number of players) and Player i sees all bits except those in the i-th row, and the players communicate by broadcast in order to evaluate a specified function f at A. We discover new computational power when k exceeds log m. We give a protocol with communication cost poly-logarithmic in m, for block composed functions with limited block width. These are functions of the form f o g where f is a symmetric b-variate function, and g is a (kr)-variate function and (f o g)(A) is defined, for a k times (br) matrix to be f(g(A-1),...,g(A-b)) where A-i is the i-th (k times r) block of A. Our protocol works provided that k > 1+ ln b + (2 to the power of r). Ada et al. (ICALP'2012) previously obtained simultaneous and deterministic efficient protocols for composed functions of block-width one. The new protocol is the first to work for block composed functions with block-width greather than one. Moreover, it is simultaneous, with vanishingly small error probability, if public coin randomness is allowed. The deterministic and zero-error version barely uses interaction.
Cite as

Arkadev Chattopadhyay and Michael E. Saks. The Power of Super-logarithmic Number of Players. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 596-603, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2014)

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@InProceedings{chattopadhyay_et_al:LIPIcs.APPROX-RANDOM.2014.596,
  author =	{Chattopadhyay, Arkadev and Saks, Michael E.},
  title =	{{The Power of Super-logarithmic Number of Players}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)},
  pages =	{596--603},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-74-3},
  ISSN =	{1868-8969},
  year =	{2014},
  volume =	{28},
  editor =	{Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.596},
  URN =		{urn:nbn:de:0030-drops-47243},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2014.596},
  annote =	{Keywords: Communication complexity, Number-On-Forehead model, composed functions}
}
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