Volume
OASIcs, Volume 61
1st Symposium on Simplicity in Algorithms (SOSA 2018)
- Part of: Series: Open Access Series in Informatics (OASIcs)
Event
SOSA 2018, January 7-10, 2018, New Orleans, LA, USAEditor
Publication Details
- published at: 2018-01-05
- Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- ISBN: 978-3-95977-064-4
- DBLP: db/conf/soda/sosa2018
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Document
Complete Volume
OASIcs, Volume 61, SOSA'18, Complete Volume
Abstract
OASIcs, Volume 61, SOSA'18, Complete Volume
Cite as
1st Symposium on Simplicity in Algorithms (SOSA 2018). Open Access Series in Informatics (OASIcs), Volume 61, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@Proceedings{seidel:OASIcs.SOSA.2018,
title = {{OASIcs, Volume 61, SOSA'18, Complete Volume}},
booktitle = {1st Symposium on Simplicity in Algorithms (SOSA 2018)},
series = {Open Access Series in Informatics (OASIcs)},
ISBN = {978-3-95977-064-4},
ISSN = {2190-6807},
year = {2018},
volume = {61},
editor = {Seidel, Raimund},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.SOSA.2018},
URN = {urn:nbn:de:0030-drops-84405},
doi = {10.4230/OASIcs.SOSA.2018},
annote = {Keywords: Algorithm design and analysis}
}
Document
Front Matter
Front Matter, Table of Contents, Preface, Conference Organization
Abstract
Front Matter, Table of Contents, Preface, Conference Organization
Cite as
1st Symposium on Simplicity in Algorithms (SOSA 2018). Open Access Series in Informatics (OASIcs), Volume 61, pp. 0:i-0:xii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{seidel:OASIcs.SOSA.2018.0,
author = {Seidel, Raimund},
title = {{Front Matter, Table of Contents, Preface, Conference Organization}},
booktitle = {1st Symposium on Simplicity in Algorithms (SOSA 2018)},
pages = {0:i--0:xii},
series = {Open Access Series in Informatics (OASIcs)},
ISBN = {978-3-95977-064-4},
ISSN = {2190-6807},
year = {2018},
volume = {61},
editor = {Seidel, Raimund},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.SOSA.2018.0},
URN = {urn:nbn:de:0030-drops-82942},
doi = {10.4230/OASIcs.SOSA.2018.0},
annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
A Naive Algorithm for Feedback Vertex Set
Abstract
Given a graph on n vertices and an integer k, the feedback vertex set problem asks for the deletion of at most k vertices to make the graph acyclic. We show that a greedy branching algorithm, which always branches on an undecided vertex with the largest degree, runs in single-exponential time, i.e., O(c^k n^2) for some constant c.
Cite as
Yixin Cao. A Naive Algorithm for Feedback Vertex Set. In 1st Symposium on Simplicity in Algorithms (SOSA 2018). Open Access Series in Informatics (OASIcs), Volume 61, pp. 1:1-1:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{cao:OASIcs.SOSA.2018.1,
author = {Cao, Yixin},
title = {{A Naive Algorithm for Feedback Vertex Set}},
booktitle = {1st Symposium on Simplicity in Algorithms (SOSA 2018)},
pages = {1:1--1:9},
series = {Open Access Series in Informatics (OASIcs)},
ISBN = {978-3-95977-064-4},
ISSN = {2190-6807},
year = {2018},
volume = {61},
editor = {Seidel, Raimund},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.SOSA.2018.1},
URN = {urn:nbn:de:0030-drops-82961},
doi = {10.4230/OASIcs.SOSA.2018.1},
annote = {Keywords: greedy algorithm, analysis of algorithms, branching algorithm, parameterized computation -- graph modification problem}
}
Document
A Note on Iterated Rounding for the Survivable Network Design Problem
Abstract
In this note we consider the survivable network design problem (SNDP) in undirected graphs. We make two contributions. The first is a new counting argument in the iterated rounding based 2-approximation for edge-connectivity SNDP (EC-SNDP) originally due to Jain. The second contribution is to make some connections between hypergraphic version of SNDP (Hypergraph-SNDP) introduced by Zhao, Nagamochi and Ibaraki, and edge and node-weighted versions of EC-SNDP and element-connectivity SNDP (Elem-SNDP). One useful consequence is a 2-approximation for Elem-SNDP that avoids the use of set-pair based relaxation and analysis.
Cite as
Chandra Chekuri and Thapanapong Rukkanchanunt. A Note on Iterated Rounding for the Survivable Network Design Problem. In 1st Symposium on Simplicity in Algorithms (SOSA 2018). Open Access Series in Informatics (OASIcs), Volume 61, pp. 2:1-2:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{chekuri_et_al:OASIcs.SOSA.2018.2,
author = {Chekuri, Chandra and Rukkanchanunt, Thapanapong},
title = {{A Note on Iterated Rounding for the Survivable Network Design Problem}},
booktitle = {1st Symposium on Simplicity in Algorithms (SOSA 2018)},
pages = {2:1--2:10},
series = {Open Access Series in Informatics (OASIcs)},
ISBN = {978-3-95977-064-4},
ISSN = {2190-6807},
year = {2018},
volume = {61},
editor = {Seidel, Raimund},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.SOSA.2018.2},
URN = {urn:nbn:de:0030-drops-82976},
doi = {10.4230/OASIcs.SOSA.2018.2},
annote = {Keywords: survivable network design, iterated rounding, approximation, element connectivity}
}
Document
Congestion Minimization for Multipath Routing via Multiroute Flows
Abstract
Congestion minimization is a well-known routing problem for which
there is an O(log n/loglog n)-approximation via randomized rounding due to Raghavan and Thompson. Srinivasan formally introduced the low-congestion multi-path routing problem as a generalization of the (single-path) congestion minimization problem. The goal is to route multiple disjoint paths for each pair, for the sake of fault tolerance. Srinivasan developed a dependent randomized scheme for a special case of the multi-path problem when the input consists of a given set of disjoint paths for each pair and the goal is to select a given subset of them. Subsequently Doerr gave a different dependentrounding scheme and derandomization. Doerr et al. considered the problem where the paths have to be chosen, and applied the dependent rounding technique and evaluated it experimentally. However, their algorithm does not maintain the required disjointness property without which the problem easily reduces to the standard congestion minimization problem.
In this note we show a simple algorithm that solves the problem
correctly without the need for dependent rounding --- standard
independent rounding suffices. This is made possible via the notion
of multiroute flows originally suggested by Kishimoto et al. One advantage of the simpler rounding is an improved bound on the congestion when the path lengths are short.
Cite as
Chandra Chekuri and Mark Idleman. Congestion Minimization for Multipath Routing via Multiroute Flows. In 1st Symposium on Simplicity in Algorithms (SOSA 2018). Open Access Series in Informatics (OASIcs), Volume 61, pp. 3:1-3:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{chekuri_et_al:OASIcs.SOSA.2018.3,
author = {Chekuri, Chandra and Idleman, Mark},
title = {{Congestion Minimization for Multipath Routing via Multiroute Flows}},
booktitle = {1st Symposium on Simplicity in Algorithms (SOSA 2018)},
pages = {3:1--3:12},
series = {Open Access Series in Informatics (OASIcs)},
ISBN = {978-3-95977-064-4},
ISSN = {2190-6807},
year = {2018},
volume = {61},
editor = {Seidel, Raimund},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.SOSA.2018.3},
URN = {urn:nbn:de:0030-drops-82984},
doi = {10.4230/OASIcs.SOSA.2018.3},
annote = {Keywords: multipath routing, congestion minimization, multiroute flows}
}
Document
Better and Simpler Error Analysis of the Sinkhorn-Knopp Algorithm for Matrix Scaling
Abstract
Given a non-negative real matrix A, the matrix scaling problem is to determine if it is possible to scale the rows and columns so that each row and each column sums to a specified target value for it.
The matrix scaling problem arises in many algorithmic applications, perhaps most notably as a preconditioning step in solving linear system of equations. One of the most natural and by now classical approach to matrix scaling is the Sinkhorn-Knopp algorithm (also known as the RAS method) where one alternately scales either all rows or all columns to meet the target values. In addition to being extremely simple and natural, another appeal of this procedure is that it easily lends itself to parallelization. A central question is to understand the rate of convergence of the Sinkhorn-Knopp algorithm.
Specifically, given a suitable error metric to measure deviations from target values, and an error bound epsilon, how quickly does the Sinkhorn-Knopp algorithm converge to an error below epsilon? While there are several non-trivial convergence results known about the Sinkhorn-Knopp algorithm, perhaps somewhat surprisingly, even for natural error metrics such as ell_1-error or ell_2-error, this is not entirely understood.
In this paper, we present an elementary convergence analysis for the Sinkhorn-Knopp algorithm that improves upon the previous best bound. In a nutshell, our approach is to show (i) a simple bound on the number of iterations needed so that the KL-divergence between the current row-sums and the target row-sums drops below a specified threshold delta, and (ii) then show that for a suitable choice of delta, whenever KL-divergence is below delta, then the ell_1-error or the ell_2-error is below epsilon. The well-known Pinsker's inequality immediately allows us to translate a bound on the KL divergence to a bound on ell_1-error. To bound the ell_2-error in terms of the KL-divergence, we establish a new inequality, referred to as (KL vs ell_1/ell_2) inequality in the paper. This new inequality is a strengthening of the Pinsker's inequality that we believe is of independent interest. Our analysis of ell_2-error significantly improves upon the best previous convergence bound for ell_2-error.
The idea of studying Sinkhorn-Knopp convergence via KL-divergence is not new and has indeed been previously explored. Our contribution is an elementary, self-contained presentation of this approach and an interesting new inequality that yields a significantly stronger convergence guarantee for the extensively studied ell_2-error.
Cite as
Deeparnab Chakrabarty and Sanjeev Khanna. Better and Simpler Error Analysis of the Sinkhorn-Knopp Algorithm for Matrix Scaling. In 1st Symposium on Simplicity in Algorithms (SOSA 2018). Open Access Series in Informatics (OASIcs), Volume 61, pp. 4:1-4:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{chakrabarty_et_al:OASIcs.SOSA.2018.4,
author = {Chakrabarty, Deeparnab and Khanna, Sanjeev},
title = {{Better and Simpler Error Analysis of the Sinkhorn-Knopp Algorithm for Matrix Scaling}},
booktitle = {1st Symposium on Simplicity in Algorithms (SOSA 2018)},
pages = {4:1--4:11},
series = {Open Access Series in Informatics (OASIcs)},
ISBN = {978-3-95977-064-4},
ISSN = {2190-6807},
year = {2018},
volume = {61},
editor = {Seidel, Raimund},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.SOSA.2018.4},
URN = {urn:nbn:de:0030-drops-83045},
doi = {10.4230/OASIcs.SOSA.2018.4},
annote = {Keywords: Matrix Scaling, Entropy Minimization, KL Divergence Inequalities}
}
Document
Approximation Schemes for 0-1 Knapsack
Abstract
We revisit the standard 0-1 knapsack problem. The latest polynomial-time approximation scheme by Rhee (2015) with approximation factor 1+eps has running time near O(n+(1/eps)^{5/2}) (ignoring polylogarithmic factors), and is randomized. We present a simpler algorithm which achieves the same result and is deterministic.
With more effort, our ideas can actually lead to an improved time bound near O(n + (1/eps)^{12/5}), and still further improvements for small n.
Cite as
Timothy M. Chan. Approximation Schemes for 0-1 Knapsack. In 1st Symposium on Simplicity in Algorithms (SOSA 2018). Open Access Series in Informatics (OASIcs), Volume 61, pp. 5:1-5:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{chan:OASIcs.SOSA.2018.5,
author = {Chan, Timothy M.},
title = {{Approximation Schemes for 0-1 Knapsack}},
booktitle = {1st Symposium on Simplicity in Algorithms (SOSA 2018)},
pages = {5:1--5:12},
series = {Open Access Series in Informatics (OASIcs)},
ISBN = {978-3-95977-064-4},
ISSN = {2190-6807},
year = {2018},
volume = {61},
editor = {Seidel, Raimund},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.SOSA.2018.5},
URN = {urn:nbn:de:0030-drops-82994},
doi = {10.4230/OASIcs.SOSA.2018.5},
annote = {Keywords: knapsack problem, approximation algorithms, optimization, (min,+)-convolution}
}
Document
Counting Solutions to Polynomial Systems via Reductions
Abstract
This paper provides both positive and negative results for counting solutions to systems of polynomial equations over a finite field. The general idea is to try to reduce the problem to counting solutions to a single polynomial, where the task is easier. In both cases, simple methods are utilized that we expect will have wider applicability (far beyond algebra).
First, we give an efficient deterministic reduction from approximate counting for a system of (arbitrary) polynomial equations to approximate counting for one equation, over any finite field. We apply this reduction to give a deterministic poly(n,s,log p)/eps^2 time algorithm for approximately counting the fraction of solutions to a system of s quadratic n-variate polynomials over F_p (the finite field of prime order p) to within an additive eps factor, for any prime p. Note that uniform random sampling would already require Omega(s/eps^2) time, so our algorithm behaves as a full derandomization of uniform sampling. The approximate-counting algorithm yields efficient approximate counting for other well-known problems, such as 2-SAT, NAE-3SAT, and 3-Coloring. As a corollary, there is a deterministic algorithm (with analogous running time) for producing solutions to such systems which have at least eps p^n solutions.
Second, we consider the difficulty of exactly counting solutions to a single polynomial of constant degree, over a finite field. (Note that finding a solution in this case is easy.) It has been known for over 20 years that this counting problem is already NP-hard for degree-three polynomials over F_2; however, all known reductions increased the number of variables by a considerable amount. We give a subexponential-time reduction from counting solutions to k-CNF formulas to counting solutions to a degree-k^{O(k)} polynomial (over any finite field of O(1) order) which exactly preserves the number of variables. As a corollary, the Strong Exponential Time Hypothesis (even its weak counting variant #SETH) implies that counting solutions to constant-degree polynomials (even over F_2) requires essentially 2^n time. Similar results hold for counting orthogonal pairs of vectors over F_p.
Cite as
R. Ryan Williams. Counting Solutions to Polynomial Systems via Reductions. In 1st Symposium on Simplicity in Algorithms (SOSA 2018). Open Access Series in Informatics (OASIcs), Volume 61, pp. 6:1-6:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{williams:OASIcs.SOSA.2018.6,
author = {Williams, R. Ryan},
title = {{Counting Solutions to Polynomial Systems via Reductions}},
booktitle = {1st Symposium on Simplicity in Algorithms (SOSA 2018)},
pages = {6:1--6:15},
series = {Open Access Series in Informatics (OASIcs)},
ISBN = {978-3-95977-064-4},
ISSN = {2190-6807},
year = {2018},
volume = {61},
editor = {Seidel, Raimund},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.SOSA.2018.6},
URN = {urn:nbn:de:0030-drops-83078},
doi = {10.4230/OASIcs.SOSA.2018.6},
annote = {Keywords: counting complexity, polynomial equations, finite field, derandomization, strong exponential time hypothesis}
}
Document
On Sampling Edges Almost Uniformly
Abstract
We consider the problem of sampling an edge almost uniformly from an unknown graph, G = (V, E). Access to the graph is provided via queries of the following types: (1) uniform vertex queries, (2) degree queries, and (3) neighbor queries. We describe a new simple algorithm that returns a random edge e in E using \tilde{O}(n/sqrt{eps m}) queries in expectation, such that each edge e is sampled with probability (1 +/- eps)/m. Here, n = |V| is the number of vertices, and m = |E| is the number of edges. Our algorithm is optimal in the sense that any algorithm that samples an edge from an almost-uniform distribution must perform Omega(n/sqrt{m}) queries.
Cite as
Talya Eden and Will Rosenbaum. On Sampling Edges Almost Uniformly. In 1st Symposium on Simplicity in Algorithms (SOSA 2018). Open Access Series in Informatics (OASIcs), Volume 61, pp. 7:1-7:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{eden_et_al:OASIcs.SOSA.2018.7,
author = {Eden, Talya and Rosenbaum, Will},
title = {{On Sampling Edges Almost Uniformly}},
booktitle = {1st Symposium on Simplicity in Algorithms (SOSA 2018)},
pages = {7:1--7:9},
series = {Open Access Series in Informatics (OASIcs)},
ISBN = {978-3-95977-064-4},
ISSN = {2190-6807},
year = {2018},
volume = {61},
editor = {Seidel, Raimund},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.SOSA.2018.7},
URN = {urn:nbn:de:0030-drops-83001},
doi = {10.4230/OASIcs.SOSA.2018.7},
annote = {Keywords: Sublinear Algorithms, Graph Algorithms, Sampling Edges, Query Complexity}
}
Document
A Simple PTAS for the Dual Bin Packing Problem and Advice Complexity of Its Online Version
Abstract
Recently, Renault (2016) studied the dual bin packing problem in the per-request advice model of online algorithms. He showed that given O(1/eps) advice bits for each input item allows approximating the dual bin packing problem online to within a factor of 1+\eps. Renault asked about the advice complexity of dual bin packing in the tape-advice model of online algorithms. We make progress on this question. Let s be the maximum bit size of an input item weight. We present a conceptually simple online algorithm that with total advice O((s + log n)/eps^2) approximates the dual bin packing to within a 1+eps factor. To this end, we describe and analyze a simple offline PTAS for the dual bin packing problem. Although a PTAS for a more general problem was known prior to our work (Kellerer 1999, Chekuri and Khanna 2006), our PTAS is arguably simpler to state and analyze. As a result, we could easily adapt our PTAS to obtain the advice-complexity result.
We also consider whether the dependence on s is necessary in our algorithm. We show that if s is unrestricted then for small enough eps > 0 obtaining a 1+eps approximation to the dual bin packing requires Omega_eps(n) bits of advice. To establish this lower bound we analyze an online reduction that preserves the advice complexity and approximation ratio from the binary separation problem due to Boyar et al. (2016). We define two natural advice complexity classes that capture the distinction similar to the Turing machine world distinction between pseudo polynomial time algorithms and polynomial time algorithms. Our results on the dual bin packing problem imply the separation of the two classes in the advice complexity world.
Cite as
Allan Borodin, Denis Pankratov, and Amirali Salehi-Abari. A Simple PTAS for the Dual Bin Packing Problem and Advice Complexity of Its Online Version. In 1st Symposium on Simplicity in Algorithms (SOSA 2018). Open Access Series in Informatics (OASIcs), Volume 61, pp. 8:1-8:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{borodin_et_al:OASIcs.SOSA.2018.8,
author = {Borodin, Allan and Pankratov, Denis and Salehi-Abari, Amirali},
title = {{A Simple PTAS for the Dual Bin Packing Problem and Advice Complexity of Its Online Version}},
booktitle = {1st Symposium on Simplicity in Algorithms (SOSA 2018)},
pages = {8:1--8:12},
series = {Open Access Series in Informatics (OASIcs)},
ISBN = {978-3-95977-064-4},
ISSN = {2190-6807},
year = {2018},
volume = {61},
editor = {Seidel, Raimund},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.SOSA.2018.8},
URN = {urn:nbn:de:0030-drops-83033},
doi = {10.4230/OASIcs.SOSA.2018.8},
annote = {Keywords: dual bin packing, PTAS, tape-advice complexity}
}
Document
Simple and Efficient Leader Election
Abstract
We provide a simple and efficient population protocol for leader election that uses O(log n) states and elects exactly one leader in O(n (log n)^2) interactions with high probability and in expectation. Our analysis is simple and based on fundamental stochastic arguments. Our protocol combines the tournament based leader elimination by Alistarh and Gelashvili, ICALP'15, with the synthetic coin introduced by Alistarh et al., SODA'17.
Cite as
Petra Berenbrink, Dominik Kaaser, Peter Kling, and Lena Otterbach. Simple and Efficient Leader Election. In 1st Symposium on Simplicity in Algorithms (SOSA 2018). Open Access Series in Informatics (OASIcs), Volume 61, pp. 9:1-9:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{berenbrink_et_al:OASIcs.SOSA.2018.9,
author = {Berenbrink, Petra and Kaaser, Dominik and Kling, Peter and Otterbach, Lena},
title = {{Simple and Efficient Leader Election}},
booktitle = {1st Symposium on Simplicity in Algorithms (SOSA 2018)},
pages = {9:1--9:11},
series = {Open Access Series in Informatics (OASIcs)},
ISBN = {978-3-95977-064-4},
ISSN = {2190-6807},
year = {2018},
volume = {61},
editor = {Seidel, Raimund},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.SOSA.2018.9},
URN = {urn:nbn:de:0030-drops-83029},
doi = {10.4230/OASIcs.SOSA.2018.9},
annote = {Keywords: population protocols, leader election, distributed, randomized}
}
Document
A Simple Algorithm for Approximating the Text-To-Pattern Hamming Distance
Abstract
The algorithmic task of computing the Hamming distance between a given pattern of length m and each location in a text of length n, both over a general alphabet \Sigma, is one of the most fundamental algorithmic tasks in string algorithms. The fastest known runtime for exact computation is \tilde O(n\sqrt m). We recently introduced a complicated randomized algorithm for obtaining a (1 +/- eps) approximation for each location in the text in O( (n/eps) log(1/eps) log n log m log |\Sigma|) total time, breaking a barrier that stood for 22 years. In this paper, we introduce an elementary and simple randomized algorithm that takes O((n/eps) log n log m) time.
Cite as
Tsvi Kopelowitz and Ely Porat. A Simple Algorithm for Approximating the Text-To-Pattern Hamming Distance. In 1st Symposium on Simplicity in Algorithms (SOSA 2018). Open Access Series in Informatics (OASIcs), Volume 61, pp. 10:1-10:5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{kopelowitz_et_al:OASIcs.SOSA.2018.10,
author = {Kopelowitz, Tsvi and Porat, Ely},
title = {{A Simple Algorithm for Approximating the Text-To-Pattern Hamming Distance}},
booktitle = {1st Symposium on Simplicity in Algorithms (SOSA 2018)},
pages = {10:1--10:5},
series = {Open Access Series in Informatics (OASIcs)},
ISBN = {978-3-95977-064-4},
ISSN = {2190-6807},
year = {2018},
volume = {61},
editor = {Seidel, Raimund},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.SOSA.2018.10},
URN = {urn:nbn:de:0030-drops-83089},
doi = {10.4230/OASIcs.SOSA.2018.10},
annote = {Keywords: Pattern Matching, Hamming Distance, Approximation Algorithms}
}
Document
Compact LP Relaxations for Allocation Problems
Abstract
We consider the restricted versions of Scheduling on Unrelated Machines and the Santa Claus problem. In these problems we are given a set of jobs and a set of machines. Every job j has a size p_j and a set of allowed machines \Gamma(j), i.e., it can only be assigned to those machines. In the first problem, the objective is to minimize the maximum load among all machines; in the latter problem it is to maximize the minimum load. For these problems, the strongest LP relaxation known is the configuration LP. The configuration LP has an exponential number of variables and it cannot be solved exactly unless P = NP.
Our main result is a new LP relaxation for these problems. This LP has only O(n^3) variables and constraints. It is a further relaxation of the configuration LP, but it obeys the best bounds known for its integrality gap (11/6 and 4).
For the configuration LP these bounds were obtained using two local search algorithm. These algorithms, however, differ significantly in presentation. In this paper, we give a meta algorithm based on the local search ideas. With an instantiation for each objective function, we prove the bounds for the new compact LP relaxation (in particular, for the configuration LP). This way, we bring out many analogies between the two proofs, which were not apparent before.
Cite as
Klaus Jansen and Lars Rohwedder. Compact LP Relaxations for Allocation Problems. In 1st Symposium on Simplicity in Algorithms (SOSA 2018). Open Access Series in Informatics (OASIcs), Volume 61, pp. 11:1-11:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{jansen_et_al:OASIcs.SOSA.2018.11,
author = {Jansen, Klaus and Rohwedder, Lars},
title = {{Compact LP Relaxations for Allocation Problems}},
booktitle = {1st Symposium on Simplicity in Algorithms (SOSA 2018)},
pages = {11:1--11:19},
series = {Open Access Series in Informatics (OASIcs)},
ISBN = {978-3-95977-064-4},
ISSN = {2190-6807},
year = {2018},
volume = {61},
editor = {Seidel, Raimund},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.SOSA.2018.11},
URN = {urn:nbn:de:0030-drops-83012},
doi = {10.4230/OASIcs.SOSA.2018.11},
annote = {Keywords: Linear programming, unrelated machines, makespan, max-min, restricted assignment, santa claus}
}
Document
Just Take the Average! An Embarrassingly Simple 2^n-Time Algorithm for SVP (and CVP)
Abstract
We show a 2^{n+o(n)}-time (and space) algorithm for the Shortest Vector Problem on lattices (SVP) that works by repeatedly running an embarrassingly simple "pair and average" sieving-like procedure on a list of lattice vectors. This matches the running time (and space) of the current fastest known algorithm, due to Aggarwal, Dadush, Regev, and Stephens-Davidowitz (ADRS, in STOC, 2015), with a far simpler algorithm. Our algorithm is in fact a modification of the ADRS algorithm, with a certain careful rejection sampling step removed.
The correctness of our algorithm follows from a more general "meta-theorem," showing that such rejection sampling steps are unnecessary for a certain class of algorithms and use cases. In particular, this also applies to the related 2^{n + o(n)}-time algorithm for the Closest Vector Problem (CVP), due to Aggarwal, Dadush, and Stephens-Davidowitz (ADS, in FOCS, 2015), yielding a similar embarrassingly simple algorithm for gamma-approximate CVP for any gamma = 1+2^{-o(n/log n)}. (We can also remove the rejection sampling procedure from the 2^{n+o(n)}-time ADS algorithm for exact CVP, but the resulting algorithm is still quite complicated.)
Cite as
Divesh Aggarwal and Noah Stephens-Davidowitz. Just Take the Average! An Embarrassingly Simple 2^n-Time Algorithm for SVP (and CVP). In 1st Symposium on Simplicity in Algorithms (SOSA 2018). Open Access Series in Informatics (OASIcs), Volume 61, pp. 12:1-12:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{aggarwal_et_al:OASIcs.SOSA.2018.12,
author = {Aggarwal, Divesh and Stephens-Davidowitz, Noah},
title = {{Just Take the Average! An Embarrassingly Simple 2^n-Time Algorithm for SVP (and CVP)}},
booktitle = {1st Symposium on Simplicity in Algorithms (SOSA 2018)},
pages = {12:1--12:19},
series = {Open Access Series in Informatics (OASIcs)},
ISBN = {978-3-95977-064-4},
ISSN = {2190-6807},
year = {2018},
volume = {61},
editor = {Seidel, Raimund},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.SOSA.2018.12},
URN = {urn:nbn:de:0030-drops-83062},
doi = {10.4230/OASIcs.SOSA.2018.12},
annote = {Keywords: Lattices, SVP, CVP}
}
Document
Complex Semidefinite Programming and Max-k-Cut
Abstract
In a second seminal paper on the application of semidefinite
programming to graph partitioning problems, Goemans and Williamson
showed in 2004 how to formulate and round a complex semidefinite program to give what is to date still the best-known approximation guarantee of .836008 for Max-3-Cut. (This approximation ratio was also achieved independently around the same time by De Klerk et
al..) Goemans and Williamson left open the problem of how to apply their techniques to Max-k-Cut for general k. They point out that it does not seem straightforward or even possible to formulate a good quality complex semidefinite program for the general Max-k-Cut problem, which presents a barrier for the further application of their techniques.
We present a simple rounding algorithm for the standard semidefinite
programmming relaxation of Max-k-Cut and show that it is equivalent to the rounding of Goemans and Williamson in the case of Max-3-Cut. This allows us to transfer the elegant analysis of Goemans and Williamson for Max-3-Cut to Max-k-Cut. For k > 3, the resulting approximation ratios are about .01 worse than the best known guarantees. Finally, we present a generalization of our rounding algorithm and conjecture (based on computational observations) that it matches the best-known guarantees of De Klerk et al.
Cite as
Alantha Newman. Complex Semidefinite Programming and Max-k-Cut. In 1st Symposium on Simplicity in Algorithms (SOSA 2018). Open Access Series in Informatics (OASIcs), Volume 61, pp. 13:1-13:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{newman:OASIcs.SOSA.2018.13,
author = {Newman, Alantha},
title = {{Complex Semidefinite Programming and Max-k-Cut}},
booktitle = {1st Symposium on Simplicity in Algorithms (SOSA 2018)},
pages = {13:1--13:11},
series = {Open Access Series in Informatics (OASIcs)},
ISBN = {978-3-95977-064-4},
ISSN = {2190-6807},
year = {2018},
volume = {61},
editor = {Seidel, Raimund},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.SOSA.2018.13},
URN = {urn:nbn:de:0030-drops-83098},
doi = {10.4230/OASIcs.SOSA.2018.13},
annote = {Keywords: Graph Partitioning, Max-k-Cut, Semidefinite Programming}
}
Document
A Simple, Space-Efficient, Streaming Algorithm for Matchings in Low Arboricity Graphs
Abstract
We present a simple single-pass data stream algorithm using O((log n)/eps^2) space that returns an (alpha + 2)(1 + eps) approximation to the size of the maximum matching in a graph of arboricity alpha.
Cite as
Andrew McGregor and Sofya Vorotnikova. A Simple, Space-Efficient, Streaming Algorithm for Matchings in Low Arboricity Graphs. In 1st Symposium on Simplicity in Algorithms (SOSA 2018). Open Access Series in Informatics (OASIcs), Volume 61, pp. 14:1-14:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{mcgregor_et_al:OASIcs.SOSA.2018.14,
author = {McGregor, Andrew and Vorotnikova, Sofya},
title = {{A Simple, Space-Efficient, Streaming Algorithm for Matchings in Low Arboricity Graphs}},
booktitle = {1st Symposium on Simplicity in Algorithms (SOSA 2018)},
pages = {14:1--14:4},
series = {Open Access Series in Informatics (OASIcs)},
ISBN = {978-3-95977-064-4},
ISSN = {2190-6807},
year = {2018},
volume = {61},
editor = {Seidel, Raimund},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.SOSA.2018.14},
URN = {urn:nbn:de:0030-drops-82959},
doi = {10.4230/OASIcs.SOSA.2018.14},
annote = {Keywords: data streams, matching, planar graphs, arboricity}
}
Document
Simple Analyses of the Sparse Johnson-Lindenstrauss Transform
Abstract
For every n-point subset X of Euclidean space and target distortion 1+eps for 0<eps<1, the Sparse Johnson Lindenstrauss Transform (SJLT) of (Kane, Nelson, J. ACM 2014) provides a linear dimensionality-reducing map f:X-->l_2^m where f(x) = Ax for A a matrix with m rows where (1) m = O((log n)/eps^2), and (2) each column of A is sparse, having only O(eps m) non-zero entries. Though the constructions given for such A in (Kane, Nelson, J. ACM 2014) are simple, the analyses are not, employing intricate combinatorial arguments. We here give two simple alternative proofs of their main result, involving no delicate combinatorics. One of these proofs has already been tested pedagogically, requiring slightly under forty minutes by the third author at a casual pace to cover all details in a blackboard course lecture.
Cite as
Michael B. Cohen, T.S. Jayram, and Jelani Nelson. Simple Analyses of the Sparse Johnson-Lindenstrauss Transform. In 1st Symposium on Simplicity in Algorithms (SOSA 2018). Open Access Series in Informatics (OASIcs), Volume 61, pp. 15:1-15:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{cohen_et_al:OASIcs.SOSA.2018.15,
author = {Cohen, Michael B. and Jayram, T.S. and Nelson, Jelani},
title = {{Simple Analyses of the Sparse Johnson-Lindenstrauss Transform}},
booktitle = {1st Symposium on Simplicity in Algorithms (SOSA 2018)},
pages = {15:1--15:9},
series = {Open Access Series in Informatics (OASIcs)},
ISBN = {978-3-95977-064-4},
ISSN = {2190-6807},
year = {2018},
volume = {61},
editor = {Seidel, Raimund},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.SOSA.2018.15},
URN = {urn:nbn:de:0030-drops-83056},
doi = {10.4230/OASIcs.SOSA.2018.15},
annote = {Keywords: dimensionality reduction, Johnson-Lindenstrauss, Sparse Johnson-Lindenstrauss Transform}
}