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Documents authored by Esmer, Barış Can

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DOI: 10.4230/LIPIcs.IPEC.2023.17
Approximate Monotone Local Search for Weighted Problems

Authors: Barış Can Esmer, Ariel Kulik, Dániel Marx, Daniel Neuen, and Roohani Sharma

Published in: LIPIcs, Volume 285, 18th International Symposium on Parameterized and Exact Computation (IPEC 2023)

Abstract
In a recent work, Esmer et al. describe a simple method - Approximate Monotone Local Search - to obtain exponential approximation algorithms from existing parameterized exact algorithms, polynomial-time approximation algorithms and, more generally, parameterized approximation algorithms. In this work, we generalize those results to the weighted setting. More formally, we consider monotone subset minimization problems over a weighted universe of size n (e.g., Vertex Cover, d-Hitting Set and Feedback Vertex Set). We consider a model where the algorithm is only given access to a subroutine that finds a solution of weight at most α ⋅ W (and of arbitrary cardinality) in time c^k ⋅ n^{𝒪(1)} where W is the minimum weight of a solution of cardinality at most k. In the unweighted setting, Esmer et al. determine the smallest value d for which a β-approximation algorithm running in time dⁿ ⋅ n^{𝒪(1)} can be obtained in this model. We show that the same dependencies also hold in a weighted setting in this model: for every fixed ε > 0 we obtain a β-approximation algorithm running in time 𝒪((d+ε)ⁿ), for the same d as in the unweighted setting. Similarly, we also extend a β-approximate brute-force search (in a model which only provides access to a membership oracle) to the weighted setting. Using existing approximation algorithms and exact parameterized algorithms for weighted problems, we obtain the first exponential-time β-approximation algorithms that are better than brute force for a variety of problems including Weighted Vertex Cover, Weighted d-Hitting Set, Weighted Feedback Vertex Set and Weighted Multicut.
Cite as

Barış Can Esmer, Ariel Kulik, Dániel Marx, Daniel Neuen, and Roohani Sharma. Approximate Monotone Local Search for Weighted Problems. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 17:1-17:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{esmer_et_al:LIPIcs.IPEC.2023.17,
  author =	{Esmer, Bar{\i}\c{s} Can and Kulik, Ariel and Marx, D\'{a}niel and Neuen, Daniel and Sharma, Roohani},
  title =	{{Approximate Monotone Local Search for Weighted Problems}},
  booktitle =	{18th International Symposium on Parameterized and Exact Computation (IPEC 2023)},
  pages =	{17:1--17:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-305-8},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{285},
  editor =	{Misra, Neeldhara and Wahlstr\"{o}m, Magnus},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2023.17},
  URN =		{urn:nbn:de:0030-drops-194360},
  doi =		{10.4230/LIPIcs.IPEC.2023.17},
  annote =	{Keywords: parameterized approximations, exponential approximations, monotone local search}
}
Document
DOI: 10.4230/LIPIcs.IPEC.2022.12
Computing Generalized Convolutions Faster Than Brute Force

Authors: Barış Can Esmer, Ariel Kulik, Dániel Marx, Philipp Schepper, and Karol Węgrzycki

Published in: LIPIcs, Volume 249, 17th International Symposium on Parameterized and Exact Computation (IPEC 2022)

Abstract
In this paper, we consider a general notion of convolution. Let D be a finite domain and let Dⁿ be the set of n-length vectors (tuples) of D. Let f : D × D → D be a function and let ⊕_f be a coordinate-wise application of f. The f-Convolution of two functions g,h : Dⁿ → {-M,…,M} is (g ⊛_f h)(v) := ∑_{v_g,v_h ∈ D^n s.t. v = v_g ⊕_f v_h} g(v_g) ⋅ h(v_h) for every 𝐯 ∈ Dⁿ. This problem generalizes many fundamental convolutions such as Subset Convolution, XOR Product, Covering Product or Packing Product, etc. For arbitrary function f and domain D we can compute f-Convolution via brute-force enumeration in 𝒪̃(|D|^{2n} ⋅ polylog(M)) time. Our main result is an improvement over this naive algorithm. We show that f-Convolution can be computed exactly in 𝒪̃((c ⋅ |D|²)ⁿ ⋅ polylog(M)) for constant c := 5/6 when D has even cardinality. Our main observation is that a cyclic partition of a function f : D × D → D can be used to speed up the computation of f-Convolution, and we show that an appropriate cyclic partition exists for every f. Furthermore, we demonstrate that a single entry of the f-Convolution can be computed more efficiently. In this variant, we are given two functions g,h : Dⁿ → {-M,…,M} alongside with a vector 𝐯 ∈ Dⁿ and the task of the f-Query problem is to compute integer (g ⊛_f h)(𝐯). This is a generalization of the well-known Orthogonal Vectors problem. We show that f-Query can be computed in 𝒪̃(|D|^{(ω/2)n} ⋅ polylog(M)) time, where ω ∈ [2,2.373) is the exponent of currently fastest matrix multiplication algorithm.
Cite as

Barış Can Esmer, Ariel Kulik, Dániel Marx, Philipp Schepper, and Karol Węgrzycki. Computing Generalized Convolutions Faster Than Brute Force. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 12:1-12:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{esmer_et_al:LIPIcs.IPEC.2022.12,
  author =	{Esmer, Bar{\i}\c{s} Can and Kulik, Ariel and Marx, D\'{a}niel and Schepper, Philipp and W\k{e}grzycki, Karol},
  title =	{{Computing Generalized Convolutions Faster Than Brute Force}},
  booktitle =	{17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
  pages =	{12:1--12:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-260-0},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{249},
  editor =	{Dell, Holger and Nederlof, Jesper},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.12},
  URN =		{urn:nbn:de:0030-drops-173685},
  doi =		{10.4230/LIPIcs.IPEC.2022.12},
  annote =	{Keywords: Generalized Convolution, Fast Fourier Transform, Fast Subset Convolution}
}
Document
DOI: 10.4230/LIPIcs.ESA.2022.50
Faster Exponential-Time Approximation Algorithms Using Approximate Monotone Local Search

Authors: Barış Can Esmer, Ariel Kulik, Dániel Marx, Daniel Neuen, and Roohani Sharma

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Abstract
We generalize the monotone local search approach of Fomin, Gaspers, Lokshtanov and Saurabh [J.ACM 2019], by establishing a connection between parameterized approximation and exponential-time approximation algorithms for monotone subset minimization problems. In a monotone subset minimization problem the input implicitly describes a non-empty set family over a universe of size n which is closed under taking supersets. The task is to find a minimum cardinality set in this family. Broadly speaking, we use approximate monotone local search to show that a parameterized α-approximation algorithm that runs in c^k⋅n^𝒪(1) time, where k is the solution size, can be used to derive an α-approximation randomized algorithm that runs in dⁿ⋅n^𝒪(1) time, where d is the unique value in (1, 1+{c-1}/α) such that 𝒟(1/α‖{d-1}/{c-1}) = {ln c}/α and 𝒟(a‖b) is the Kullback-Leibler divergence. This running time matches that of Fomin et al. for α = 1, and is strictly better when α > 1, for any c > 1. Furthermore, we also show that this result can be derandomized at the expense of a sub-exponential multiplicative factor in the running time. We use an approximate variant of the exhaustive search as a benchmark for our algorithm. We show that the classic 2ⁿ⋅n^𝒪(1) exhaustive search can be adapted to an α-approximate exhaustive search that runs in time (1+exp(-α⋅ℋ(1/(α))))ⁿ⋅n^𝒪(1), where ℋ is the entropy function. Furthermore, we provide a lower bound stating that the running time of this α-approximate exhaustive search is the best achievable running time in an oracle model. When compared to approximate exhaustive search, and to other techniques, the running times obtained by approximate monotone local search are strictly better for any α ≥ 1, c > 1. We demonstrate the potential of approximate monotone local search by deriving new and faster exponential approximation algorithms for Vertex Cover, 3-Hitting Set, Directed Feedback Vertex Set, Directed Subset Feedback Vertex Set, Directed Odd Cycle Transversal and Undirected Multicut. For instance, we get a 1.1-approximation algorithm for Vertex Cover with running time 1.114ⁿ⋅n^𝒪(1), improving upon the previously best known 1.1-approximation running in time 1.127ⁿ⋅n^𝒪(1) by Bourgeois et al. [DAM 2011].
Cite as

Barış Can Esmer, Ariel Kulik, Dániel Marx, Daniel Neuen, and Roohani Sharma. Faster Exponential-Time Approximation Algorithms Using Approximate Monotone Local Search. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 50:1-50:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{esmer_et_al:LIPIcs.ESA.2022.50,
  author =	{Esmer, Bar{\i}\c{s} Can and Kulik, Ariel and Marx, D\'{a}niel and Neuen, Daniel and Sharma, Roohani},
  title =	{{Faster Exponential-Time Approximation Algorithms Using Approximate Monotone Local Search}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{50:1--50:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.50},
  URN =		{urn:nbn:de:0030-drops-169887},
  doi =		{10.4230/LIPIcs.ESA.2022.50},
  annote =	{Keywords: parameterized approximations, exponential approximations, monotone local search}
}
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