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A Simple Gap-producing Reduction for the Parameterized Set Cover Problem [article]

Bingkai Lin
2019 arXiv   pre-print
This paper presents a simple gap-producing reduction which, given a set cover instance I=(S,U,E) and two integers k<h< (1-o(1))√( |S|/ |S|), outputs a new set cover instance I'=(S,U',E') with |U'|=|U|^  ...  approximate set cover to a factor below ( n)^1/poly(k,e(ϵ)) for some function e.  ...  The author wishes to thank the anonymous referees for their detailed comments.  ... 
arXiv:1902.03702v2 fatcat:vyekexrvwjfojpqup3nzjfx3lq

A Simple Gap-Producing Reduction for the Parameterized Set Cover Problem

Bingkai Lin, Michael Wagner
2019 International Colloquium on Automata, Languages and Programming  
This paper presents a simple gap-producing reduction which, given a set cover instance I = (S, U, E) and two integers k < h ≤ (1 − o( 1 )) k log |S|/ log log |S|, outputs a new set cover instance 1) time  ...  Given an n-vertex bipartite graph I = (S, U, E), the goal of set cover problem is to find a minimum sized subset of S such that every vertex in U is adjacent to some vertex of this subset.  ...  910 1) -time such that I C A L P 2 0 1 Simple Gap-Producing Reduction for the Parameterized Set Cover Problem an independent set for all i ∈ [k], outputs a set cover instance I = (S, U, E) with |U |  ... 
doi:10.4230/lipics.icalp.2019.81 dblp:conf/icalp/Lin19 fatcat:mymdpzeugbanbikrj5vsr7hboi

Parameterized Inapproximability of Exact Cover and Nearest Codeword [article]

Venkatesan Guruswami, Patrick Lin
2019 arXiv   pre-print
The $k$-ExactCover problem is a parameterized version of the ExactCover problem, in which we are given a universe $U$, a collection $S$ of subsets of $U$, and an integer $k$, and the task is to determine  ...  cover of size $k$ and the case where every set cover of $I$ has size at least $\frac14 \sqrt[k]{\frac{\log N}{\log \log N}}$.  ...  We thank Pasin Manurangsi for some useful discussions on ExactCover and the recent literature in parameterized inapproximability.  ... 
arXiv:1905.06503v2 fatcat:ewnbq3n4dzfjjauovlj7yv52o4

Lossy kernelization

Daniel Lokshtanov, Fahad Panolan, M. S. Ramanujan, Saket Saurabh
2017 Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing - STOC 2017  
In this paper we propose a new framework for analyzing the performance of pre-processing algorithms. Our framework builds on the notion of kernelization from parameterized complexity.  ...  On the negative side we prove that L P parameterized by the length of the path and S C parameterized by the universe size do not admit even an α-approximate kernel of polynomial size, for any α ≥ 1, unless  ...  ACKNOWLEDGEMENTS e authors thank Dániel Marx and for enlightening discussions on related work in the literature, and Magnus Wahlström for pointing out the remark about randomized pre-processing algorithms  ... 
doi:10.1145/3055399.3055456 dblp:conf/stoc/LokshtanovPRS17 fatcat:tfxcsrgswjdvjkiqo6mmdhnypm

Tight Running Time Lower Bounds for Strong Inapproximability of Maximum k-Coverage, Unique Set Cover and Related Problems (via t-Wise Agreement Testing Theorem) [article]

Pasin Manurangsi
2019 arXiv   pre-print
) for any constant ϵ > 0. - Any constant factor approximation for k-Unique Set Cover, k-Nearest Codeword Problem and k-Closest Vector Problem. - (1 + δ)-approximation for k-Minimum Distance Problem and  ...  We show, assuming the (randomized) Gap Exponential Time Hypothesis (Gap-ETH), that the following tasks cannot be done in T(k) · N^o(k)-time for any function T where N denote the input size: - (1 - 1/e  ...  The problem is particularly interesting because the reduction from Label Cover cannot give totally FPT inapproximability for the k-Unique Set Cover problem.  ... 
arXiv:1910.11850v1 fatcat:fv7s5nhedncvhh6gfl3jxk23dm

Lossy Kernelization [article]

Daniel Lokshtanov, Fahad Panolan, M. S. Ramanujan, Saket Saurabh
2016 arXiv   pre-print
On the negative side we prove that Longest Path parameterized by the length of the path and Set Cover parameterized by the universe size do not admit even an α-approximate kernel of polynomial size, for  ...  In this paper we propose a new framework for analyzing the performance of preprocessing algorithms. Our framework builds on the notion of kernelization from parameterized complexity.  ...  The authors thank Dániel Marx for enlightening discussions on related work in the literature, and Magnus Wahlström for pointing out the remark about randomized pre-processing algorithms following Definition  ... 
arXiv:1604.04111v2 fatcat:y22ajuxiufgcxfjdqcszmvvsou

A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms [article]

Andreas Emil Feldmann, Karthik C. S., Euiwoong Lee, Pasin Manurangsi
2020 arXiv   pre-print
Parameterization and approximation are two popular ways of coping with NP-hard problems. More recently, the two have also been combined to derive many interesting results.  ...  We survey developments in the area both from the algorithmic and hardness perspectives, with emphasis on new techniques and potential future research directions.  ...  Conflicts of Interest: The authors declare no conflict of interest.  ... 
arXiv:2006.04411v1 fatcat:hjgu7f3s7zbydkcnioq3qlzgza

Parameterized Intractability of Even Set and Shortest Vector Problem from Gap-ETH

Arnab Bhattacharyya, Suprovat Ghoshal, Karthik C. S., Pasin Manurangsi, Michael Wagner
2018 International Colloquium on Automata, Languages and Programming  
Similar to k-Even Set, this problem is also a long-standing open problem in the field of Parameterized Complexity.  ...  The k-Even Set problem is a parameterized variant of the Minimum Distance Problem of linear codes over F 2 , which can be stated as follows: given a generator matrix A and an integer k, determine whether  ...  The parameterized intractability of GapMDP 1.99 is obtained by setting γ = 400 in the above reduction. Gap Amplification for GapMDP 1.99 .  ... 
doi:10.4230/lipics.icalp.2018.17 dblp:conf/icalp/BhattacharyyaGS18 fatcat:hrycg3ktk5code4aclejrfkozi

A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

Andreas Emil Feldmann, Karthik C. S., Euiwoong Lee, Pasin Manurangsi
2020 Algorithms  
Parameterization and approximation are two popular ways of coping with NP-hard problems. More recently, the two have also been combined to derive many interesting results.  ...  We survey developments in the area both from the algorithmic and hardness perspectives, with emphasis on new techniques and potential future research directions.  ...  Conflicts of Interest: The authors declare no conflict of interest.  ... 
doi:10.3390/a13060146 fatcat:2u2vv3uksfguvj6473t2gsq42a

A Survey on Parameterized Inapproximability: k-Clique, k-SetCover, and More [article]

Xuandi Ren
2021 arXiv   pre-print
In the past a few years, many interesting inapproximability results have been obtained from the parameterized perspective.  ...  This article surveys some of such results, with a focus on k-Clique, k-SetCover, and other related problems.  ...  Bingkai Lin, who brought me into the beautiful world of hardness of approximation, discussed with me regularly and guided me patiently.  ... 
arXiv:2112.04669v2 fatcat:uytklhfuuzetpkrs576zav7jb4

Constant Approximating Parameterized k-SetCover is W[2]-hard [article]

Bingkai Lin, Xuandi Ren, Yican Sun, Xiuhan Wang
2022 arXiv   pre-print
Our technique could also be applied to rule out polynomial time o(log n/loglog n) ratio approximation algorithms for the non-parameterized k-SetCover problem, assuming W[1]FPT.  ...  Furthermore, our reduction results in a k-SetCover instance with k as small as O(log^2 n·loglog n).  ...  At a very high level, this method first constructs a graph with some threshold properties then combines this graph with the input instance to produce a gap instance of the desired problem.  ... 
arXiv:2202.04377v1 fatcat:xq23esja3nbozoplmrny4e4u6a

Parameterized Intractability of Even Set and Shortest Vector Problem from Gap-ETH [article]

Arnab Bhattacharyya, Suprovat Ghoshal, Karthik C. S., Pasin Manurangsi
2018 arXiv   pre-print
Similar to k-Even Set, this problem is also a long-standing open problem in the field of Parameterized Complexity.  ...  The k-Even Set problem is a parameterized variant of the Minimum Distance Problem of linear codes over F_2, which can be stated as follows: given a generator matrix A and an integer k, determine whether  ...  Pasin would like to thank Danupon Nanongkai for introducing him to the k-Even Set problem and for subsequent useful discussions.  ... 
arXiv:1803.09717v1 fatcat:ymowevvnkvg3dktbbzbpf3cezi

The Complexity Ecology of Parameters: An Illustration Using Bounded Max Leaf Number

Michael Fellows, Daniel Lokshtanov, Neeldhara Misra, Matthias Mnich, Frances Rosamond, Saket Saurabh
2009 Theory of Computing Systems  
In the framework of parameterized complexity, exploring how one parameter affects the complexity of a different parameterized (or unparameterized problem) is of general interest.  ...  The reason why such investigations are of general interest is that real-world input distributions for computational problems often inherit structure from the natural computational processes that produce  ...  Note that the situation between two consecutive boxes has a simple structure: there is just a set of edge-paths between the two boxes. We refer to the situation between two boxes as a gap.  ... 
doi:10.1007/s00224-009-9167-9 fatcat:ezqsz2ggzzgd5iicukjjenzoai

The Complexity Ecology of Parameters: An Illustration Using Bounded Max Leaf Number [chapter]

Michael Fellows, Frances Rosamond
2007 Lecture Notes in Computer Science  
In the framework of parameterized complexity, exploring how one parameter affects the complexity of a different parameterized (or unparameterized problem) is of general interest.  ...  The reason why such investigations are of general interest is that real-world input distributions for computational problems often inherit structure from the natural computational processes that produce  ...  Note that the situation between two consecutive boxes has a simple structure: there is just a set of edge-paths between the two boxes. We refer to the situation between two boxes as a gap.  ... 
doi:10.1007/978-3-540-73001-9_28 fatcat:6ctmbq7kgrbfpeer6mpc5pqaja

Parameterized Complexity and Approximation Algorithms

D. Marx
2007 Computer journal  
We discuss the different ways parameterized complexity can be extended to approximation algorithms, survey results of this type, and propose directions for future research.  ...  Approximation algorithms and parameterized complexity are usually considered to be two separate ways of dealing with hard algorithmic problems.  ...  ACKNOWLEDGMENTS I would like to thank Mike Fellows for asking me to write this survey, and for the many ideas, comments, and references that were essential in compiling the material.  ... 
doi:10.1093/comjnl/bxm048 fatcat:6yukdt7tvja3jf76zyuxfm536q
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