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

2019
*
arXiv
*
pre-print

This paper presents

arXiv:1902.03702v2
fatcat:vyekexrvwjfojpqup3nzjfx3lq
*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. ...##
###
A Simple Gap-Producing Reduction for the Parameterized Set Cover Problem

2019
*
International Colloquium on Automata, Languages and Programming
*

This paper presents

doi:10.4230/lipics.icalp.2019.81
dblp:conf/icalp/Lin19
fatcat:mymdpzeugbanbikrj5vsr7hboi
*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 | ...##
###
Parameterized Inapproximability of Exact Cover and Nearest Codeword
[article]

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. ...

##
###
Lossy kernelization

2017
*
Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing - STOC 2017
*

In this paper we propose

doi:10.1145/3055399.3055456
dblp:conf/stoc/LokshtanovPRS17
fatcat:tfxcsrgswjdvjkiqo6mmdhnypm
*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 ...##
###
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]

2019
*
arXiv
*
pre-print

)

arXiv:1910.11850v1
fatcat:fv7s5nhedncvhh6gfl3jxk23dm
*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*. ...##
###
Lossy Kernelization
[article]

2016
*
arXiv
*
pre-print

On

arXiv:1604.04111v2
fatcat:y22ajuxiufgcxfjdqcszmvvsou
*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 ...##
###
A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms
[article]

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. ...

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

2018
*
International Colloquium on Automata, Languages and Programming
*

Similar to k-Even

doi:10.4230/lipics.icalp.2018.17
dblp:conf/icalp/BhattacharyyaGS18
fatcat:hrycg3ktk5code4aclejrfkozi
*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 . ...##
###
A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

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. ...

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

2021
*
arXiv
*
pre-print

In

arXiv:2112.04669v2
fatcat:uytklhfuuzetpkrs576zav7jb4
*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. ...##
###
Constant Approximating Parameterized k-SetCover is W[2]-hard
[article]

2022
*
arXiv
*
pre-print

Our technique could also be applied to rule out polynomial time o(log n/loglog n) ratio approximation algorithms

arXiv:2202.04377v1
fatcat:xq23esja3nbozoplmrny4e4u6a
*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*. ...##
###
Parameterized Intractability of Even Set and Shortest Vector Problem from Gap-ETH
[article]

2018
*
arXiv
*
pre-print

Similar to k-Even

arXiv:1803.09717v1
fatcat:ymowevvnkvg3dktbbzbpf3cezi
*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. ...##
###
The Complexity Ecology of Parameters: An Illustration Using Bounded Max Leaf Number

2009
*
Theory of Computing Systems
*

In

doi:10.1007/s00224-009-9167-9
fatcat:ezqsz2ggzzgd5iicukjjenzoai
*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*. ...##
###
The Complexity Ecology of Parameters: An Illustration Using Bounded Max Leaf Number
[chapter]

2007
*
Lecture Notes in Computer Science
*

In

doi:10.1007/978-3-540-73001-9_28
fatcat:6ctmbq7kgrbfpeer6mpc5pqaja
*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*. ...##
###
Parameterized Complexity and Approximation Algorithms

2007
*
Computer journal
*

We discuss

doi:10.1093/comjnl/bxm048
fatcat:6yukdt7tvja3jf76zyuxfm536q
*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. ...
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