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The union of minimal hitting sets: Parameterized combinatorial bounds and counting

2009
*
Journal of Discrete Algorithms
*

We study

doi:10.1016/j.jda.2009.01.003
fatcat:yccolf6rjvb6tjgcwgd4lsznbq
*the**union**of*all inclusion-*minimal*k-*hitting**sets*in hypergraphs*of*rank r (where*the*rank is*the*maximum size*of*hyperedges). ... We show that this*union*is relevant for certain*combinatorial*inference problems*and*give worst-case*bounds*on its size, depending on r*and*k. ... Partial support also came from*the*Swedish Research Council (Vetenskapsrådet), grant no. 2007-6437, "*Combinatorial*inference algorithms -*parameterization**and*clustering". ...##
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Parameterizations of hitting set of bundles and inverse scope

2013
*
Journal of combinatorial optimization
*

This contrasts to

doi:10.1007/s10878-013-9629-5
fatcat:6a7icbjphfc5blkr2yy67rocia
*the*to*the*corresponding*parameterized**Hitting**Set*version which is in FPT. ... First we prove that*Hitting**Set**of*Bundles, with*the*number*of*hyperedges*and**the*solution size as parameter, is W [1]-complete. ... Acknowledgment This work has been supported by*the*Swedish Research Council (Vetenskapsrådet), through*the*project grant 2010-4661, "Generalized*and*fast search strategies for*parameterized*problems". ...##
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Pairs Covered by a Sequence of Sets
[chapter]

2015
*
Lecture Notes in Computer Science
*

Enumerating

doi:10.1007/978-3-319-22177-9_17
fatcat:pqt6xxvex5g3tmml2lv4u6vqg4
*minimal*new combinations*of*elements in a sequence*of**sets*is interesting, e.g., for novelty detection in a stream*of*texts.*The**sets*are*the*bags*of*words occuring in*the*texts. ... We give a greedy heuristic based on*the*twin graph, a succinct description*of**the*pairs covered by a*set*family,*and*on finding good candidate*sets*by random sampling. ...*The*author also wishes to thank*the*referees for careful reading. ...##
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Efficiently Enumerating Hitting Sets of Hypergraphs Arising in Data Profiling
[article]

2021
*
arXiv
*
pre-print

More generally, we present an algorithm that on n-vertex, m-edge hypergraphs has delay O(m^k^*+1 n^2)

arXiv:1805.01310v3
fatcat:zm626hj75javxjnay5it4ligse
*and*uses O(mn) space, where k^* is*the*maximum size*of*any*minimal**hitting**set*. ...*The*transversal hypergraph problem is*the*task*of*enumerating*the**minimal**hitting**sets**of*a hypergraph. It is a long-standing open question whether this can be done in output-polynomial time. ...*The*authors would like to thank Felix Naumann*and*Thorsten Papenbrock for*the*many fruitful discussions about data profiling,*and*Erik Kohlros for conducting additional experiments. ...##
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Semi-bandit Optimization in the Dispersed Setting
[article]

2020
*
arXiv
*
pre-print

We develop online optimization algorithms capable

arXiv:1904.09014v3
fatcat:hw3syqdlnvh7rgyjcrjf6ckw7y
*of*using this kind*of*extra information by working in*the*semi-bandit feedback*setting*. ... Across many fields in AI, science,*and*engineering, practitioners will often fix a family*of**parameterized*algorithms*and*then optimize those parameters to obtain good performance on example instances ... Say a subset*of*C is*hit*by a line segment if*the*subset is exactly*the**set**of*curves in C which intersect*the*segment,*and**hit*by a line if some segment*of**the*line*hits**the*subset. ...##
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On Directed Feedback Vertex Set parameterized by treewidth
[article]

2017
*
arXiv
*
pre-print

We study

arXiv:1707.01470v2
fatcat:44rsxejoynbyzo4iccx3r7pysa
*the*Directed Feedback Vertex*Set*problem*parameterized*by*the*treewidth*of**the*input graph. ... We prove that unless*the*Exponential Time Hypothesis fails,*the*problem cannot be solved in time 2^o(t t)· n^O(1) on general directed graphs, where t is*the*treewidth*of**the*underlying undirected graph ... More precisely, we give a*parameterized*reduction from*the*k × k*Hitting**Set*with thin*sets*problem, for which a lower*bound*excluding running time 2 o(k log k) ·n O(1) under ETH was given in [18] . ...##
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The Parity of Set Systems Under Random Restrictions with Applications to Exponential Time Problems
[chapter]

2015
*
Lecture Notes in Computer Science
*

A structural result in

doi:10.1007/978-3-662-47672-7_19
fatcat:4ali2rhiffdtbo5soefqdg3kxm
*parameterized*complexity: We define*the**parameterized*complexity class ⊕W[1]*and*prove that it is at least as hard as W[1] under randomized fpt-reductions with*bounded*one-sided ... When compared to previously known reductions*of*this type, ours excel in their simplicity: For graph problems, restricting elements*of**the*ground*set*usually corresponds to simple deletion*and*contraction ... variant*of**the*isolation lemma,*and*Ryan Williams for pointing us to Vassilevska Williams et al. 2015 . ...##
###
On Directed Feedback Vertex Set Parameterized by Treewidth
[chapter]

2018
*
Lecture Notes in Computer Science
*

We study

doi:10.1007/978-3-030-00256-5_6
fatcat:fvwtkzsjvvcenlqgm5idfaxczy
*the*Directed Feedback Vertex*Set*problem*parameterized*by*the*treewidth*of**the*input graph. ... We prove that unless*the*Exponential Time Hypothesis fails,*the*problem cannot be solved in time 2 o(t log t) · n O(1) on general directed graphs, where t is*the*treewidth*of**the*underlying undirected ... More precisely, we give a*parameterized*reduction from*the*k × k*Hitting**Set*with thin*sets*problem, for which a lower*bound*excluding running time 2 o(k log k) ·n O(1) under ETH was given in [18] . ...##
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Geometric Hitting Set for Segments of Few Orientations
[article]

2016
*
arXiv
*
pre-print

We study several natural instances

arXiv:1603.06077v2
fatcat:vxlfbon5crckjf5vaoln5hsykm
*of**the*geometric*hitting**set*problem for input consisting*of**sets**of*line segments (*and*rays, lines) having a small number*of*distinct slopes. ... We give hardness*and*hardness*of*approximation results for these problems. We prove that*the**hitting**set*problem for vertical lines*and*horizontal rays is polynomially solvable. ... For each such*union*i, we let S i be*the**set**of*points covering*the**union*, while H i*and*V i are*the**sets**of*points covering*the*horizontal segment*and*vertical segment respectively. ...##
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Simultaneous Feedback Vertex Set: A Parameterized Perspective
[article]

2015
*
arXiv
*
pre-print

In

arXiv:1510.01557v1
fatcat:uhci5tmlhjgipp3dpqzaszppnq
*the*latter problem, we are given an α-edge-colored graph G*and**the*goal is to find a*set*S*of*at most k vertices such that each graph G_i ∖ S, where G_i = (V, E_i)*and*1 ≤ i ≤α, is in F. ... Our positive results answer one*of**the*open problems posed by Cai*and*Ye (MFCS 2014). ... For*the*base case, each G i is a disjoint*union**of*cycles (though not G)*and*to find an α-simultaneous feedback vertex*set*for G we cast*the*remaining problem as an instance*of**Hitting**Set**parameterized*...##
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Upper Dominating Set: Tight Algorithms for Pathwidth and Sub-Exponential Approximation
[article]

2021
*
arXiv
*
pre-print

An upper dominating

arXiv:2101.07550v1
fatcat:w4wzquzq2vbh3hma4gowxcvuuu
*set*is a*minimal*dominating*set*in a graph. In*the*Upper Dominating*Set*problem,*the*goal is to find an upper dominating*set**of*maximum size. ... We study*the*complexity*of**parameterized*algorithms for Upper Dominating*Set*, as well as its sub-exponential approximation. ... So*the*size*of*any*minimal**hitting**set*is*bounded*by: t · α(G) d + |V (G)| = O(n d+dε r 1/d+2ε ) We have now construct a graph G ′*of**the*Maximum*Minimal**Hitting**Set*problem where*the*gap between*the*values ...##
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Strict group testing and the set basis problem

2014
*
Journal of combinatorial theory. Series A
*

Besides other

doi:10.1016/j.jcta.2014.04.005
fatcat:zihi3zdvcjdtrpcoa5bheby2nu
*combinatorial*tools we generalize d-disjunct matrices to any candidate hypergraphs,*and*we reveal connections to*the**set*basis problem*and*communication complexity. ... Group testing is*the*problem to identify up to d defectives out*of*n elements, by testing subsets for*the*presence*of*defectives. ... Acknowledgments Research*of**the*first two authors has been supported by*the*Swedish Research Council (Vetenskapsrådet) through*the*grant 2010-4661, "Generalized*and*fast search strategies for*parameterized*...##
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Rank-determining sets of metric graphs

2011
*
Journal of combinatorial theory. Series A
*

In addition, we investigate

doi:10.1016/j.jcta.2011.03.002
fatcat:c2b2altgofcudnfbryjpm7sza4
*the*properties*of*rank-determining*sets*in general*and*formulate a criterion for rank-determining*sets*. ... We define a rank-determining*set**of*a metric graph Γ to be a subset A*of*Γ such that*the*rank*of*a divisor D on Γ is always equal to*the*rank*of*D restricted on A. ... Acknowledgments I would most*of*all like to thank Matthew Baker for introducing me to this topic, for his encouragement*of*further study on general properties*of*rank-determining*sets*,*and*for many valuable ...##
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Parameterized exact and approximation algorithms for maximumk-set cover and related satisfiability problems

2016
*
RAIRO - Theoretical Informatics and Applications
*

This problem is W[2]-hard when

doi:10.1051/ita/2016022
fatcat:urg3gnvtobbajnkjijvq57pmru
*parameterized*by k,*and*FPT when*parameterized*by p. We investigate*the**parameterized*approximability*of**the*problem with respect to parameters k*and*p. ... Given a family*of*subsets S over a*set**of*elements X*and*two integers p*and*k, max k-*set*cover consists*of*finding a subfamily T ⊆ S*of*cardinality at most k, covering at least p elements*of*X. ...*The*pertinent suggestions*and*comments*of*two anonymous referees have greatly improved*the*quality*of*this paper. ...##
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Near-Optimal Disjoint-Path Facility Location Through Set Cover by Pairs
[article]

2016
*
arXiv
*
pre-print

Although computing this lower

arXiv:1611.01210v1
fatcat:oi4wpouv4rckhbqxx36eamblt4
*bound*requires solving*the*NP-hard*Hitting**Set*problem*and*can underestimate*the*optimal value by a linear factor in*the*worst case, it can be computed quickly by CPLEX,*and*... For*the**set*-disjoint variant, we validate our claim*of*near-optimality via a new lower-*bounding*integer programming formulation. ...*The*authors thank David Applegate for helpful discussions*and*in particular for simplifying*the*statement*and*proof*of*Theorem 5. ...
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