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

Peter Damaschke, Leonid Molokov
2009 Journal of Discrete Algorithms  
We study 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".  ... 
doi:10.1016/j.jda.2009.01.003 fatcat:yccolf6rjvb6tjgcwgd4lsznbq

Parameterizations of hitting set of bundles and inverse scope

Peter Damaschke
2013 Journal of combinatorial optimization  
This contrasts to 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".  ... 
doi:10.1007/s10878-013-9629-5 fatcat:6a7icbjphfc5blkr2yy67rocia

Pairs Covered by a Sequence of Sets [chapter]

Peter Damaschke
2015 Lecture Notes in Computer Science  
Enumerating 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.  ... 
doi:10.1007/978-3-319-22177-9_17 fatcat:pqt6xxvex5g3tmml2lv4u6vqg4

Efficiently Enumerating Hitting Sets of Hypergraphs Arising in Data Profiling [article]

Thomas Bläsius, Tobias Friedrich, Julius Lischeid, Kitty Meeks, Martin Schirneck
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) 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.  ... 
arXiv:1805.01310v3 fatcat:zm626hj75javxjnay5it4ligse

Semi-bandit Optimization in the Dispersed Setting [article]

Maria-Florina Balcan, Travis Dick, Wesley Pegden
2020 arXiv   pre-print
We develop online optimization algorithms capable 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.  ... 
arXiv:1904.09014v3 fatcat:hw3syqdlnvh7rgyjcrjf6ckw7y

On Directed Feedback Vertex Set parameterized by treewidth [article]

Marthe Bonamy, Łukasz Kowalik, Jesper Nederlof, Michał Pilipczuk, Arkadiusz Socała, Marcin Wrochna
2017 arXiv   pre-print
We study 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] .  ... 
arXiv:1707.01470v2 fatcat:44rsxejoynbyzo4iccx3r7pysa

The Parity of Set Systems Under Random Restrictions with Applications to Exponential Time Problems [chapter]

Andreas Björklund, Holger Dell, Thore Husfeldt
2015 Lecture Notes in Computer Science  
A structural result in 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 .  ... 
doi:10.1007/978-3-662-47672-7_19 fatcat:4ali2rhiffdtbo5soefqdg3kxm

On Directed Feedback Vertex Set Parameterized by Treewidth [chapter]

Marthe Bonamy, Łukasz Kowalik, Jesper Nederlof, Michał Pilipczuk, Arkadiusz Socała, Marcin Wrochna
2018 Lecture Notes in Computer Science  
We study 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] .  ... 
doi:10.1007/978-3-030-00256-5_6 fatcat:fvwtkzsjvvcenlqgm5idfaxczy

Geometric Hitting Set for Segments of Few Orientations [article]

Sándor P. Fekete, Kan Huang, Joseph S. B. Mitchell, Ojas Parekh, Cynthia A. Phillips
2016 arXiv   pre-print
We study several natural instances 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.  ... 
arXiv:1603.06077v2 fatcat:vxlfbon5crckjf5vaoln5hsykm

Simultaneous Feedback Vertex Set: A Parameterized Perspective [article]

Akanksha Agrawal, Daniel Lokshtanov, Amer E. Mouawad, Saket Saurabh
2015 arXiv   pre-print
In 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  ... 
arXiv:1510.01557v1 fatcat:uhci5tmlhjgipp3dpqzaszppnq

Upper Dominating Set: Tight Algorithms for Pathwidth and Sub-Exponential Approximation [article]

Louis Dublois, Michael Lampis, Vangelis Th. Paschos
2021 arXiv   pre-print
An upper dominating 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  ... 
arXiv:2101.07550v1 fatcat:w4wzquzq2vbh3hma4gowxcvuuu

Strict group testing and the set basis problem

Peter Damaschke, Azam Sheikh Muhammad, Gábor Wiener
2014 Journal of combinatorial theory. Series A  
Besides other 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  ... 
doi:10.1016/j.jcta.2014.04.005 fatcat:zihi3zdvcjdtrpcoa5bheby2nu

Rank-determining sets of metric graphs

Ye Luo
2011 Journal of combinatorial theory. Series A  
In addition, we investigate 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  ... 
doi:10.1016/j.jcta.2011.03.002 fatcat:c2b2altgofcudnfbryjpm7sza4

Parameterized exact and approximation algorithms for maximumk-set cover and related satisfiability problems

Édouard Bonnet, Vangelis Th. Paschos, Florian Sikora
2016 RAIRO - Theoretical Informatics and Applications  
This problem is W[2]-hard when 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.  ... 
doi:10.1051/ita/2016022 fatcat:urg3gnvtobbajnkjijvq57pmru

Near-Optimal Disjoint-Path Facility Location Through Set Cover by Pairs [article]

David S. Johnson, Lee Breslau, Ilias Diakonikolas, Nick Duffield, Yu Gu, MohammadTaghi Hajiaghayi, Howard Karloff, Mauricio G. C. Resende, and Subhabrata Sen
2016 arXiv   pre-print
Although computing this lower 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.  ... 
arXiv:1611.01210v1 fatcat:oi4wpouv4rckhbqxx36eamblt4
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