Hitting Set Research Articles

Theoretical and Empirical Analysis of Cost-Function Merging for Implicit Hitting Set WCSP Solving

The Implicit Hitting Set (HS) approach has shown very effective for MaxSAT solving. However, only preliminary promising results have been obtained for the very similar Weighted CSP framework. In this paper we contribute towards both a better theoretical understanding of the HS approach and a more effective HS-based solvers for WCSP. First, we bound the minimum number of iterations of HS thanks to what we call distinguished cores. Then, we show a source of inefficiency by introducing two simple problems where HS is unfeasible. Next, we propose two reformulation methods that merge cost-functions to overcome the problem. We provide a theoretical analysis that quantifies the magnitude of the improvement of each method with respect to the number of iterations of the algorithm. In particular, we show that the reformulations can bring an exponential number of iterations down to a constant number in our working examples. Finally, we complement our theoretical analysis with two sets of experiments. First, we show that our results are aligned with real executions. Second, and most importantly, we conduct experiments on typical benchmark problems and show that cost-function merging may be heuristically applied and it may accelerate HS algorithms by several orders of magnitude. In some cases, it even outperforms state-of-the-art solvers.

Constrained hitting set problem with intervals: Hardness, FPT and approximation algorithms

We study a constrained version of the Geometric Hitting Set problem where we are given a set of points, partitioned into pairwise disjoint subsets, and a set of intervals. The objective is to hit all the intervals with a minimum number of points such that if we select a point from a subset, we must select all the points from that subset. We consider two special cases of the problem where each subset can have at most 2 and 3 points. If each subset contains at most 2 points and the intervals are disjoint, we show that the problem admits a polynomial-time algorithm. On the contrary, if each subset contains at most t points, where t≥2, and the intervals are overlapping, we show that the problem becomes NP-Hard. Further, when each subset contains at most t points where t≥3, and the intervals are disjoint, we prove that the problem is NP-Hard, and we provide two constant factor approximation algorithms for this problem. We also study the problem from the parameterized complexity perspective. If the intervals are disjoint, then we prove that the problem is in FPT when parameterized by the size of the solution. We also complement this result by giving a lower bound in the size of the kernel for disjoint intervals, and we also provide a polynomial kernel when the size of all subsets is bounded by a constant.

A parameterized algorithm for subset feedback vertex set in tournaments

The Subset Feedback Vertex Set problem (SFVS) takes an n-vertex graph G=(V,E), a terminal set T⊆V, and an integer k as the input. The goal is to determine whether there exists a subset S⊆V of at most k vertices whose removal makes no terminal in T contained in a cycle in the remaining graph. When T=V, SFVS degenerates to the classical Feedback Vertex Set problem (FVS). Both SFVS and FVS have been extensively studied in parameterized algorithms. In this paper, we study parameterized algorithms for Subset Feedback Vertex Set in Tournaments (SFVST), i.e., SFVS with the restriction that the input graph is always a tournament. By using the iterative compression method and a novel dynamic programming, we show that SFVST can be solved in 2k+o(k)nO(1) time, improving the bound obtained from 3-Hitting Set.

Minimum Hitting Set of Interval Bundles Problem: Computational Complexity and Approximability

The minimum hitting set of bundles problem (Mhsb) is a natural generalization of the minimum hitting set problem, where instead of hitting single elements, bundles of elements are hit. More specifically, we are given a ground set of elements and a family of sets. Every set in this family contains bundles of elements, which are subsets of the ground set. The task is to find a collection of elements of minimum size such that at least one bundle of every set in the family is hit. Motivated by several applications, we consider Mhsb restricted to interval and 2-dimensional interval bundles. We study the computational complexity and give polynomial-time algorithms for several classes of instances with these special structured bundles.

An Improved Deterministic Parameterized Algorithm for Cactus Vertex Deletion

A cactus is a connected graph that does not contain K4 − e as a minor. Given a graph G = (V,E) and an integer k ≥ 0, Cactus Vertex Deletion (also known as Diamond Hitting Set) is the problem of deciding whether G has a vertex set of size at most k whose removal leaves a forest of cacti. The previously best deterministic parameterized algorithm for this problem was due to Bonnet et al. [WG 2016], which runs in time 26knO(1), where n is the number of vertices of G. In this paper, we design a deterministic algorithm for Cactus Vertex Deletion, which runs in time 17.64knO(1). As an almost straightforward application of our algorithm, we also give a deterministic 17.64knO(1)-time algorithm for Even Cycle Transversal, which improves the previous running time 50knO(1) of the known deterministic parameterized algorithm due to Misra et al. [WG 2012].

Minimum Membership Covering and Hitting

Set Cover is a well-studied problem with application in many fields. A well-known variant of this problem is the Minimum Membership Set Cover problem: Given a set of points and a set of objects, the objective is to cover all points while minimizing the maximum number of objects that contain any one point. A dual of this problem is the Minimum Membership Hitting Set problem: Given a set of points and a set of objects, the objective is to stab all of the objects while minimizing the maximum number of points that an object contains. We study both of these variants in a geometric setting with various types of geometric objects in the plane, including axis-parallel line segments, axis-parallel strips, rectangles that are anchored on a horizontal line from one side, rectangles that are stabbed by a horizontal line, and rectangles that are anchored on one of two horizontal lines (i.e., each rectangle shares its top or its bottom edge (or both) with one of the input horizontal lines). For each of these problems we either prove NP-hardness or we give a polynomial-time algorithm. In particular, we show that it is NP-complete to decide whether there exists a solution with depth exactly 1 for either the Minimum Membership Set Cover or the Minimum Membership Hitting Set problem. In addition, we study a generalized version of the Minimum Membership Hitting Set problem.

A Minimum Hitting Set Algorithm With Prejudging Mechanism for Model-Based Fault Diagnosis in Distribution Networks

Fault diagnosis has a critical role in maintaining the reliability and stability of distribution networks. Compared with common diagnosis methods, such as expert systems that depend on the professional experience and information of fault, the model-based diagnosis (MBD) has many distinct advantages. However, the efficiency and accuracy of the calculation of the minimum hitting set (MHS) limit the performance of MBD. To overcome these limitations, an MHS algorithm with a prejudging mechanism is proposed in this article. This mechanism is constructed based on the distribution characteristics of MHS. The use of this mechanism prior to MHS calculation enables the removal of most non-HS particles and all super-MHS particles and enables the compression of the MHS calculation solution space several times. Moreover, after compression, only two types of particles remain in the left solution space, which are non-HS and MHS. As a result, the MHS can be picked out directly through the HS fitness function and the HS criterion and by avoiding the participation of the MHS ensured strategy (MHSES). Three cases studies have been introduced to test the effect of the prejudging mechanism on the performance of the MHS calculation. In addition, a discussion on the MBD of the distribution networks has been presented to validate the efficiency and accuracy of the proposed algorithm.

A divide-and-conquer method for scalable phylogenetic network inference from multilocus data.

MotivationReticulate evolutionary histories, such as those arising in the presence of hybridization, are best modeled as phylogenetic networks. Recently developed methods allow for statistical inference of phylogenetic networks while also accounting for other processes, such as incomplete lineage sorting. However, these methods can only handle a small number of loci from a handful of genomes.ResultsIn this article, we introduce a novel two-step method for scalable inference of phylogenetic networks from the sequence alignments of multiple, unlinked loci. The method infers networks on subproblems and then merges them into a network on the full set of taxa. To reduce the number of trinets to infer, we formulate a Hitting Set version of the problem of finding a small number of subsets, and implement a simple heuristic to solve it. We studied their performance, in terms of both running time and accuracy, on simulated as well as on biological datasets. The two-step method accurately infers phylogenetic networks at a scale that is infeasible with existing methods. The results are a significant and promising step towards accurate, large-scale phylogenetic network inference.Availability and implementationWe implemented the algorithms in the publicly available software package PhyloNet (https://bioinfocs.rice.edu/PhyloNet).Supplementary information Supplementary data are available at Bioinformatics online.

Succinct Hitting Sets and Barriers to Proving Lower Bounds for Algebraic Circuits

We formalize a framework of algebraically natural lower bounds for algebraic circuits. Just as with the natural proofs notion of Razborov and Rudich (1997) for Boolean circuit lower bounds, our notion of algebraically natural lower bounds captures nearly all lower bound techniques known. However, unlike in the Boolean setting, there has been no concrete evidence demonstrating that this is a barrier to obtaining super-polynomial lower bounds for general algebraic circuits, as there is little understanding whether algebraic circuits are expressive enough to support “cryptography” secure against algebraic circuits. Following a similar result of Williams (2016) in the Boolean setting, we show that the existence of an algebraic natural proofs barrier is equivalent to the existence of succinct derandomization of the polynomial identity testing problem, that is, to the existence of a hitting set for the class of poly(N)-degree poly(N)-size circuits which consists of coefficient vectors of polynomials of polylog(N) degree with polylog(N)-size circuits. Further, we give an explicit universal construction showing that if such a succinct hitting set exists, then our universal construction suffices. Further, we assess the existing literature constructing hitting sets for restricted classes of algebraic circuits and observe that none of them are succinct as given. Yet, we show how to modify some of these constructions to obtain succinct hitting sets. This constitutes the first evidence supporting the existence of an algebraic natural proofs barrier. Our framework is similar to the Geometric Complexity Theory (GCT) program of Mulmuley and Sohoni (2001), except that here we emphasize constructiveness of the proofs while the GCT program emphasizes symmetry. Nevertheless, our succinct hitting sets have relevance to the GCT program as they imply lower bounds for the complexity of the defining equations of polynomials computed by small circuits. A conference version of this paper appeared in the Proceedings of the 49th Annual ACM Symposium on Theory of Computing (STOC 2017).

A multivariate framework for weighted FPT algorithms

We introduce a multivariate approach for solving weighted parameterized problems. By allowing flexible use of parameters, our approach defines a framework for applying the classic bounded search trees technique. In our model, given an instance of size n of a minimization/maximization problem, and a parameter W≥1, we seek a solution of weight at most/at least W. We demonstrate the usefulness of our approach in solving Vertex Cover, 3-Hitting Set, Edge Dominating Set and Max Internal Out-Branching. While the best known algorithms for these problems admit running times of the form cWnO(1), for some c>1, our framework yields running times of the form csnO(1), where s≤W is the minimum size of a solution of weight at most/at least W. If no such solution exists, s=min⁡{W,m}, where m is the maximum size of a solution. In addition, we analyze the parameter t≤s, the minimum size of a solution.

Quick but Odd Growth of Cacti

Let $${\mathcal {F}}$$F be a family of graphs. Given an n-vertex input graph G and a positive integer k, testing whether G has a vertex subset S of size at most k, such that $$G-S$$G-S belongs to $${\mathcal {F}}$$F, is a prototype vertex deletion problem. These type of problems have attracted a lot of attention in recent times in the domain of parameterized complexity. In this paper, we study two such problems; when $${\mathcal {F}}$$F is either the family of forests of cacti or the family of forests of odd-cacti. A graph H is called a forest of cacti if every pair of cycles in H intersect on at most one vertex. Furthermore, a forest of cacti H is called a forest of odd cacti, if every cycle of H is of odd length. Let us denote by $${\mathcal {C}}$$C and $${{\mathcal {C}}}_\mathsf{odd}$$Codd, the families of forests of cacti and forests of odd cacti, respectively. The vertex deletion problems corresponding to $${\mathcal {C}}$$C and $${{\mathcal {C}}}_\mathsf{odd}$$Codd are called Diamond Hitting Set and Even Cycle Transversal, respectively. In this paper we design randomized algorithms with worst case run time $$12^k n^{\mathcal {O}(1)}$$12knO(1) for both these problems. Our algorithms considerably improve the running time for Diamond Hitting Set and Even Cycle Transversal, compared to what is known about them.

Hitting Forbidden Minors: Approximation and Kernelization

We study a general class of problems called $\mathcal{F}$-Deletion problems. In an $\mathcal{F}$-Deletion problem, we are asked whether a subset of at most $k$ vertices can be deleted from a graph $G$ such that the resulting graph does not contain as a minor any graph from the family ${\cal F}$ of forbidden minors. We study the problem parameterized by $k$, using $p$-$\mathcal{F}$-Deletion to refer to the parameterized version of the problem. We obtain a number of algorithmic results on the $p$-$\mathcal{F}$-Deletion problem when $\mathcal{F}$ contains a planar graph. We give a linear vertex kernel on graphs excluding $t$-claw $K_{1,t}$, the star with $t$ leaves, as an induced subgraph, where $t$ is a fixed integer and an approximation algorithm achieving an approximation ratio of $O(\log^{3/2} OPT)$, where $OPT$ is the size of an optimal solution on general undirected graphs. Finally, we obtain polynomial kernels for the case when $\cal F$ only contains graph $\theta_c$ as a minor for a fixed integer $c$. The graph $\theta_c$ consists of two vertices connected by $c$ parallel edges. Even though this may appear to be a very restricted class of problems it already encompasses well-studied problems such as Vertex Cover, Feedback Vertex Set, and Diamond Hitting Set. The generic kernelization algorithm is based on a nontrivial application of protrusion techniques, previously used only for problems on topological graph classes.

3-Hitting set on bounded degree hypergraphs: Upper and lower bounds on the kernel size

We study upper and lower bounds on the vertex-kernel size for the 3-HITTING SET problem on hypergraphs of degree at most 3, denoted 3-3-HS. We first show that, unless P = NP, 3-3-HS on 3-uniform hypergraphs does not have a vertex-kernel of size at most 35k/19 > 1.8421k. We then give a 4k - k0.2692 vertex-kernel for 3-3-hs that is computable in time O(k2). We do not assume that the hypergraph is 3-uniform for the vertex-kernel upper bound results. This result improves the upper bound of 4k on the vertex-kernel size for 3-3-HS, implied by the results of Wahlström.

A genetic algorithm for the minimum hitting set

In this paper the Minimum Hitting Set (MHS) problem is considered. The problem is solved by a genetic algorithms (GA) that uses binary encoding and standard genetics operators adapted to the problem. In proposed implementation all individuals are feasibility by default, so search is directed into the promising regions. The overall performance of the GA implementation is improved by caching technique. Local search optimization is used to refine the solutions explored by GAs. The algorithm is tested on standard instances from the literature. The obtained results are also compared with the results of existing methods for solving MHS in order to assess their merits.

In This Issue

In This Issue

The Implicit Hitting Set Approach to Solve Combinatorial Optimization Problems with an Application to Multigenome Alignment

We develop a novel framework, the implicit hitting set approach, for solving a class of combinatorial optimization problems. The explicit hitting set problem is as follows: given a set U and a family S of subsets of U, find a minimum-cardinality set that intersects (hits) every set in S. In the implicit hitting set problem (IHSP), the family of subsets S is not explicitly listed (its size is, generally, exponential in terms of the size of U); instead, it is given via a polynomial-time oracle that verifies if a given set H is a hitting set or returns a set in S that is not hit by H. Many NP-hard problems can be straightforwardly formulated as implicit hitting set problems. We show that the implicit hitting set approach is valuable in developing exact and heuristic algorithms for solving this class of combinatorial optimization problems. Specifically, we provide a generic algorithmic strategy, which combines efficient heuristics and exact methods, to solve any IHSP. Given an instance of an IHSP, the proposed algorithmic strategy gives a sequence of feasible solutions and lower bounds on the optimal solution value and ultimately yields an optimal solution. We specialize this algorithmic strategy to solve the multigenome alignment problem and present computational results that illustrate the effectiveness of the implicit hitting set approach.

Parameterized reductions and algorithms for a graph editing problem that generalizes vertex cover

We study a novel generalization of the Vertex Cover problem which is motivated by, e.g., error correction (data cleaning) prior to inference of chemical mixtures by their observable reaction products. We focus on the important case of deciding on one of two candidate substances. This problem has nice graph-theoretic formulations situated between Vertex Cover and 3-Hitting Set. In order to characterize its parameterized complexity we devise parameter-preserving reductions, and we show that some minimum solution can be computed faster than by solving 3-Hitting Set in general. More explicitly, we introduce the Union Editing problem: In a hypergraph with red and blue vertices, edit the colors so that the red set becomes exactly the union of some hyperedges. The case of degree 2 is equivalent to Star Editing: in a graph with red and blue edges, edit the colors so that the red set becomes exactly the union of some stars, i.e., vertices with all their incident edges. Our time bound is O∗(1.84c) where c denotes the total number of recolored edges.

BOUNDED SEARCH TREE ALGORITHMS FOR PARAMETRIZED COGRAPH DELETION: EFFICIENT BRANCHING RULES BY EXPLOITING STRUCTURES OF SPECIAL GRAPH CLASSES

Many fixed-parameter tractable algorithms using a bounded search tree have been repeatedly improved, often by describing a larger number of branching rules involving an increasingly complex case analysis. We introduce a novel and general search strategy that branches on the forbidden subgraphs of a graph class relaxation. By using the class of P4-sparse graphs as the relaxed graph class, we obtain efficient bounded search tree algorithms for several parametrized deletion problems. We give the first non-trivial bounded search tree algorithms for the cograph edge-deletion problem and the trivially perfect edge-deletion problems. For the cograph vertex deletion problem, a refined analysis of the runtime of our simple bounded search algorithm gives a faster exponential factor than those algorithms designed with the help of complicated case distinctions and non-trivial running time analysis [R. Niedermeier and P. Rossmanith, An efficient fixed-parameter algorithm for 3-hitting set, J. Discrete Algorithms1(1) (2003) 89–102] and computer-aided branching rules [J. Gramm, J. Guo, F. Hüffner and R. Niedermeier, Automated generation of search tree algorithms for hard graph modification problems, Algorithmica39(4) (2004) 321–347].

Approximating vertex cover in dense hypergraphs

We consider the Minimum Vertex Cover problem in hypergraphs in which every hyperedge has size k (also known as Minimum Hitting Set problem, or minimum set cover with element frequency k). Simple algorithms exist that provide k-approximations, and this is believed to be the best approximation achievable in polynomial time. We show how to exploit density and regularity properties of the input hypergraph to break this barrier. In particular, we provide a randomized polynomial-time algorithm with approximation factor k/(1+(k−1)d¯kΔ), where d¯ and Δ are the average and maximum degree, and Δ must be Ω(nk−1/logn). The proposed algorithm generalizes the recursive sampling technique of Imamura and Iwama (SODAʼ05) for vertex cover in dense graphs. As a corollary, we obtain an approximation factor arbitrarily close to k/(2−1/k) for subdense regular hypergraphs, which is shown to be the best possible under the Unique Games conjecture.

Parameterized algorithmics for d-Hitting Set

Based on the methodology we presented in earlier work on parameterized algorithms for 3-Hitting Set, we develop simple search tree-based algorithms for d-Hitting Set. We considerably improve on the bounds that were elsewhere derived for these problems.