Hitting Set Problem Research Articles

An approximation algorithm for the k-prize-collecting hitting set problem

An approximation algorithm for the k-prize-collecting hitting set problem

Approximation Algorithms for the Submodular Hitting Set Problem

In this paper, we study the submodular hitting set problem (SHSP), which is a variant of the hitting set problem. In the SHSP, we are given a supergraph H = ( V , C ) and a nonnegative submodular function on the set 2 V . The objective is to determine a vertex subset to cover all hyperedges such that the cost of submodular covering is minimized. Our main work is to present a rounding algorithm and a primal-dual algorithm respectively for the SHSP and prove that they both have the approximation ratio k , where k is the maximum number of vertices in all hyperedges.

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.

Deployment Optimization of Dual-Functional UAVs for Integrated Localization and Communication

In emergency scenarios, unmanned aerial vehicles (UAVs) can be deployed to assist localization and communication services for ground terminals. In this paper, we propose a new integrated air-ground networking paradigm that uses dual-functional UAVs to assist the ground networks for improving both communication and localization performance. We investigate the optimization problem of deploying the minimal number of UAVs to satisfy the communication and localization requirements of ground users. The problem has several technical difficulties including the cardinality minimization, the non-convexity of localization performance metric regarding UAV location, and the association between user and communication terminal. To tackle the difficulties, we adopt <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</i> -optimality as the localization performance metric, and derive the geometric characteristics of the feasible UAV hovering regions in 2D and 3D based on accurate approximation values. We solve the simplified 2D projection deployment problem by transforming the problem into a minimum hitting set problem, and propose a low-complexity algorithm to solve it. Through numerical simulations, we compare our proposed algorithm with benchmark methods. The number of UAVs required by the proposed algorithm is close to the optimal solution, while other benchmark methods require much more UAVs to accomplish the same task.

Serial and parallel kernelization of Multiple Hitting Set parameterized by the Dilworth number, implemented on the GPU

The NP-hard Multiple Hitting Set problem is the problem of finding a minimum-cardinality set intersecting each of the sets in a given input collection a given number of times. Generalizing a well-known data reduction algorithm due to Weihe, we show a problem kernel for Multiple Hitting Set parameterized by the Dilworth number, a graph parameter introduced by Foldes and Hammer in 1978 yet seemingly so far unexplored in the context of parameterized complexity theory. Using matrix multiplication, we speed up the algorithm to quadratic sequential time and logarithmic parallel time. We experimentally evaluate our algorithms. By implementing our algorithm on GPUs, we show the feasibility of realizing kernelization algorithms on SIMD (Single Instruction, Multiple Data) architectures.

A bicriteria approximation algorithm for the minimum hitting set problem in measurable range spaces

We consider the problem of finding a minimum-size hitting set in a range space F=(Q,R) defined by a measure on a family of subsets of an infinite set R. We observe that, under reasonably general assumptions, the infinite-dimensional convex relaxation can be solved (approximately) efficiently by multiplicative weight updates. As a consequence, we get an algorithm that finds, for any δ>0, a set of size O(αFzF⁎) that hits a (1−δ)-fraction of the ranges in R (with respect to the given measure) in time proportional to log⁡1δ, where zF⁎ is the value of the fractional optimal solution and αF is the integrality gap of the standard LP relaxation of the hitting set problem for the restriction of F on a set of points of size O(zF⁎log⁡1δ). Some applications of this result in Computational Geometry are given.

Almost optimal query algorithm for hitting set using a subset query

In this paper, we focus on Hitting-Set, a fundamental problem in combinatorial optimization, through the lens of sublinear time algorithms. Given access to the hypergraph through a subset query oracle in the query model, we give sublinear time algorithms for Hitting-Set with almost tight parameterized query complexity. In parameterized query complexity, we estimate the number of queries to the oracle based on the parameter k, the size of the Hitting-Set. The subset query oracle we use in this paper is called Generalized d-partite Independent Set query oracle (GPIS) and it was introduced by Bishnu et al. (ISAAC'18). GPIS is a generalization to hypergraphs of the Bipartite Independent Set query oracle (BIS) introduced by Beame et al. (ITCS'18 and TALG'20) for estimating the number of edges in graphs. Since its introduction GPIS query oracle has been used for estimating the number of hyperedges independently by Dell et al. (SODA'20 and SICOMP'22) and Bhattacharya et al. (STACS'22), and for estimating the number of triangles in a graph by Bhattacharya et al. (ISAAC'19 and TOCS'21). Formally, GPIS is defined as follows: GPISoracle for a d-uniform hypergraphHtakes as input d pairwise disjoint non-empty subsetsA1,…,Adof vertices inHand answers whether there is a hyperedge inHthat intersects each setAi, wherei∈{1,2,…,d}. For d=2, the GPIS oracle is nothing but BIS oracle.We show that d-Hitting-Set, the hitting set problem for d-uniform hypergraphs, can be solved using O˜d(kdlog⁡n)GPIS queries. Additionally, we also showed that d-Decision-Hitting-Set, the decision version of d-Hitting-Set can be solved with O˜d(min⁡{kdlog⁡n, k2d2})GPIS queries. We complement these parameterized upper bounds with an almost matching parameterized lower bound that states that any algorithm that solves d-Decision-Hitting-Set requires Ω((k+dd))GPIS queries.

Covering and packing of triangles intersecting a straight line

We study four geometric optimization problems: set cover , hitting set , piercing set , and independent set with right-triangles (a triangle is a right-triangle whose base is parallel to the x -axis, perpendicular is parallel to the y -axis, and the slope of the hypotenuse is − 1 ). The input triangles are constrained to be intersecting a straight line . The straight line can either be a horizontal or an inclined line (a line whose slope is − 1 ). A right-triangle is said to be a λ -right-triangle , if the length of both its base and perpendicular is λ . For 1-right-triangles where the triangles intersect an inclined line, we prove that the set cover and hitting set problems are NP -hard, whereas the piercing set and independent set problems are in P . The same results hold for 1-right-triangles where the triangles are intersecting a horizontal line instead of an inclined line. We prove that the piercing set and independent set problems with right-triangles intersecting an inclined line are NP -hard. Finally, we give an n O ( ⌈ log c ⌉ + 1 ) time exact algorithm for the independent set problem with λ -right-triangles intersecting a straight line such that λ takes more than one value from [ 1 , c ] , for some integer c . We also present O ( n 2 ) -time dynamic programming algorithms for the independent set problem with 1-right-triangles where the triangles intersect a horizontal line and an inclined line.

Exponentially Faster Shortest Paths in the Congested Clique

We present improved deterministic algorithms for approximating shortest paths in the Congested Clique model of distributed computing. We obtain poly(log log n )-round algorithms for the following problems in unweighted undirected n -vertex graphs: ( 1 + ϵ )-approximation of multi-source shortest paths (MSSP) from O (√ n ) sources. (2 + ϵ )-approximation of all pairs shortest paths (APSP). (1 + ϵ , β)-approximation of APSP where β = O (log log n / ϵ ) log log n . These bounds improve exponentially over the state-of-the-art poly-logarithmic bounds due to [Censor-Hillel et al., PODC19]. It also provides the first nearly-additive bounds for the APSP problem in sub-polynomial time. Our approach is based on distinguishing between short and long distances based on some distance threshold t = O ( β / ϵ ) where β = O (log log n / ϵ ) log log n . Handling the long distances is done by devising a new algorithm for computing a sparse (1 + ϵ , β ) emulator with O ( n log log n ) edges. For the short distances, we provide distance-sensitive variants for the distance tool-kit of [Censor-Hillel et al., PODC19]. By exploiting the fact that this tool-kit should be applied only on local balls of radius t , their round complexities get improved from poly (log n ) to poly (log t ). Finally, our deterministic solutions for these problems are based on a derandomization scheme of a novel variant of the hitting set problem, which might be of independent interest.

Dynamic Kernels for Hitting Sets and Set Packing

Computing small kernels for the hitting set problem is a well-studied computational problem where we are given a hypergraph with n vertices and m hyperedges, each of size d for some small constant d, and a parameter k. The task is to compute a new hypergraph, called a kernel, whose size is polynomial with respect to the parameter k and which has a size-k hitting set if, and only if, the original hypergraph has one. State-of-the-art algorithms compute kernels of size k^d (which is a polynomial as d is a constant), and they do so in time mcdot 2^d {text {poly}}(d) for a small polynomial {text {poly}}(d) (which is linear in the hypergraph size for d fixed). We generalize this task to the dynamic setting where hyperedges may continuously be added or deleted and one constantly has to keep track of a size-k^d kernel. This paper presents a deterministic solution with worst-case time 3^d {text {poly}}(d) for updating the kernel upon inserts and time 5^d {text {poly}}(d) for updates upon deletions. These bounds nearly match the time 2^d {text {poly}}(d) needed by the best static algorithm per hyperedge. Let us stress that for constant d our algorithm maintains a hitting set kernel with constant, deterministic, worst-case update time that is independent of n, m, and the parameter k. As a consequence, we also get a deterministic dynamic algorithm for keeping track of size-k hitting sets in d-hypergraphs with update times O(1) and query times O(c^k) where c = d - 1 + O(1/d) equals the best base known for the static setting.

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.

Scheduling with gaps: new models and algorithms

We consider scheduling problems for unit jobs with release times, where the number or size of the gaps in the schedule is taken into consideration, either in the objective function or as a constraint. Except for several papers on minimum-energy scheduling, there is no work in the scheduling literature that uses performance metrics depending on the gap structure of a schedule. One of our objectives is to initiate the study of such scheduling problems. We focus on the model with unit-length jobs. First we examine scheduling problems with deadlines, where we consider two variants of minimum-gap scheduling: maximizing throughput with a budget for the number of gaps and minimizing the number of gaps with a throughput requirement. We then turn to other objective functions. For example, in some scenarios gaps in a schedule may be actually desirable, leading to the problem of maximizing the number of gaps. A related problem involves minimizing the maximum gap size. The second part of the paper examines the model without deadlines, where we focus on the tradeoff between the number of gaps and the total or maximum flow time. For all these problems we provide polynomial time algorithms, with running times ranging from O(nlog n) for some problems to O(n^7) for other. The solutions involve a spectrum of algorithmic techniques, including different dynamic programming formulations, speed-up techniques based on searching Monge arrays, searching X+Y matrices, or implicit binary search. Throughout the paper, we also draw a connection between gap scheduling problems and their continuous analogues, namely hitting set problems for intervals of real numbers. As it turns out, for some problems the continuous variants provide insights leading to efficient algorithms for the corresponding discrete versions, while for other problems completely new techniques are needed to solve the discrete version.

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.

Dominating set of rectangles intersecting a straight line

The Minimum Dominating Set (MDS) problem is one of the well-studied problems in computer science. It is well-known that this problem is $$\mathsf {NP}$$ -hard for simple geometric objects; unit disks, axis-parallel unit squares, and axis-parallel rectangles to name a few. An interesting variation of the MDS problem with rectangles is when there exists a straight line that intersects each of the given rectangles. In the recent past researchers have studied the maximum independent set, minimum hitting set problems on this setting with different geometric objects. We study the MDS problem with axis-parallel rectangles, unit-height rectangles, and unit squares in the plane. These geometric objects are constrained to be intersected by a straight line. For axis-parallel rectangles, we prove that this problem is $$\mathsf {NP}$$ -hard. When the objects are axis-parallel unit squares, we present a polynomial time algorithm using dynamic programming. We provide a polynomial time algorithm for unit-height rectangles as well. For unit squares that touch the straight line at a single point from either side of the straight line, we show that there is an $$O(n\log n)$$ -time algorithm.

Generic Detectability and Isolability of Topology Failures in Networked Linear Systems

This paper studies the possibility of detecting and isolating topology failures (including link failures and node failures) of a networked system from subsystem measurements, in which subsystems are of fixed high-order linear dynamics, and the exact interaction weights among them are unknown. We prove that in such class of networked systems with the same network topologies, the detectability and isolability of a given topology failure (set) are generic properties, indicating that it is the network topology that dominates the property of being detectable or isolable for a failure (set). We first give algebraic conditions for detectability and isolability of arbitrary parameter perturbations for a lumped plant, and then derive graph-theoretical necessary and sufficient conditions for generic detectability and isolability of topology failures for the networked systems. On the basis of these results, we consider the problems of deploying the smallest set of sensors for generic detectability and isolability. We reduce the associated sensor placement problems to the hitting set problems, which can be effectively solved by greedy algorithms with guaranteed approximation performances.

An approximation algorithm for submodular hitting set problem with linear penalties

The hitting set problem is a generalization of the vertex cover problem to hypergraphs. Xu et al. (Theor Comput Sci 630:117–125, 2016) presented a primal-dual algorithm for the submodular vertex cover problem with linear/submodular penalties. Motivated by their work, we study the submodular hitting set problem with linear penalties (SHSLP). The goal of the SHSLP is to select a vertex subset in the hypergraph to cover some hyperedges and penalize the uncovered ones such that the total cost of covering and penalty is minimized. Based on the primal-dual scheme, we obtain a k-approximation algorithm for the SHSLP, where k is the maximum number of vertices in all hyperedges.

Dynamic Parameterized Problems and Algorithms

Fixed-parameter algorithms and kernelization are two powerful methods to solve NP-hard problems. Yet so far those algorithms have been largely restricted to static inputs. In this article, we provide fixed-parameter algorithms and kernelizations for fundamental NP-hard problems with dynamic inputs. We consider a variety of parameterized graph and hitting set problems that are known to have f ( k ) n 1+o(1) time algorithms on inputs of size n , and we consider the question of whether there is a data structure that supports small updates (such as edge/vertex/set/element insertions and deletions) with an update time of g ( k ) n o(1) ; such an update time would be essentially optimal. Update and query times independent of n are particularly desirable. Among many other results, we show that F EEDBACK V ERTEX S ET and k -P ATH admit dynamic algorithms with f ( k )log O(1) update and query times for some function f depending on the solution size k only. We complement our positive results by several conditional and unconditional lower bounds. For example, we show that unlike their undirected counterparts, D IRECTED F EEDBACK V ERTEX S ET and D IRECTED k -P ATH do not admit dynamic algorithms with n o(1) update and query times even for constant solution sizes k ≤ 3 , assuming popular hardness hypotheses. We also show that unconditionally, in the cell probe model, D IRECTED F EEDBACK V ERTEX S ET cannot be solved with update time that is purely a function of k .

Computing Hitting Set Kernels By AC0-Circuits

Given a hypergraph H = (V,E), what is the smallest subset $X \subseteq V$ such that e ∩ X≠∅ holds for all e ∈ E? This problem, known as the hitting set problem, is a basic problem in combinatorial optimization and has been studied extensively in both classical and parameterized complexity theory. There are well-known kernelization algorithms for it, which get a hypergraph H and a number k as input and output a hypergraph H′ such that (1) H has a hitting set of size k if and only if $H^{\prime }$ has such a hitting set and (2) the size of $H^{\prime }$ depends only on k and on the maximum cardinality d of hyperedges in H. The algorithms run in polynomial time and can be parallelized to a certain degree: one can easily compute hitting set kernels in parallel time O(k) and not-so-easily in time O(d) – but it was conjectured that these are the best parallel algorithms possible. We refute this conjecture and show how hitting set kernels can be computed in constant parallel time. For our proof, we introduce a new, generalized notion of hypergraph sunflowers and show how iterated applications of the color coding technique can sometimes be collapsed into a single application.

Approximability and hardness of geometric hitting set with axis-parallel rectangles

We show that geometric hitting set with axis-parallel rectangles is APX-hard even when all rectangles share a common point. We also show that geometric hitting set problem is APX-hard for several classes of objects given in [3] such as axis-parallel ellipses with a shared point, axis-parallel cubes sharing a common point, etc.Further, we give a polynomial time approximation scheme (PTAS) for the weighted hitting set problem with axis-parallel rectangles when all rectangles have integer side lengths bounded by a constant C. Finally, we show that the problem is NP-hard for rectangles of size 1×2 and 2×1 even when every rectangle intersects a given unit height horizontal strip.

Formula omitted]-hardness of geometric set cover and hitting set with rectangles containing a common point

We prove that both set cover and hitting set problems with points and axis-parallel rectangles are NP-hard even when (i) all rectangles share a common point and (ii) there are only two types of rectangles a×b and b×a for some positive integers a and b.