Raising The Bar For Vertex Cover: Fixed-parameter Tractability Above A Higher Guarantee

Parameterized approximation algorithms for weighted vertex cover

A pseudoforest is a graph whose connected components have at most one cycle. Let X be a pseudoforest modulator of graph G, i.e. a vertex subset of G such that $$G-X$$ is a pseudoforest. We show that Vertex Cover admits a polynomial kernel being parameterized by the size of the pseudoforest modulator. In other words, we provide a polynomial time algorithm that for an input graph G and integer k, outputs a graph $$G'$$ and integer $$k'$$, such that $$G'$$ has $$\mathcal {O}|X|^{12}$$ vertices and G has a vertex cover of size k if and only if $$G'$$ has vertex cover of size $$k'$$. We complement our findings by proving that there is no polynomial kernel for Vertex Cover parameterized by the size of a modulator to a mock forest a graph where no cycles share a vertex unless $$\text {NP}\subseteq \text {coNP/poly}$$. In particular, this also rules out polynomial kernels when parameterized by the size of a modulator to cactus graphs.

Many of the currently best-known approximation algorithms for NP-hard optimization problems are based on Linear Programming (LP) and Semi-definite Programming (SDP) relaxations. Given its power, this class of algorithms seems to contain the most favourable candidates for outperforming the current state-of-the-art approximation guarantees for NP-hard problems, for which there still exists a gap between the inapproximability results and the approximation guarantees that we know how to achieve in polynomial time. In this thesis, we address both the power and the limitations of these relaxations, as well as the connection between the shortcomings of these relaxations and the inapproximability of the underlying problem. In the first part, we study the limitations of LP relaxations of well-known graph problems such as the Vertex Cover problem and the Independent Set problem. We prove that any small LP relaxation for the aforementioned problems, cannot have an integrality gap strictly better than $2$ and $\omega(1)$, respectively. Furthermore, our lower bound for the Independent Set problem also holds for any SDP relaxation. Prior to our work, it was only known that such LP relaxations cannot have an integrality gap better than $1.5$ for the Vertex Cover Problem, and better than $2$ for the Independent Set problem. In the second part, we study the so-called knapsack cover inequalities that are used in the current best relaxations for numerous combinatorial optimization problems of covering type. In spite of their widespread use, these inequalities yield LP relaxations of exponential size, over which it is not known how to optimize exactly in polynomial time. We address this issue and obtain LP relaxations of quasi-polynomial size that are at least as strong as that given by the knapsack cover inequalities. In the last part, we show a close connection between structural hardness for k-partite graphs and tight inapproximability results for scheduling problems with precedence constraints. This connection is inspired by a family of integrality gap instances of a certain LP relaxation. Assuming the hardness of an optimization problem on k-partite graphs, we obtain a hardness of $2-\varepsilon$ for the problem of minimizing the makespan for scheduling with preemption on identical parallel machines, and a super constant inapproximability for the problem of scheduling on related parallel machines. Prior to this result, it was only known that the first problem does not admit a PTAS, and the second problem is NP-hard to approximate within a factor strictly better than 2, assuming the Unique Games Conjecture.

An efficient local search framework for the minimum weighted vertex cover problem

Mean analysis of an online algorithm for the vertex cover problem

The eternal vertex cover problem is a variant of the classical vertex cover problem where a set of guards on the vertices have to be dynamically reconfigured from one vertex cover to another in every round of an attacker-defender game. The minimum number of guards required to protect a graph from an infinite sequence of attacks is the eternal vertex cover number (evc) of the graph. It is known that, given a graph G and an integer k, checking whether \({\text {evc}}(G) \le k\) is NP-Hard. However, for any graph G, \({\text {mvc}}(G) \le {\text {evc}}(G) \le 2 {\text {mvc}}(G)\), where \({\text {mvc}}(G)\) is the minimum vertex cover number of G. Precise value of eternal vertex cover number is known only for certain very basic graph classes like trees, cycles and grids. Though a characterization is known for graphs for which \({\text {evc}}(G) = 2{\text {mvc}}(G)\), a characterization of graphs for which \({\text {evc}}(G) = {\text {mvc}}(G)\) remained open. Here, we achieve such a characterization for a class of graphs that includes chordal graphs and internally triangulated planar graphs. For some graph classes including biconnected chordal graphs, our characterization leads to a polynomial time algorithm to precisely determine \({\text {evc}}(G)\) and to determine a safe strategy of guard movement in each round of the game with \({\text {evc}}(G)\) guards.

The generalized vertex cover problem, an extension of classic minimum vertex cover problem, is an important NP-hard combinatorial optimization problem with a wide range of applications. The aim of this paper is to design an efficient local search algorithm with tabu strategy and perturbation mechanism to solve this problem. Firstly, we use tabu strategy to prevent the local search from immediately returning to a previously visited candidate solution and avoiding the cycling problem. Secondly, we propose the flip gain for each vertex, and then the tabu strategy is combined with the flip gain for vertex selecting. Finally, we apply a simple perturbation mechanism to help the search to escape from deep local optima and to bring diversification into the search. The experiments are carried on random instances with up to 1000 vertexes and 450,000 edges. The experimental results show that our algorithm performs better than a state-of-art algorithm in terms of both solution quality and computational efficiency in most instances.

We introduce two generalizations of core percolation in graphs to hypergraphs, related to the minimum hyperedge cover problem and the minimum vertex cover problem on hypergraphs, respectively. We offer analytical solutions of these two core percolations for uncorrelated random hypergraphs whose vertex degree and hyperedge cardinality distributions are arbitrary but have nondiverging moments. We find that for several real-world hypergraphs their two cores tend to be much smaller than those of their null models, suggesting that covering problems in those real-world hypergraphs can actually be solved in polynomial time.

To introduce our question and the parameterization, consider the classical Vertex Cover problem. In this problem, the input is a graph G on n vertices and a positive integer \(\ell \), and the goal is to find a vertex subset S of size at most \(\ell \) such that \(G-S\) is an independent set. Further, we want that G[S] is highly connected. That is, G[S] should be \(n-k\) edge-connected. Clearly, the problem is NP-complete, as substituting \(k=n-1\), we obtain the Connected Vertex Cover problem. A simple observation also shows that the problem admits an algorithm with running time \(n^{{\mathcal O}(k)}\). Since the problem is polynomial-time solvable for every fixed integer k, a natural parameter is the integer k. In all the problems we consider, the parameter is k, and the goal is to find a solution S of size at most \(\ell \), such that G[S] is \(n-k\) edge-connected and \(G-S\) satisfies a property. We show that this version of well-known problems such as Vertex Cover, Feedback Vertex Set, Odd Cycle Transversal and Multiway Cut admit an algorithm with running time \(f(k)\cdot n^{{\mathcal O}(1)}\), that is, they are FPT with the parameter k. One of our main subroutines to obtain these algorithms is an FPT algorithm for \(n-k\) edge connected Steiner Subgraph, which could be of an independent interest. Finally, we also show that such an algorithm is not possible for Multicut.Keywordssubset problemsparameterized algorithmsconnectivity

Parameterized algorithms for finding highly connected solution

The minimum vertex cover (MVC) problem is a well-studied NP-Complete problem and has various applications. In this paper, a new heuristic approach has been proposed to find the minimum vertex cover of a graph. The proposed algorithm has been tested on random graphs and BHOSLIB instances. The results have shown that the proposed algorithm can yield better solutions especially on dense graphs for solving the minimum vertex cover problem.

Multiple-interval graphs are a natural generalization of interval graphs where each vertex may have more then one interval associated with it. We initiate the study of optimization problems in multiple-interval graphs by considering three classical problems: MINIMUM VERTEX COVER, MINIMUM DOMINATING SET, and MAXIMUM CLIQUE. We describe applications for each one of these problems, and then proceed to discuss approximation algorithms for them. Our results can be summarized as follows: Let t be the number of intervals associated with each vertex in a given multiple-interval graph. For MINIMUM VERTEX COVER, we give a (2--1/t)-approximation algorithm which equals the best known ratio for 2t--1 bounded degree graphs. Since these graphs are known to be included in multiple-interval graphs with t intervals associated to each vertex, this ratio is in some sense tight. Following this, we give a t2-approximation algorithm for MINIMUM DOMINATING SET which adapts well to more general variants of the problem. We then proceed to prove that MAXIMUM CLIQUE is NP-complete for the case of t = 3, and provide a (t2--t+1)/2-approximation algorithm for the problem, using recent bounds proven for the so-called transversal number of t-interval families.

The vertex cover problem is a classic NP-complete problem for which the best worst-case approximation ratio is roughly 2. In this paper, we use a collection of simple reductions, each of which guarantees an approximation ratio of \(\frac{3}{2}\), to find approximate vertex covers for a large collection of test graphs from various sources. We explain these reductions and explore the interaction between them. These reductions are extremely fast and even though they, by themselves are not guaranteed to find a vertex cover, we manage to find a 3/2-approximate vertex cover for every single graph in our large collection of test examples.KeywordsApproximation RatioVertex CoverExtended NetworkOriginal GraphLinear Programming RelaxationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Map labeling is a classical problem in cartography and geographic information systems (GIS) that asks to place labels for area, line, and point features, with the goal to select and place the maximum number of independent, i.e., overlap-free, labels. A practically interesting case is point labeling with axis-parallel rectangular labels of common size. In a fully dynamic setting, at each time step, either a new label appears or an existing label disappears. Then, the challenge is to maintain a maximum cardinality subset of pairwise independent labels with sub-linear update time. Motivated by this, we study the maximal independent set ((MIS)) and maximum independent set (Max-IS) problems on fully dynamic (insertion/deletion model) sets of axis-parallel rectangles of two types -- (i) uniform height and width and (ii) uniform height and arbitrary width; both settings can be modeled as rectangle intersection graphs. We present the first deterministic algorithm for maintaining an MIS (and thus a 4-approximate Max-IS) of a dynamic set of uniform rectangles with polylogarithmic update time. This breaks the natural barrier of $\Omega(\Delta)$ update time (where $\Delta$ is the maximum degree in the graph) for \emph{vertex updates} presented by Assadi et al. (STOC 2018). We continue by investigating Max-IS and provide a series of deterministic dynamic approximation schemes with approximation factors between 2 and 4 and corresponding running-time trade-offs. We have implemented our algorithms and reported the results of an experimental comparison exploring the trade-off between solution quality and update time for synthetic and real-world map labeling instances.

We consider a natural generalization of the Partial Vertex Cover problem. Here an instance consists of a graph G = (V,E), a cost function c: V → ℤ + , a partition P 1, …, P r of the edge set E, and a parameter k i for each partition P i . The goal is to find a minimum cost set of vertices which cover at least k i edges from the partition P i . We call this the Partition-VC problem. In this paper, we give matching upper and lower bound on the approximability of this problem. Our algorithm is based on a novel LP relaxation for this problem. This LP relaxation is obtained by adding knapsack cover inequalities to a natural LP relaxation of the problem. We show that this LP has integrality gap of O(logr), where r is the number of sets in the partition of the edge set. We also extend our result to more general settings.