Showing papers on "Vertex cover published in 2000"

The Capacitated K -Center Problem

Abstract: The capacitated K-center problem is a basic facility location problem, where we are asked to locate K facilities in a graph and to assign vertices to facilities, so as to minimize the maximum distance from a vertex to the facility to which it is assigned. Moreover, each facility may be assigned at most L vertices. This problem is known to be NP-hard. We give polynomial time approximation algorithms for two different versions of this problem that achieve approximation factors of 5 and 6. We also study some generalizations of this problem.

Number of guards needed by a museum: a phase transition in vertex covering of random graphs.

Abstract: In this Letter we study the $\mathrm{NP}$-complete vertex cover problem on finite connectivity random graphs When the allowed size of the cover set is decreased, a discontinuous transition in solvability and typical-case complexity occurs This transition is characterized by means of exact numerical simulations as well as by analytical replica calculations The replica symmetric phase diagram is in excellent agreement with numerical findings up to average connectivity $e$, where replica symmetry becomes locally unstable

PBS: A Backtrack-Search Pseudo-Boolean Solver and Optimizer

Abstract: Optimized solvers for the Boolean Satisfiability (SAT) problem [5, 14, 15, 17, 19, 23, 24] found many applications in areas such as hardware and software verification, FPGA routing, planning in AI, etc. Further uses are complicated by the need to express “counting constraints” in conjunctive normal form (CNF). Expressing such constraints by pure CNF leads to more complex SAT instances. Alternatively, those constraints can be handled by Integer Linear Programming (ILP), but off-the-shelf ILP solvers tend to ignore the Boolean nature of 0-1 variables. This work attempts to generalize recent highly successful SAT techniques to new applications. First, we extend the basic Davis-Putnam framework to handle counting constraints and apply it to solve routing problems. Our implementation outperforms previously reported solvers for the satisfiability with “pseudo-Boolean” constraints and shows significant speed-up over best SAT solvers when such constraints are translated into CNF,. Additionally, we solve instances of the Max-ONEs optimization problem which seeks to maximize the number of “true” values over all satisfying assignments. This, and the related Min-ONEs problem are important due to reductions from Max-Clique and Min Vertex Cover. Our experimental results for various benchmarks are superior to all approaches reported earlier.

One for the Price of Two: a Unified Approach for Approximating Covering Problems

Abstract: We present a simple and unified approach for developing and analyzing approximation algorithms for covering problems. We illustrate this on approximation algorithms for the following problems: Vertex Cover, Set Cover, Feedback Vertex Set, Generalized Steiner Forest, and related problems. The main idea can be phrased as follows: iteratively, pay two dollars (at most) to reduce the total optimum by one dollar (at least), so the rate of payment is no more than twice the rate of the optimum reduction. This implies a total payment (i.e., approximation cost) ≤ twice the optimum cost. Our main contribution is based on a formal definition for covering problems, which includes all the above fundamental problems and others. We further extend the Bafna et al. extension of the Local-Ratio theorem. Our extension eventually yields a short generic r -approximation algorithm which can generate most known approximation algorithms for most covering problems. Another extension of the Local-Ratio theorem to randomized algorithms gives a simple proof of Pitt's randomized approximation for Vertex Cover. Using this approach, we develop a modified greedy algorithm, which for Vertex Cover gives an expected performance ratio ≤ 2 .

On Efficient Fixed Parameter Algorithms for WEIGHTED VERTEX COVER

Abstract: We investigate the fixed parameter complexity of one of the most popular problems in combinatorial optimization, WEIGHTED VERTEX COVER. Given a graph G = (V, E), a weight function ω: V → R+, and k ∈ R+, WEIGHETD VERTEX COVER (WVC for short) asks for a subset C of vertices in V of weight at most k such that every edge of G has at least one endpoint in C. WVC and its variants have all been shown to be NP-complete. We show that, when restricting the range of ω to positive integers, the so-called INTEGER-WVC can be solved as fast as unweighted VERTEX COVER. Our main result is that if the range of ω is restricted to positive reals ≥ 1, then so-called REAL-WVC can be solved in time O(1.3954k + k|V|). If we modify the problem in such a way that k is not the weight of the vertex cover we are looking for, but the number of vertices in a minimum weight vertex cover, then the same running time can be obtained. If the weights are arbitrary (referred to by GENERAL-WVC), however, the problem is not fixed parameter tractable unless P = NP.

Approximation algorithms for submodular set cover with applications

Abstract: The main problem considered is submodular set cover, the problem of minimizing a linear function under a nondecreasing submodular constraint, which generalizes both wellknown set cover and minimum matroid base problems. The problem is NP-hard, and two natural greedy heuristics are introduced along with analysis of their performance. As applications of these heuristics we consider various special cases of submodular set cover, including partial cover variants of set cover and vertex cover, and node-deletion problems for hereditary and matroidal properties. An approximation bound derived for each of them is either matching or generalizing the best existing bounds. key words: submodular function, set cover, approximation algorithms, greedy heuristics

Improved approximation algorithms for the vertex cover problem in graphs and hypergraphs

Abstract: We obtain improved algorithms for finding small vertex covers in bounded degree graphs and hypergraphs. We use semidefinite programming to relax the problems and introduce new rounding techniques for these relaxations. On graphs with maximum degree at most ∆, the algorithm achieves a performance ratio of 2 − (1 − o(1)) 2 ln ln∆ ln∆ for large ∆, which improves the previously known ratio of 2− log∆+O(1) ∆ obtained by Halldorsson and Radhakrishnan. Using similar techniques, we also present improved approximations for the vertex cover problem in hypergraphs. For k-uniform hypergraphs with n vertices, we achieve a ratio of k − (1 − o(1)) k ln lnn lnn for large n, and for k-uniform hypergraphs with maximum degree at most ∆ the algorithm achieves a ratio of k − (1 − o(1)) k(k−1) ln ln∆ ln∆ for large ∆. These results considerably improve the previous best ratio of k(1 − c/∆ 1 k−1 ) for bounded degree k-uniform hypergraphs, and k(1 − c/n k−1 k ) for general k-uniform hypergraphs, both obtained by Krivelevich. Using similar techniques, we also obtain an approximation algorithm for the weighted independent set problem, matching a recent result of Halldorsson.

A Polynomial Approximation Algorithm for the Minimum Fill-In Problem

Abstract: In the minimum fill-in problem, one wishes to find a set of edges of smallest size, whose addition to a given graph will make it chordal. The problem has important applications in numerical algebra and has been studied intensively since the 1970s. We give the first polynomial approximation algorithm for the problem. Our algorithm constructs a triangulation whose size is at most eight times the optimum size squared. The algorithm builds on the recent parameterized algorithm of Kaplan, Shamir, and Tarjan for the same problem. For bounded degree graphs we give a polynomial approximation algorithm with a polylogarithmic approximation ratio. We also improve the parameterized algorithm.

Coordinatized Kernels and Catalytic Reductions: An Improved FPT Algorithm for Max Leaf Spanning Tree and Other Problems

Abstract: We describe some new, simple and apparently general methods for designing FPT algorithms, and illustrate how these can be used to obtain a significantly improved FPT algorithm for the MAXIMUM LEAF SPANNING TREE problem. Furthermore, we sketch how the methods can be applied to a number of other well-known problems, including the parametric dual of DOMINATING SET (also known as NONBLOCKER), MATRIX DOMINATION, EDGE DOMINATING SET, and FEEDBACK VERTEX SET FOR UNDIRECTED GRAPHS. The main payoffs of these new methods are in improved functions f(k) in the FPT running times, and in general systematic approaches that seem to apply to a wide variety of problems.

An 8-Approximation Algorithm for the Subset Feedback Vertex Set Problem

Abstract: We present an 8-approximation algorithm for the problem of finding a minimum weight subset feedback vertex set (or subset-fvs, in short). The input in this problem consists of an undirected graph G=(V,E) with vertex weights c(v) and a subset of vertices S called special vertices. A cycle is called interesting if it contains at least one special vertex. A subset of vertices is called a subset-fvs with respect to S if it intersects every interesting cycle. The goal is to find a minimum weight subset-fvs. The best previous algorithm for the general case provided only a logarithmic approximation factor. The minimum weight subset-fvs problem generalizes two NP-complete problems: the minimum weight feedback vertex set problem in undirected graphs and the minimum weight multiway vertex cut problem. The main tool that we use in our algorithm and its analysis is a new version of multicommodity flow, which we call relaxed multicommodity flow. Relaxed multicommodity flow is a hybrid of multicommodity flow and multiterminal flow.

Totally balanced combinatorial optimization games

Abstract: Combinatorial optimization games deal with cooperative games for which the value of every subset of players is obtained by solving a combinatorial optimization problem on the resources collectively owned by this subset. A solution of the game is in the core if no subset of players is able to gain advantage by breaking away from this collective decision of all players. The game is totally balanced if and only if the core is non-empty for every induced subgame of it.¶We study the total balancedness of several combinatorial optimization games in this paper. For a class of the partition game [5], we have a complete characterization for the total balancedness. For the packing and covering games [3], we completely clarify the relationship between the related primal/dual linear programs for the corresponding games to be totally balanced. Our work opens up the question of fully characterizing the combinatorial structures of totally balanced packing and covering games, for which we present some interesting examples: the totally balanced matching, vertex cover, and minimum coloring games.

Self-stabilizing Vertex Coloring of Arbitrary Graphs

Abstract: A self-stabilizing algorithm, regardless of the initial system state, converges in finite time to a set of states that satisfy a legitimacy predicate without the need for explicit exception handler of backward recovery. The vertex coloration problem consists in ensuring that every node in the system has a color that is different from any of its neighbors. We provide three self-stabilizing solutions to the vertex coloration problem under unfair scheduling that are based on a greedy technique. We use at most $B+1$ different colors (in complete graphs), where $B$ is the graph degree, and ensure stabilization within $O(n\times B)$ processor atomic steps. Two of our algorithms deal with uniform networks where nodes do not have identifiers. Our solutions lead to directed acyclic orientation and maximal independent set construction at no additional cost.

Improvement on Vertex Cover for Low-Degree Graphs

Abstract: We present an improved algorithm for the Vertex Cover problem on graphs of degree bounded by 3 (3DVC). We show that the 3DVC problem can be solved in time O(1.2192 k k), where k is the number of vertices in a minimum vertex cover of the graph. Our algorithm also improves previous algorithms on the Independent Set problem on graphs with degree bounded by 3.

A 2 1/10-Approximation Algorithm for a Generalization of the Weighted Edge-Dominating Set Problem

Abstract: We study the approximability of the weighted edge-dominating set problem. Although even the unweighted case is NP-Complete, in this case a solution of size at most twice the minimum can be efficiently computed due to its close relationship with minimum maximal matching; however, in the weighted case such a nice relationship is not known to exist. In this paper, after showing that weighted edge domination is as hard to approximate as the well studied weighted vertex cover problem, we consider a natural strategy, reducing edge-dominating set to edge cover. Our main result is a simple 2 1/10 -approximation algorithm for the weighted edge-dominating set problem, improving the existing ratio, due to a simple reduction to weighted vertex cover, of 2rWVC, where rWVC is the approximation guarantee of any polynomial-time weighted vertex cover algorithm. The best value of rWVC currently stands at 2- log log |V|/2 log |V|. Furthermore we establish that the factor of 2 1/10 is tight in the sense that it coincides with the integrality gap incurred by a natural linear programming relaxation of the problem.

Approximation Algorithms for the Label-CoverMAX and Red-Blue Set Cover Problems

Abstract: This paper presents approximation algorithms for two problems. First, a randomized algorithm guaranteeing approximation ratio √n with high probability is proposed for the Max-Rep problem of [Kor98], or the Label-CoverMAX problem (cf. [Hoc95]), where n is the number of vertices in the graph. This algorithm is then generalized into a 4√n- ratio algorithm for the nonuniform version of the problem. Secondly, it is shown that the Red-Blue Set Cover problem of [CDKM00] can be approximated with ratio 2√n log β, where n is the number of sets and β is the number of blue elements. Both algorithms can be adapted to the weighted variants of the respective problems, yielding the same approximation ratios.

On Some Approximation Algorithms for Dense Vertex Cover Problem

Abstract: In this work we study the performance of some algorithms approximating the vertex cover problem (VCP) with respect to density constraints. We consider a modification of the algorithm of Karpinski and Zelikovsky [9] for the weighted VCP and prove a performance guarantee for this algorithm in terms of a new density parameter. Also we investigate the hardness of approximation for the VCP on everywhere e-dense graph and show that it is NP-hard to approximate within a factor less than.

On the Hardness of Approximating NP Witnesses

Abstract: The search version for NP-complete combinatorial optimization problems asks for finding a solution of optimal value. Such a solution is called a witness. We follow a recent paper by Kumar and Sivakumar, and study a relatively new notion of approximate solutions that ignores the value of a solution and instead considers its syntactic representation (under some standard encoding scheme). The results that we present are of a negative nature. We show that for many of the well known NP-complete problems (such as 3-SAT, CLIQUE, 3-COLORING, SET COVER) it is NP-hard to produce a solution whose Hamming distance from an optimal solution is substantially closer than what one would obtain by just taking a random solution. In fact, we have been able to show similar results for most of Karp’s 21 original NP-complete problems. (At the moment, our results are not tight only for UNDIRECTED HAMILTONIAN CYCLE and FEEDBACK EDGE SET.)

Modeling and analysis of genetic algorithm with tournament selection

Abstract: In this paper we propose a mathematical model of a simplified version of genetic algorithm (GA) based on mutation and tournament selection and obtain upper and lower bounds on expected proportion of the individuals with the fitness above certain threshold. As an illustration we consider a GA optimizing the bit-counting function and a GA for the vertex cover problem on graphs of a special structure. The theoretical estimates obtained are compared with experimental results.

On Approximating Minimum Vertex Cover for Graphs with Perfect Matching

Abstract: It has been a challenging open problem whether there is a polynomial time approximation algorithm for the VERTEX COVER problem whose approximation ratio is bounded by a constant less than 2. In this paper, we study the VERTEX COVER problem on graphs with perfect matching (shortly, VC-PM). We show that if the VC-PM problem has a polynomial time approximation algorithm with approximation ratio bounded by a constant less than 2, then so does the VERTEX COVER problem on general graphs. Approximation algorithms for VC-PM are developed, which induce improvements over previously known algorithms on sparse graphs. For example, for graphs of average degree 5, the approximation ratio of our algorithm is 1.414, compared with the previously best ratio 1.615 by Halldorsson and Radhakrishnan.

A Generalized Cover Time for Random Walks on Graphs

Abstract: Given a random walk on a graph, the cover time is the first time (number of steps) that every vertex has been hit (covered) by the walk. Define the marking time for the walk as follows. When the walk reaches vertex vi, a coin is flipped and with probability pi the vertex is marked (or colored). We study the time that every vertex is marked. (When all the pi’s are equal to 1, this gives the usual cover time problem.) General formulas are given for the marking time of a graph. Connections are made with the generalized coupon collector’s problem. Asymptotics for small p i ’s are given. Techniques used include combinatorics of random walks, theory of determinants, analysis and probabilistic considerations.

Solving vertex covering problems using hybrid genetic algorithms

Abstract: Presents a hybrid genetic algorithm for the vertex cover problems in which scan-repair and local improvement techniques are used for local optimization. With the hybrid approach, genetic algorithms are used to perform. Global exploration among a population, while neighborhood search methods are used to perform local exploitation around the chromosomes. The experimental results indicate that hybrid genetic algorithms can obtain solutions of excellent quality of problem instances with different size. The results are compared among the three algorithms presented. The pure genetic algorithms are outperformed by the neighborhood search heuristics procedures combined with genetic algorithms.

The Minor-Order Obstructions for The Graphs of Vertex Cover Six

Abstract: We provide for the first time a complete list of forbidden minors (obstructions) for the family of graphs with vertex cover 6. This paper shows how to limit both the search space of graphs and improve the efficiency of an obstruction checking algorithm when restricted to k–Vertex Cover graph families. In particular, our upper bounds 2k + 1 (2k + 2) on the maximum number of vertices for connected (disconnected) obstructions are shown to be sharp for all k > 0.

Improved approximations for tour and tree covers

Abstract: A tree (tour) cover of an edge-weighted graph is a set of edges which forms a tree (closed walk) and covers every other edge in the graph. Arkin, Halldorsson and Hassin (Information Processing Letters 47:275- 282, 1993) give approximation algorithms with ratio 3.55 (tree cover) and 5.5 (tour cover). We present algorithms with worst-case ratio 3 for both problems.

On Approximability of the Independent/Connected Edge Dominating Set Problems

Abstract: We investigate polynomial-time approximability of the problems related to edge dominating sets of graphs. When edges are unit-weighted, the edge dominating set problem is polynomially equivalent to the minimum maximal matching problem, in either exact or approximate computation, and the former problem was recently found to be approximable within a factor of 2 even with arbitrary weights. It will be shown, in contrast with this, that the minimum weight maximal matching problem cannot be approximated within any polynomially computable factor unless P=NP. The connected edge dominating set problem and the connected vertex cover problem also have the same approximability when edges/vertices are unit-weighted, and the former problem is known to be approximable, even with general edge weights, within a factor of 3.55. We will show that, when general weights are allowed, 1) the connected edge dominating set problem can be approximated within a factor of 3 + Ɛ, and 2) the connected vertex cover problem is approximable within a factor of ln n + 3 but cannot be within (1 - Ɛ) ln n for any Ɛ > 0 unless NP ⊂ DTIME(nO(log log n)).

Instant recognition of polynominal time solvability, half integrality and 2-approximations

Abstract: We describe here a technique applicable to integer programming problems which we refer to as IP2. An IP2 problem has linear constraints where each constraint has up to three variables with nonzero coefficients, and one of the three variables appear in that constraint only. The technique is used to identify either polynomial time solvability of the problem in time that is required to solve a minimum cut problem on an associated graph; or in case the problem is NP-hard the technique is used to generate superoptimal solution all components of which are integer multiple of \( \tfrac{1} {2} \) In some of the latter cases, for minimization problems, the half integral solution may be rounded to a feasible solution that is provably within a factor of 2 of the optimum.