Showing papers on "Vertex cover published in 1999"

A 2-Approximation Algorithm for the Undirected Feedback Vertex Set Problem

Abstract: A feedback vertex set of a graph is a subset of vertices that contains at least one vertex from every cycle in the graph. The problem considered is that of finding a minimum feedback vertex set given a weighted and undirected graph. We present a simple and efficient approximation algorithm with performance ratio of at most 2, improving previous best bounds for either weighted or unweighted cases of the problem. Any further improvement on this bound, matching the best constant factor known for the vertex cover problem, is deemed challenging. The approximation principle, underlying the algorithm, is based on a generalized form of the classical local ratio theorem, originally developed for approximation of the vertex cover problem, and a more flexible style of its application.

Vertex Cover: Further Observations and Further Improvements

Abstract: Recently, there have been increasing interests and progresses in lowering the worst case time complexity for well-known NP-hard problems, in particular for the VERTEX COVER problem. In this paper, new properties for the VERTEX COVER problem are indicated and several new techniques are introduced, which lead to a simpler and improved algorithm of time complexity O(kn + 1:271kk2) for the problem. Our algorithm also induces improvement on previous algorithms for the INDEPENDENT SET problem on graphs of small degree.

On Some Tighter Inapproximability Results (Extended Abstract)

Abstract: We give a number of improved inapproximability results, including the best up to date explicit approximation thresholds for bounded occurence satisfiability problems like MAX-2SAT and E2-LIN-2, and the bounded degree graph problems, like MIS, Node Cover, and MAX CUT. We prove also for the first time inapproximability of the problem of Sorting by Reversals and display an explicit approximation threshold.

Algorithmic Aspects of the Core of Combinatorial Optimization Games

Abstract: We discuss an integer programming formulation for a class of cooperative games. We focus on algorithmic aspects of the core, one of the most important solution concepts in cooperative game theory. Central to our study is a simple but very useful observation that the core for this class is nonempty if and only if an associated linear program has an integer optimal solution. Based on this, we study the computational complexity and algorithms to answer important questions about the cores of various games on graphs, such as maximum flow, connectivity, maximum matching, minimum vertex cover, minimum edge cover, maximum independent set, and minimum coloring.

Hardness Results for the Power Range Assignmet Problem in Packet Radio Networks

Abstract: The minimum range assignment problem consists of assigning transmission ranges to the stations of a multi-hop packet radio network so as to minimize the total power consumption provided that the transmission range assigned to the stations ensures the strong connectivity of the network (i.e. each station can communicate with any other station by multi-hop transmission). The complexity of this optimization problem was studied by Kirousis, Kranakis, Krizanc, and Pelc (1997). In particular, they proved that, when the stations are located in a 3-dimensional Euclidean space, the problem is NP-hard and admits a 2-approximation algorithm. On the other hand, they left the complexity of the 2-dimensional case as an open problem.

On Approximation Properties of the Independent Set Problem for Low Degree Graphs

Abstract: The main problem we consider in this paper is the Independent Set problem for bounded degree graphs. It is shown that the problem remains MAX SNP-complete when the maximum degree is bounded by 3. Some related problems are also shown to be MAX SNP-complete at the lowest possible degree bounds. Next we study better poly-time approximation of the problem for degree 3 graphs, and improve the previously best ratio, 5/4, to arbitrarily close to 6/5. This result also provides improved poly-time approximation ratios, B+3/5+e, for odd degree B.

Wavelength conversion in optical networks

Abstract: In many models of optical routing, we are given a set of communication paths in a network, and we must assign a wavelength to each path so that paths sharing an edge receive different wavelengths. The goal is to assign as few wavelengths as possible in order to make as efficient use as possible of the optical bandwidth. A lot of work in the area of optical networks has considered the use of wavelength converters: if a node of a network contains a wavelength converter, any path that passes through this node may change its wavelength. Having converters at some of the nodes can reduce the number of wavelengths required for routing, down to the following natural congestion bound: even with converters, we will always need at least as many wavelengths as the maximum number of paths sharing a single edge. Thus Wilfong and Winkler defined a set S of nodes in a network to be sufficient if, placing converters at the nodes in S, every set of paths can be routed with a number of wavelengths equal to its congestion bound. They showed that finding a sufficient set of minimum size is NP-complete, even in planar graphs.In this paper, we provide a polynomial-time algorithm to find a sufficient set for an arbitrary directed network whose size is within a factor of 2 of minimum. We also observe that improving on the factor of 2 would lead to a corresponding improvement for the vertex cover problem. For the case of planar graphs, we provide a polynomial-time approximation scheme. The algorithms are based on connections between the minimum sufficient set problem and the undirected feedback vertex set problem. In particular, as a component of the algorithm on planar graphs, we develop the first polynomial-time approximation scheme for the undirected feedback vertex set problem in planar graphs, a result that we feel to be of interest in its own right.

Upper bounds for Vertex Cover further improved

Abstract: The problem instance of Vertex Cover consists of an undirected graph G = (V, E) and a positive integer k, the question is whether there exists a subset C C V of vertices such that each edge in E has at least one of its endpoints in C with |C| ≤ k. We improve two recent worst case upper bounds for Vertex Cover. First, Balasubramanian et al. showed that Vertex Cover can be solved in time O(kn + 1.32472 k k 2 ), where n is the number of vertices in G. Afterwards, Downey et al. improved this to O(kn + 1.31951 k k 2 ). Bringing the exponential base significantly below 1.3, we present the new upper bound O(kn +1.29175 k k 2 ).

An improved fixed parameter tractable algorithm for vertex cover

Abstract: Given a graph G V E Vertex Cover asks for a smallest subset V V such that for each edge a b in G a V or b V We present an improved xed parameter tractable algorithm when the problem is parameterized by the size k of V The algorithm has a complexity of O kn maxf k k kg We improve the klam value by to k

Using homogenous weights for approximating the partial cover problem

Abstract: In this paper we consider the natural generalizations of two fundamental problems, the Set-Cover problem and the Min-Knapsack problem. We are given a hypergraph, each vertex of which has a nonnegative weight, and each edge of which has a nonnegative length. For a given threshold ??, our objective is to find a subset of the vertices with minimum total cost, such that at least a length of ?? of the edges is covered. This problem is called the partial set cover problem. We present an O(|V|2+|H|)-time, ?E-approximation algorithm for this problem, where ?E?2 is an upper bound on the edge cardinality of the hypergraph and |H| is the size of the hypergraph (i.e., the sum of all its edges cardinalities). The special case where ?E=2 is called the partial vertex cover problem. For this problem a 2-approximation was previously known, however, the time complexity of our solution, i.e., O(|V|2), is a dramatic improvement.We show that if the weights are homogeneous (i.e., proportional to the potential coverage of the sets) then any minimal cover is a good approximation. Now, using the local-ratio technique, it is sufficient to repeatedly subtract a homogeneous weight function from the given weight function.

New Upper Bounds for MaxSat

Abstract: Given a boolean formula F in conjunctive normal form and an integer k, is there a truth assignment satisfying at least k clauses? This is the decision version of the Maximum Satisfiability (MaxSat) problem we study in this paper. We improve upper bounds on the worst case running time for MaxSat. First, Cai and Chen showed that MaxSat can be solved in time |F|2O(k) when the clause size is bounded by a constant. Imposing no restrictions on clause size, Mahajan and Raman and, independently, Dantsin et al. improved this to O(|F|Φk), where Φ ≅ 1:6181 is the golden ratio. We present an algorithm running in time O(|F|1:3995k). The result extends to finding an optimal assignment and has several applications, in particular, for parameterized complexity and approximation algorithms. Moreover, if F has K clauses, we can find an optimal assignment in O(|F|1:3972K) steps and in O(1:1279|F|) steps, respectively. These are the fastest algorithm in the number of clauses and the length of the formula, respectively.

A Min-Max Theorem on Feedback Vertex Sets

Abstract: We establish a necessary and sufficient condition for the linear system {x : Hx ≥ e,x ≥ 0} associated with a bipartite tournament to be TDI, where H is the cycle-vertex incidence matrix and e is the all-one vector. The consequence is a min-max relation on packing and covering cycles, together with strongly polynomial time algorithms for the feedback vertex set problem and the cycle packing problem on the corresponding bipartite tournaments. In addition, we show that the feedback vertex set problem on general bipartite tournaments is NP-complete and approximable within 3.5 based on the max-min theorem.

Optimal placement of wavelength converters in trees and trees of rings

Abstract: In wavelength-routed optical networks, wavelength converters can potentially reduce the requirement on the number of wavelengths. The problem of placing a minimum number of wavelength converters in a WDM network so that any routing can be satisfied using no more wavelengths than if there were wavelength converters at every node was raised in Wilfong et al. (1998) and shown to be NP-complete in general WDM networks. Recently, it was proved in Kleinberg et al. (1999) that this problem is as hard as the well-known minimum vertex cover problem. In this paper, we further their study in two topologies that are of more practical concrete relevance to the telecommunications industry: trees and tree of rings. We show that the optimal wavelength converter placement problem in these two practical topologies are tractable. Efficient polynomial-time algorithms are presented.

Approximation algorithms for clustering problems

Abstract: Clustering is a ubiquitous problem that arises in many applications in different fields such as data mining, image processing, machine learning, and bioinformatics. Clustering problems have been extensively studied as optimization problems with various objective functions in the Operations Research and Computer Science literature. We focus on a class of objective functions more commonly referred to as facility location problems. These problems arise in a wide range of applications such as, plant or warehouse location problems, cache placement problems, and network design problems where the costs obey economies of scale. In the simplest of these problems, the uncapacitated facility location (UFL) problem, we want to open facilities at some subset of a given set of locations and assign each client in a given set D to an open facility so as to minimize the sum of the facility opening costs and client assignment costs. This a very well-studied problem; however it fails to address many of the requirements of real applications. In this thesis we consider various problems that build upon UFL and capture additional issues that arise in applications such as, uncertainties in the data, clients with different service needs, and facilities with interconnectivity requirements. By focusing initially on facility location problems in these new models, we develop new algorithmic techniques that will find application in a wide range of settings. We consider a widely used paradigm in stochastic programming to model settings where the underlying data, for example, the locations or demands of the clients, may be uncertain: the 2-stage with recourse model that involves making some initial decisions, observing additional information, and then augmenting the initial decisions, if necessary, by taking recourse actions. We present a randomized polynomial time algorithm that solves a large class of 2-stage stochastic linear programs (LPs) to near-optimality with high probability. We exploit this tool to devise the first approximation algorithms for various 2-stage discrete stochastic problems such as the stochastic versions of the set cover, vertex cover, and facility location problems, when the underlying random data is only given as a “black box” and no restrictions are placed on the cost structure. We introduce the facility location with service installation costs problem to model applications involving clients with different service requirements. if the service requested by it has been installed at the facility (incurring a service installation cost). The connected facility location problem captures settings where the open facilities want to communicate with each other or with a central authority; we model this by requiring that the open facilities be interconnected by a Steiner tree. We give intuitive and efficient algorithms for both these problems. We use these algorithms to obtain approximation algorithms for the κ-median variants of these problems, where in addition to all of the constraints of the problem, a bound of κ is imposed on the number of facilities that may be opened.

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.

Stability of Approximation Algorithms for Hard Optimization Problems

Abstract: To specify the set of tractable (practically solvable) computing problems is one of the few main research tasks of theoretical computer science Because of this the investigation of the possibility or the impossibility to efficiently compute approximations of hard optimization problems becomes one of the central and most fruitful areas of recent algorithm and complexity theory The current point of view is that optimization problems are considered to be tractable if there exist polynomial-time randomized approximation algorithms that solve them with a reasonable approximation ratio If a optimization problem does not admit such a polynomial-time algorithm, then the problem is considered to be not tractableThe main goal of this paper is to relativize this specification of tractability The main reason for this attempt is that we consider the requirement for the tractability to be strong because of the definition of the complexity as the "worst-case" complexity This definition is also related to the approximation ratio of approximation algorithms and then an optimization problem is considered to be intractable because some subset of problem instances is hard But in the practice we often have the situation that the hard problem instances do not occur The general idea of this paper is to try to partition the set of all problem instances of a hard optimization problem into a (possibly infinite) spectrum of subclasses according to their polynomial-time approximability Searching for a method enabling such a fine problem analysis (classification of problem instances) we introduce the concept of stability of approximation To show that the application of this concept may lead to a "fine" characterization of the hardness of particular problem instances we consider the traveling salesperson problem and the knapsack problem

Upper bounds for vertex cover further improved

Abstract: The problem instance of Vertex Cover consists of an undirected graph G = (V, E) and a positive integer k, the question is whether there exists a subset C ⊆ V of vertices such that each edge in E has at least one of its endpoints in C with |C| ≤ k. We improve two recent worst case upper bounds for Vertex Cover. First, Balasubramanian et al. showed that Vertex Cover can be solved in time O(kn+1:32472kk2), where n is the number of vertices in G. Afterwards, Downey et al. improved this to O(kn+1:31951kk2). Bringing the exponential base significantly below 1:3, we present the new upper bound O(kn+1:29175kk2).

An approximation of the minimum vertex cover in a graph

Abstract: For a given undirected graphG withn vertices andm edges, we present an approximation algorithm for the minimum vertex cover problem. Our algorithm finds a vertex cover within $$2 - \frac{{8m}}{{13n^2 + 8m}}$$ of the optimal size inO(nm) time.

On the complexity of approximating colored-graph problems extended abstract

Abstract: In this paper we prove explicit lower bounds on the approximability of some graph problems restricted to instances which are already colored with a constant number of colors. As far as we know, this is the first time these problems are explicitily defined and analyzed. This allows us to drastically improve the previously known inapproximability results which were mainly a consequence of the analysis of bounded-degree graph problems. Moreover, we apply one of these results to obtain new lower bounds on the approximabiluty of the minimum delay schedule problem on store-and-forward networks of bounded diameter. Finally, we propose a generalization of our analysis of the complexity of approximating colored-graph problems to the complexity of approximating approximated optimization problems.

An Efficient Exact Algorithm for Constraint Bipartite Vertex Cover

Abstract: The "Constraint Bipartite Vertex Cover" problem (CBVC for short) is: given a bipartite graph G with n vertices and two positive integers k1, k2, is there a vertex cover taking at most k1 vertices from one and at most k2 vertices from the other vertex set of G? CBVC is NP-complete. It formalizes the spare allocation problem for reconfigurable axrays, an important problem from VLSI manufacturing. We provide the first nontrivial so-called "fixed parameter" algorithm for CBVC, running in time O(1.3999k1+k2 + (k1+k2n). Our algorithm is efficient for small values of k1 and k2, as occurring in applications.

Efficient Algorithms for Integer Programs with Two Variables per Constraint

Abstract: Given a bounded integer program with n variables and m constraints each with 2 variables we present an O(mU) time and O(m) space feasibility algorithm for such integer programs (where U is the maximal variable range size). We show that with the same complexity we can find an optimal solution for the positively weighted minimization problem for monotone systems. Using the localratio technique we develop an O(nmU) time and O(m) space 2-approximation algorithm for the positively weighted minimization problem for the general case. We further generalize all results to non linear constraints (called axis-convex constraints) and to non linear (but monotone) weight functions.Our algorithms are not only better in complexity than other known algorithms, but they are also considerably simpler, and contribute to the understanding of these very fundamental problems.

Optimal vertex adjustment method using a Viterbi algorithm for vertex-based shape coding

Abstract: An optimal vertex adjustment method is presented for vertex-based shape coding. The method optimally shifts the vertices, selected by a conventional vertex selection method, within the window that satisfies the allowable maximum distortion using a Viterbi algorithm. Experimental results show that the proposed method reduces the approximation distortion of the contour by /spl sim/30% for the same rate over a wide range of distortion, compared with the conventional method.

Designing regular graphs with the use of evolutionary algorithms

Abstract: The paper deals with a fundamental problem arising in the design of optimal networks-the maximization of the number of spanning trees. To make the problem computationally tractable, we consider a class of regular graphs. The problem is solved with the use of the evolutionary algorithm and compared to the 2-opt method. The problem-specific genetic operators are introduced, Various experiments with different graph structures have been performed, the results are reported and discussed. The influence of introducing some preliminary knowledge about the problem on the algorithm effectiveness is studied.