A fundamental facility location problem is to choose the location of facilities, such as industrial plants and warehouses, to minimize the cost of satisfying the demand for some commodity. There are associated costs for locating the facilities, as well as transportation costs for distributing the commodities. We assume that the transportation costs form a metric. This problem is commonly referred to as the uncapacitated facility location (UFL) problem. Applications to bank account location and clustering, as well as many related pieces of work, are discussed by Cornuejols, Nemhauser and Wolsey [2]. Recently, the first constant factor approximation algorithm for this problem was obtained by Shmoys, Tardos and Aardal [16]. We show that a simple greedy heuristic combined with the algorithm by Shmoys, Tardos and Aardal, can be used to obtain an approximation guarantee of 2.408. We discuss a few variants of the problem, demonstrating better approximation factors for restricted versions of the problem. We also show that the problem is Max SNP-hard. However, the inapproximability constants derived from the Max SNP hardness are very close to one. By relating this problem to Set Cover, we prove a lower bound of 1.463 on the best possible approximation ratio assuming NP / ∈ DT IME[n O(log log n)]. 1

We present an algorithm that constructively produces a solution to the k-dominating set problem for planar graphs in time O(c . To obtain this result, we show that the treewidth of a planar graph with domination number (G) is O( (G)), and that such a tree decomposition can be found in O( (G)n) time. The same technique can be used to show that the k-face cover problem ( find a size k set of faces that cover all vertices of a given plane graph) can be solved in O(c n) time, where c 1 = 3 and k is the size of the face cover set. Similar results can be obtained in the planar case for some variants of k-dominating set, e.g., k-independent dominating set and k-weighted dominating set.

We present a multi-scale layout algorithm for the aesthetic drawing of undirected graphs with straight-line edges. The algorithm is extremely fast, and is capable of drawing graphs that are substantially larger than those we have encountered in prior work. For example, the paper contains a drawing of a graph with over 15,000 vertices. Also we achieve "nice" drawings of 1000 vertex graphs in about 1 second. The proposed algorithm embodies a new multi-scale scheme for drawing graphs, which was motivated by the earlier multi-scale algorithm of Hadany and Harel [HH99]. In principle, it could significantly improve the speed of essentially any force-directed method (regardless of that method's ability of drawing weighted graphs or the continuity of its cost-function).

We present a novel approach to the aesthetic drawing of undirected graphs. The method has two phases: first embed the graph in a very high dimension and then project it into the 2-D plane using PCA. Experiments we have carried out show the ability of the method to draw graphs of 10 nodes in few seconds. The new method appears to have several advantages over classical methods, including a significantly better running time, a useful inherent capability to exhibit the graph in various dimensions, and an effective means for interactive exploration of large graphs.

We present an algorithm that constructively produces a solution to the k-dominating set problem for planar graphs in time O(c √ kn), where c = 36√34. To obtain this result, we show that the treewidth of a planar graph with domination number γ(G) is O ( � γ(G)), and that such a tree decomposition can be found in O ( � γ(G)n) time. The same technique can be used to show that the k-face cover problem (find a size k set of faces that cover all vertices of a given plane graph) can be solved √ k in O(c1 n + n2) time, where c1 = 236√34 and k is the size of the face cover set. Similar results can be obtained in the planar case for some variants of k-dominating set, e.g., k-independent dominating set and k-weighted dominating set. Keywords. NP-complete problems, fixed parameter tractability, planar graphs, planar dominating set, face cover, outerplanarity, treewidth.

. We study the Minimum Label Spanning Tree Problem. In this problem, we are given an undirected graph whose edges are labeled with colors. The goal is to nd a spanning tree which uses as least dierent colors as possible. We present an approximation algorithm with logarithmic performance guarantee. On the other hand, our hardness results show that the problem cannot be approximated within a constant factor. 1. Introduction Problems of nding spanning trees optimizing some measure have been extensively studied in literature. Typical measures include the weight or the diameter of the tree. In communication network design, it is also often desirable to obtain a tree that is \most uniform" in some specied sense. Motivated by this observation, in [CL97] Chang and Leu introduce the minimum label spanning tree problem. In this problem, we are given a graph with colored edges, and one seeks to nd a spanning tree with the least number of colors possible. Chang and Leu proved the NP...

Speeding up approximate pattern matching is a line of research in stringology since the 80’s. Practically fast approaches belong to the class of filtration algorithms, in which text regions dissimilar to the pattern are first excluded, and the remaining regions are then compared to the pattern by dynamic programming. Among the conditions used to test similarity between the regions and the pattern, many require a minimum number of common substrings between them. When only substitutions are taken into account for measuring dissimilarity, counting spaced subwords instead of substrings improves the filtration efficiency. However, a preprocessing step is required to design one or more patterns, called spaced seeds (or gapped seeds), for the subwords, depending on the search parameters. Two distinct lines of research appear the literature: one with probabilistic formulations of seed design problems, in which one wishes for instance to compute a seed with the highest probability to detect the desired similarities (lossy filtration), a second line with combinatorial formulations, where the goal is to find a seed that detects all or a maximum number

We describe exact algorithms that provide new upper bounds for the Maximum Satisfiability problem (MaxSat). We prove that MaxSat can be solved in time O(|F | · 1.3972 K), where |F | is the length of a formula F in conjunctive normal form and K is the number of clauses in F. We also prove the time bounds O(|F | · 1.3995 k), where k is the maximum number of satisfiable clauses, and O((1.1279) |F | ) for the same problem. For Max2Sat this implies a bound of O(1.2722 K). An exponential time approximation algorithm by Dantsin et al. uses an exact algorithm for MaxSat as a building block and is therefore also improved.

The investigation of the possibility to efficiently compute approximations of hard optimization problems is one of the central and most fruitful areas of current algorithm and complexity theory. The aim of this paper is twofold. First, we introduce the notion of stability of approximation algorithms. This notion is shown to be of practical as well as of theoretical importance, especially for the real understanding of the applicability of approximation algorithms and for the determination of the border between easy instances and hard instances of optimization problems that do not admit polynomial-time approximation. Secondly, we apply our concept to the study of the traveling salesman problem. We show how to modify the Christofides algorithm for \Delta-TSP to obtain efficient approximation algorithms with constant approximation ratio for every instance of TSP that violates the triangle inequality by a multiplicative constant factor. This improves the result of Andreae and Ba...

Abstract. The spectral approach for graph visualization computes the layout of a graph using certain eigenvectors of related matrices. Some important advantages of this approach are an ability to compute optimal layouts (according to specific requirements) and a very rapid computation time. In this paper we explore spectral visualization techniques and study their properties from different points of view. We also suggest a novel algorithm for calculating spectral layouts resulting in an extremely fast computation by optimizing the layout within a small vector space.