This paper presents a new method and a constraint-based objective function to solve two problemsrelated to the design of optical telecommunication networks, namely the Synchronous Optical Net-work Ring Assignment Problem (SRAP) and the Intra-ring Synchronous Optical Network DesignProblem (IDP). These network topology problems can be represented as a graph partitioning withcapacity constraints as shown in previous works. We present here a new objective function and anew local search algorithm to solve these problems. Experiments conducted in C
A special case of the IDP problem where all the edges have the same weight, is studied in @cite_1 . This special case is called the problem. Given a simple undirected graph @math and a value @math , we want to find a partitioning of @math , @math such that @math . The authors present two linear-time-approximation algorithms with fixed performance guarantee.
@cite_1
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Given an arc-weighted directed graph G = (V, A, spl lscr ) and a pair of nodes s, t, we seek to find an s-t walk of length at most B that maximizes some given function f of the set of nodes visited by the walk. The simplest case is when we seek to maximize the number of nodes visited: this is called the orienteering problem. Our main result is a quasi-polynomial time algorithm that yields an O(log OPT) approximation for this problem when f is a given submodular set function. We then extend it to the case when a node v is counted as visited only if the walk reaches v in its time window [R(v), D(v)]. We apply the algorithm to obtain several new results. First, we obtain an O(log OPT) approximation for a generalization of the orienteering problem in which the profit for visiting each node may vary arbitrarily with time. This captures the time window problem considered earlier for which, even in undirected graphs, the best approximation ratio known [Bansal, (2004)] is O(log sup 2 OPT). The second application is an O(log sup 2 k) approximation for the k-TSP problem in directed graphs (satisfying asymmetric triangle inequality). This is the first non-trivial approximation algorithm for this problem. The third application is an O(log sup 2 k) approximation (in quasi-poly time) for the group Steiner problem in undirected graphs where k is the number of groups. This improves earlier ratios (Garg, ) by a logarithmic factor and almost matches the inapproximability threshold on trees (Halperin and Krauthgamer, 2003). This connection to group Steiner trees also enables us to prove that the problem we consider is hard to approximate to a ratio better than spl Omega (log sup 1- spl epsi OPT), even in undirected graphs. Even though our algorithm runs in quasi-poly time, we believe that the implications for the approximability of several basic optimization problems are interesting. In the Maximum Weight Independent Set of Rectangles (MWISR) problem we are given a set of n axis-parallel rectangles in the 2D-plane, and the goal is to select a maximum weight subset of pairwise non-overlapping rectangles. Due to many applications, e.g. in data mining, map labeling and admission control, the problem has received a lot of attention by various research communities. We present the first (1 + e)-approximation algorithm for the MWISR problem with quasipolynomial running time 2poly(log n e). In contrast, the best known polynomial time approximation algorithms for the problem achieve superconstant approximation ratios of O(log log n) (unweighted case) and O(log n log log n) (weighted case). Key to our results is a new geometric dynamic program which recursively subdivides the plane into polygons of bounded complexity. We provide the technical tools that are needed to analyze its performance. In particular, we present a method of partitioning the plane into small and simple areas such that the rectangles of an optimal solution are intersected in a very controlled manner. Together with a novel application of the weighted planar graph separator theorem due to [4] this allows us to upper bound our approximation ratio by 1 + e. Our dynamic program is very general and we believe that it will be useful for other settings. In particular, we show that, when parametrized properly, it provides a polynomial time (1 + e)-approximation for the special case of the MWISR problem when each rectangle is relatively large in at least one dimension. Key to this analysis is a method to tile the plane in order to approximately describe the topology of these rectangles in an optimal solution. This technique might be a useful insight to design better polynomial time approximation algorithms or even a PTAS for the MWISR problem. In particular, note that our results imply that the MWISR problem is not APX-hard, unless NP ⊆ DTIME(2polylog (n)). Let G=(V,E) be an unweighted undirected graph on n vertices. A simple argument shows that computing all distances in G with an additive one-sided error of at most 1 is as hard as Boolean matrix multiplication. Building on recent work of [SIAM J. Comput., 28 (1999), pp. 1167--1181], we describe an @math -time algorithm APASP2 for computing all distances in G with an additive one-sided error of at most 2. Algorithm APASP2 is simple, easy to implement, and faster than the fastest known matrix-multiplication algorithm. Furthermore, for every even k>2, we describe an @math -time algorithm APASPk for computing all distances in G with an additive one-sided error of at most k. We also give an @math -time algorithm @math for producing stretch 3 estimated distances in an unweighted and undirected graph on n vertices. No constant stretch factor was previously achieved in @math time. We say that a weighted graph F=(V,E') k-emulates an unweighted graph G=(V,E) if for every @math we have @math . We show that every unweighted graph on n vertices has a 2-emulator with @math edges and a 4-emulator with @math edges. These results are asymptotically tight. Finally, we show that any weighted undirected graph on n vertices has a 3-spanner with @math edges and that such a 3-spanner can be built in @math time. We also describe an @math -time algorithm for estimating all distances in a weighted undirected graph on n vertices with a stretch factor of at most 3. We study a natural extension of the Maximum Weight Independent Set Problem (MWIS), one of the most studied optimization problems in Graph algorithms. We are given a graph @math , a weight function @math , a budget function @math , and a positive integer @math . The weight (resp. budget) of a subset of vertices is the sum of weights (resp. budgets) of the vertices in the subset. A @math -budgeted independent set in @math is a subset of vertices, such that no pair of vertices in that subset are adjacent, and the budget of the subset is at most @math . The goal is to find a @math -budgeted independent set in @math such that its weight is maximum among all the @math -budgeted independent sets in @math . We refer to this problem as MWBIS. Being a generalization of MWIS, MWBIS also has several applications in Scheduling, Wireless networks and so on. Due to the hardness results implied from MWIS, we study the MWBIS problem in several special classes of graphs. We design exact algorithms for trees, forests, cycle graphs, and interval graphs. In unweighted case we design an approximation algorithm for @math -claw free graphs whose approximation ratio ( @math ) is competitive with the approximation ratio ( @math ) of MWIS (unweighted). Furthermore, we extend Baker's technique Baker83 to get a PTAS for MWBIS in planar graphs.