1,892 research outputs found

    Approximating Directed Steiner Problems via Tree Embedding

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    In the k-edge connected directed Steiner tree (k-DST) problem, we are given a directed graph G on n vertices with edge-costs, a root vertex r, a set of h terminals T and an integer k. The goal is to find a min-cost subgraph H of G that connects r to each terminal t by k edge-disjoint r,t-paths. This problem includes as special cases the well-known directed Steiner tree (DST) problem (the case k = 1) and the group Steiner tree (GST) problem. Despite having been studied and mentioned many times in literature, e.g., by Feldman et al. [SODA'09, JCSS'12], by Cheriyan et al. [SODA'12, TALG'14] and by Laekhanukit [SODA'14], there was no known non-trivial approximation algorithm for k-DST for k >= 2 even in the special case that an input graph is directed acyclic and has a constant number of layers. If an input graph is not acyclic, the complexity status of k-DST is not known even for a very strict special case that k= 2 and |T| = 2. In this paper, we make a progress toward developing a non-trivial approximation algorithm for k-DST. We present an O(D k^{D-1} log n)-approximation algorithm for k-DST on directed acyclic graphs (DAGs) with D layers, which can be extended to a special case of k-DST on "general graphs" when an instance has a D-shallow optimal solution, i.e., there exist k edge-disjoint r,t-paths, each of length at most D, for every terminal t. For the case k= 1 (DST), our algorithm yields an approximation ratio of O(D log h), thus implying an O(log^3 h)-approximation algorithm for DST that runs in quasi-polynomial-time (due to the height-reduction of Zelikovsky [Algorithmica'97]). Consequently, as our algorithm works for general graphs, we obtain an O(D k^{D-1} log n)-approximation algorithm for a D-shallow instance of the k-edge-connected directed Steiner subgraph problem, where we wish to connect every pair of terminals by k-edge-disjoint paths

    Approximating k-Forest with Resource Augmentation: A Primal-Dual Approach

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    In this paper, we study the kk-forest problem in the model of resource augmentation. In the kk-forest problem, given an edge-weighted graph G(V,E)G(V,E), a parameter kk, and a set of mm demand pairs V×V\subseteq V \times V, the objective is to construct a minimum-cost subgraph that connects at least kk demands. The problem is hard to approximate---the best-known approximation ratio is O(min{n,k})O(\min\{\sqrt{n}, \sqrt{k}\}). Furthermore, kk-forest is as hard to approximate as the notoriously-hard densest kk-subgraph problem. While the kk-forest problem is hard to approximate in the worst-case, we show that with the use of resource augmentation, we can efficiently approximate it up to a constant factor. First, we restate the problem in terms of the number of demands that are {\em not} connected. In particular, the objective of the kk-forest problem can be viewed as to remove at most mkm-k demands and find a minimum-cost subgraph that connects the remaining demands. We use this perspective of the problem to explain the performance of our algorithm (in terms of the augmentation) in a more intuitive way. Specifically, we present a polynomial-time algorithm for the kk-forest problem that, for every ϵ>0\epsilon>0, removes at most mkm-k demands and has cost no more than O(1/ϵ2)O(1/\epsilon^{2}) times the cost of an optimal algorithm that removes at most (1ϵ)(mk)(1-\epsilon)(m-k) demands

    Shorter tours and longer detours: Uniform covers and a bit beyond

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    Motivated by the well known four-thirds conjecture for the traveling salesman problem (TSP), we study the problem of {\em uniform covers}. A graph G=(V,E)G=(V,E) has an α\alpha-uniform cover for TSP (2EC, respectively) if the everywhere α\alpha vector (i.e. {α}E\{\alpha\}^{E}) dominates a convex combination of incidence vectors of tours (2-edge-connected spanning multigraphs, respectively). The polyhedral analysis of Christofides' algorithm directly implies that a 3-edge-connected, cubic graph has a 1-uniform cover for TSP. Seb\H{o} asked if such graphs have (1ϵ)(1-\epsilon)-uniform covers for TSP for some ϵ>0\epsilon > 0. Indeed, the four-thirds conjecture implies that such graphs have 8/9-uniform covers. We show that these graphs have 18/19-uniform covers for TSP. We also study uniform covers for 2EC and show that the everywhere 15/17 vector can be efficiently written as a convex combination of 2-edge-connected spanning multigraphs. For a weighted, 3-edge-connected, cubic graph, our results show that if the everywhere 2/3 vector is an optimal solution for the subtour linear programming relaxation, then a tour with weight at most 27/19 times that of an optimal tour can be found efficiently. Node-weighted, 3-edge-connected, cubic graphs fall into this category. In this special case, we can apply our tools to obtain an even better approximation guarantee. To extend our approach to input graphs that are 2-edge-connected, we present a procedure to decompose an optimal solution for the subtour relaxation for TSP into spanning, connected multigraphs that cover each 2-edge cut an even number of times. Using this decomposition, we obtain a 17/12-approximation algorithm for minimum weight 2-edge-connected spanning subgraphs on subcubic, node-weighted graphs

    The Salesman's Improved Tours for Fundamental Classes

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    Finding the exact integrality gap α\alpha for the LP relaxation of the metric Travelling Salesman Problem (TSP) has been an open problem for over thirty years, with little progress made. It is known that 4/3α3/24/3 \leq \alpha \leq 3/2, and a famous conjecture states α=4/3\alpha = 4/3. For this problem, essentially two "fundamental" classes of instances have been proposed. This fundamental property means that in order to show that the integrality gap is at most ρ\rho for all instances of metric TSP, it is sufficient to show it only for the instances in the fundamental class. However, despite the importance and the simplicity of such classes, no apparent effort has been deployed for improving the integrality gap bounds for them. In this paper we take a natural first step in this endeavour, and consider the 1/21/2-integer points of one such class. We successfully improve the upper bound for the integrality gap from 3/23/2 to 10/710/7 for a superclass of these points, as well as prove a lower bound of 4/34/3 for the superclass. Our methods involve innovative applications of tools from combinatorial optimization which have the potential to be more broadly applied

    Low Diameter Graph Decompositions by Approximate Distance Computation

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    In many models for large-scale computation, decomposition of the problem is key to efficient algorithms. For distance-related graph problems, it is often crucial that such a decomposition results in clusters of small diameter, while the probability that an edge is cut by the decomposition scales linearly with the length of the edge. There is a large body of literature on low diameter graph decomposition with small edge cutting probabilities, with all existing techniques heavily building on single source shortest paths (SSSP) computations. Unfortunately, in many theoretical models for large-scale computations, the SSSP task constitutes a complexity bottleneck. Therefore, it is desirable to replace exact SSSP computations with approximate ones. However this imposes a fundamental challenge since the existing constructions of low diameter graph decomposition with small edge cutting probabilities inherently rely on the subtractive form of the triangle inequality, which fails to hold under distance approximation. The current paper overcomes this obstacle by developing a technique termed blurry ball growing. By combining this technique with a clever algorithmic idea of Miller et al. (SPAA 2013), we obtain a construction of low diameter decompositions with small edge cutting probabilities which replaces exact SSSP computations by (a small number of) approximate ones. The utility of our approach is showcased by deriving efficient algorithms that work in the CONGEST, PRAM, and semi-streaming models of computation. As an application, we obtain metric tree embedding algorithms in the vein of Bartal (FOCS 1996) whose computational complexities in these models are optimal up to polylogarithmic factors. Our embeddings have the additional useful property that the tree can be mapped back to the original graph such that each edge is "used" only logaritmically many times, which is of interest for capacitated problems and simulating CONGEST algorithms on the tree into which the graph is embedded
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