Let f : 2 X → R + be a monotone submodular set function, and let (X, I) be a matroid. We consider the problem max S∈I f (S). It is known that the greedy algorithm yields a 1/2approximation [17] for this problem. For certain special cases, e.g. max |S|≤k f (S), the greedy algorithm yields a (1 − 1/e)-approximation. It is known that this is optimal both in the value oracle model (where the only access to f is through a black box returning f (S) for a given set S) [37], and also for explicitly posed instances assuming P = N P [13]. In this paper, we provide a randomized (1 − 1/e)-approximation for any monotone submodular function and an arbitrary matroid. The algorithm works in the value oracle model. Our main tools are a variant of the pipage rounding technique of Ageev and Sviridenko [1], and a continuous greedy process that may be of independent interest. As a special case, our algorithm implies an optimal approximation for the Submodular Welfare Problem in the value oracle model [42]. As a second application, we show that the Generalized Assignment Problem (GAP) is also a special case; although the reduction requires |X| to be exponential in the original problem size, we are able to achieve a (1 − 1/e − o(1))approximation for GAP, simplifying previously known algorithms. Additionally, the reduction enables us to obtain approximation algorithms for variants of GAP with more general constraints.

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Abstract| Energy conservation is a critical issue in ad hoc wireless networks for node and network life, as the nodes are powered by batteries only. One major approach for energy conservation is to route a communication session along the routes which requires the lowest total energy consumption. This optimization problem is referred to as minimum-energy routing. While minimum-energy unicast routing can be solved in polynomial time by shortest-path algorithms, it remains open whether minimum-energy broadcast routing can be solved in polynomial time, despite of the NP-hardness of its general graph version. Recently three greedy heuristics were proposed in 8]: MST (minimum spanning tree), SPT (shortest-path tree), and BIP (broadcasting incremental power). They have been evaluated through simulations in 8], but little is known about their analytical performance. The main contribution of this paper is the quantitative characterization of their performances in terms of approximation ratios. By exploring geometric structures of Euclidean MSTs, we h a ve been able to prove that the approximation ratio of MST is between 6 and 12, and the approximation ratio of BIP is between 13 3 and 12. On the other hand, the approximation ratio of SPT is shown to be at least n 2 , w h e r e n is the number of receiving nodes. To our best knowledge, these are the rst analytical results for minimum-energy broadcasting.

Optimizing the energy consumption in wireless sensor networks has recently become the most important performance objective. We assume the sensor network model in which sensors can interchange idle and active modes. Given monitoring regions, battery life and energy consumption rate for each sensor, we formulate the problem of maximizing sensor network lifetime, i.e., time during which the monitored area is (partially or fully) covered.Our contributions include (1) an efficient data structure to represent the monitored area with at most n 2 points guaranteeing the full coverage which is superior to the previously used approach based on grid points, (2) efficient provably good centralized algorithms for sensor monitoring schedule maximizing the total lifetime including (1 + ln(1 − q) −1 )-approximation algorithm for the case when a q-portion of the monitored area is required to cover, e.g., for the 90% area coverage our schedule guarantees to be at most 3.3 times shorter than the optimum, (4) a family of efficient distributed protocols with trade-off between communication and monitoring power consumption, (5) extensive experimental study of the proposed algorithms showing significant advantage in quality, scalability and flexibility.

Given an undirected graph wit.h edge co&s and a subset of k nodes called terminals, a multiway cut is a subset of edges whose removal disconnects each terminal from the rest. ~iULTIW.~y CUT is the problem of finding a multiway cut of minimum cost.. Previously, a very simple combinatorial algorithm due to Dahlhaus, Johnson, Papadimitriou, Seymour, and %nnr-lkakis gave a performance guarantee of 2 (1 -$), In this paper, we present a new linear programming rslax&ion for ~fULTIW&Y CUT and a new approximation dgorithm based on it. The algorithm breaks the threshold of 2 for approximating MULTIWAY CUT, achieving a performance ratio of at. most 1.5 -$. This improves the previous result for every value of k. In particular, for k = 3 we get a ratio ofZ

Abstract. In the 0-extension problem, we are given a weighted graph with some nodes marked as terminals and a semimetric on the set of terminals. Our goal is to assign the rest of the nodes to terminals so as to minimize the sum, over all edges, of the product of the edge's weight and the distance between the terminals to which its endpoints are assigned. This problem generalizes the multiway cut problem of Dahlhaus, Johnson, Papadimitriou, Seymour, and Yannakakis and is closely related to the metric labeling problem introduced by Kleinberg and Tardos.We present approximation algorithms for 0-Extension. In arbitrary graphs, we present a O(log k)-approximation algorithm, k being the number of terminals. We also give O(1)-approximation guarantees for weighted planar graphs. Our results are based on a natural metric relaxation of the problem, previously considered by Karzanov. It is similar in flavor to the linear programming relaxation of Garg, Vazirani, and Yannakakis for the multicut problem and similar to relaxations for other graph partitioning problems. We prove that the integrality ratio of the metric relaxation is at least c √ lg k for a positive c for infinitely many k. Our results improve some of the results of Kleinberg and Tardos and they further our understanding on how to use metric relaxations.

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In this paper we study the problem of assigning transmission ranges to the nodes of a static ad hoc wireless network so as to minimize the total power consumed under the constraint that enough power is provided to the nodes to ensure that the network is connected. We focus on the MIN-POWER SYMMETRIC CONNECTIVITY problem, in which the bidirectional links established by the transmission ranges are required to form a connected graph. Implicit in previous work on transmission range assignment under asymmetric connectivity requirements is the proof that MIN-POWER SYMMETRIC CONNECTIVITY is NP-hard and that the MST algorithm has a performance ratio of 2. In this paper we make the following contributions: (1) we show that the related MIN-POWER SYMMETRIC UNICAST problem can be solved efficiently by a shortest-path computation in an appropriately constructed auxiliary graph; (2) we give an exact branch and cut algorithm based on a new integer linear program formulation solving instances with up to 35-40 nodes in 1 hour; (3) we establish the similarity between MIN-POWER SYMMETRIC CONNECTIVITY and the classic STEINER TREE problem in graphs, and use this similarity to give a polynomial-time approximation scheme with performance ratio approaching 5/3 as well as a more practical approximation algorithm with approximation factor 11/6; and (4) we give the results of a comprehensive experimental study comparing new and previously proposed heuristics with the above exact and approximation algorithms.