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A general approximation technique for constrained forest problems
 SIAM J. COMPUT.
, 1995
"... We present a general approximation technique for a large class of graph problems. Our technique mostly applies to problems of covering, at minimum cost, the vertices of a graph with trees, cycles, or paths satisfying certain requirements. In particular, many basic combinatorial optimization proble ..."
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Cited by 349 (21 self)
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We present a general approximation technique for a large class of graph problems. Our technique mostly applies to problems of covering, at minimum cost, the vertices of a graph with trees, cycles, or paths satisfying certain requirements. In particular, many basic combinatorial optimization problems fit in this framework, including the shortest path, minimumcost spanning tree, minimumweight perfect matching, traveling salesman, and Steiner tree problems. Our technique produces approximation algorithms that run in O(n log n) time and come within a factor of 2 of optimal for most of these problems. For instance, we obtain a 2approximation algorithm for the minimumweight perfect matching problem under the triangle inequality. Our running time of O(n log n) time compares favorably with the best strongly polynomial exact algorithms running in O(n 3) time for dense graphs. A similar result is obtained for the 2matching problem and its variants. We also derive the first approximation algorithms for many NPcomplete problems, including the nonfixed pointtopoint connection problem, the exact path partitioning problem, and complex locationdesign problems. Moreover, for the prizecollecting traveling salesman or Steiner tree problems, we obtain 2approximation algorithms, therefore improving the previously bestknown performance guarantees of 2.5 and 3, respectively [Math. Programming, 59 (1993), pp. 413420].
THE PRIMALDUAL METHOD FOR APPROXIMATION ALGORITHMS AND ITS APPLICATION TO NETWORK DESIGN PROBLEMS
"... The primaldual method is a standard tool in the design of algorithms for combinatorial optimization problems. This chapter shows how the primaldual method can be modified to provide good approximation algorithms for a wide variety of NPhard problems. We concentrate on results from recent researc ..."
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Cited by 120 (7 self)
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The primaldual method is a standard tool in the design of algorithms for combinatorial optimization problems. This chapter shows how the primaldual method can be modified to provide good approximation algorithms for a wide variety of NPhard problems. We concentrate on results from recent research applying the primaldual method to problems in network design.
Approximation Algorithms for MinMax Tree Partition
, 1997
"... We consider the problem of partitioning the node set of a graph into p equal sized subsets. The objective is to minimize the maximum length, over these subsets, of a minimum spanning tree. We show that no polynomial algorithm with bounded Ž 2 error ratio can be given for the problem unless P � NP. W ..."
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Cited by 5 (1 self)
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We consider the problem of partitioning the node set of a graph into p equal sized subsets. The objective is to minimize the maximum length, over these subsets, of a minimum spanning tree. We show that no polynomial algorithm with bounded Ž 2 error ratio can be given for the problem unless P � NP. We present an On. time algorithm for the problem, where n is the number of nodes in the graph. Assuming that the edge lengths satisfy the triangle inequality, its error ratio is at most 2 p � 1. We also present an improved algorithm that obtains as an input a positive Ž Ž p�x. p 2 integer x. It runs in O 2 n. time, and its error ratio is at most Ž2�x� Ž x�p�1.. p.
Approximation Algorithms for Minimum Tree Partition
 Disc. Applied Math
, 1998
"... We consider a problem of locating communication centers. In this problem, it is required to partition the set of n customers into subsets minimizing the length of nets required to connect all the customers to the communication centers. Suppose that communication centers are to be placed in p of the ..."
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Cited by 4 (3 self)
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We consider a problem of locating communication centers. In this problem, it is required to partition the set of n customers into subsets minimizing the length of nets required to connect all the customers to the communication centers. Suppose that communication centers are to be placed in p of the customers locations. The number of customers each center supports is also given. The problem remains to divide a graph into sets of the given sizes, keeping the sum of the spanning trees minimal. The problem is NPComplete, and no polynomial algorithm with bounded error ratio can be given, unless P = NP . We present an approximation algorithm for the problem assuming that the edge lengths satisfy the triangle inequality. It runs in O(p 2 4 p +n 2 ) time (n = jV j) and comes within a factor of 2p \Gamma 1 of optimal. When the sets' sizes are all equal this algorithm runs in O(n 2 ) time. Next an improved algorithm is presented which obtains as an input a positive integer x (x n \Gamm...
Approximation Algorithms for Network Design: A Survey
"... In a typical instance of a network design problem, we are given a directed or undirected graph G = (V,E), nonnegative edgecosts ce for all e ∈ E, and our goal is to find a minimumcost subgraph H of G that satisfies some design criteria. For example, we may wish to find a minimumcost set of edges ..."
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Cited by 3 (0 self)
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In a typical instance of a network design problem, we are given a directed or undirected graph G = (V,E), nonnegative edgecosts ce for all e ∈ E, and our goal is to find a minimumcost subgraph H of G that satisfies some design criteria. For example, we may wish to find a minimumcost set of edges that induces a connected graph (this is the minimumcost spanning tree problem), or we might want to find a minimumcost set of arcs in a directed graph such that every vertex can reach every other vertex (this is the minimumcost strongly connected subgraph problem). This abstract model for network design problems has a large number of practical applications; the design process of telecommunication and traffic networks, and VLSI chip design are just two examples. Many practically relevant instances of network design problems are NPhard, and thus likely intractable. This survey focuses on approximation algorithms as one possible way of circumventing this impasse. Approximation algorithms are efficient (i.e., they run in polynomialtime), and they compute solutions to a given instance of an optimization problem whose objective values are close to those of the respective optimum solutions. More concretely, most of the problems discussed in this survey are minimization problems. We then say that an algorithm is an αapproximation for a given problem if the ratio of the cost of an approximate solution computed by the algorithm to that of an optimum solution is at most α over all instances. In the
Approximation Bounds for Minimum Information Loss Microaggregation
, 2009
"... The NPhard microaggregation problem seeks a partition of data points into groups of minimum specified size k, so as to minimize the sum of the squared euclidean distances of every point to its group’s centroid. One recent heuristic provides an Oðk 3 Þ guarantee for this objective function and an O ..."
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Cited by 1 (1 self)
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The NPhard microaggregation problem seeks a partition of data points into groups of minimum specified size k, so as to minimize the sum of the squared euclidean distances of every point to its group’s centroid. One recent heuristic provides an Oðk 3 Þ guarantee for this objective function and an Oðk 2 Þ guarantee for a version of the problem that seeks to minimize the sum of the distances of the points to its group’s centroid. This paper establishes approximation bounds for another microaggregation heuristic, providing better approximation guarantees of Oðk 2 Þ for the squared distance measure and OðkÞ for the distance measure.
Approximation Algorithms for RegretBounded Vehicle Routing and Applications to DistanceConstrained Vehicle Routing ∗ ABSTRACT
"... We consider vehiclerouting problems (VRPs) that incorporate the notion of regret of a client, which is a measure of the waiting time of a client relative to its shortestpath distance from the depot. Formally, we consider both the additive and multiplicative versions of, what we call, the regretbo ..."
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We consider vehiclerouting problems (VRPs) that incorporate the notion of regret of a client, which is a measure of the waiting time of a client relative to its shortestpath distance from the depot. Formally, we consider both the additive and multiplicative versions of, what we call, the regretbounded vehicle routing problem (RVRP). In these problems, we are given an undirected complete graph G = ({r} ∪ V, E) on n nodes with a distinguished root (depot) node r, edge costs {cuv} that form a metric, and a regret bound R. Given a path P rooted at r and a node v ∈ P, let cP (v) be the distance from r to v along P. The goal is to find the fewest number of paths rooted at r that cover all the nodes so that for every node v covered by (say) path P: (i) its additive regret cP (v) − crv, with respect to P is at most R in additiveRVRP; or (ii) its multiplicative regret, cP (v)/crv, with respect to P is at most R in multiplicativeRVRP. Our main result is the first constantfactor approximation algorithm for additiveRVRP. This is a substantial improvement over the previousbest O(log n)approximation. AdditiveRVRP turns out be a rather central vehiclerouting problem, whose study reveals insights into a variety of other regretrelated problems as well as the classical distanceconstrained VRP (DVRP), enabling us to obtain guarantees for these various problems by leveraging our algorithm for additiveRVRP and the underlying techniques. We obtain approximation ratios of O ( log ( R R−1)) for multiplicativeRVRP, and O ( min { OPT, log D log log D for DVRP with distance A full version is available from the CS arXiv. Work done while the author was a postdoctoral scholar at the University of Waterloo and supported in part by the second author’s grants.