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Abstract: In a scheduling problem with controllable processing times the job processing time can be compressed through incurring an additional cost. We consider the problem of scheduling n jobs on a single machine with controllable processing times. Each job has a release date when it becomes available for processing, and, after completing its processing, requires an additional delivery time. Feasible schedules are further restricted by job precedence constraints. We develop a polynomial time approximation scheme whose running time depends only linearly on the input size. This improves and generalizes the previous -approximation algorithm by Zdrzalka. Moreover, this linear complexity bound gives a substantial improvement of the best previously known polynomial bound obtained by Hall and Shmoys for the special case without controllable processing times. [Copyright &y& Elsevier]

Abstract: We consider a variety of vehicle routing problems. The input to a problem consists of a graph and edge lengths , . Customers located at the vertices have to be visited by a set of vehicles. Two important parameters are k the number of vehicles, and λ the longest distance traveled by a vehicle. We consider two types of problems. (1) Given a bound λ on the length of each path, find a minimum sized collection of paths that cover all the vertices of the graph, or all the edges from a given subset of edges of the input graph. We also consider a variation where it is desired to cover N by a minimum number of stars of length bounded by λ. (2) Given a number k find a collection of k paths that cover either the vertex set of the graph or a given subset of edges. The goal here is to minimize λ, the maximum travel distance. For all these problems we provide constant ratio approximation algorithms and prove their NP-hardness. [Copyright &y& Elsevier]

Abstract: The main result of this paper is an approximation algorithm for the weighted region optimal path problem. In this problem, a point robot moves in a planar space composed of n triangular regions, each of which is associated with a positive unit weight. The objective is to find, for given source and destination points s and t, a path from s to t with the minimum weighted length. Our algorithm, BUSHWHACK, adopts a traditional approach (see [M. Lanthier, A. Maheshwari, J.-R. Sack, Approximating weighted shortest paths on polyhedral surfaces, in: Proceedings of the 13th Annual ACM Symposium on Coputational Geometry, 1997, pp. 274–283; L. Aleksandrov, M. Lanthier, A. Maheshwari, J.-R. Sack, An ɛ-approximation algorithm for weighted shortest paths on polyhedral surfaces, in: Proceedings of the 6th Scandinavian Workshop on Algorithm Theory, in: Lecture Notes in Comput. Sci., vol. 1432, 1998, pp. 11–22; L. Aleksandrov, A. Maheshwari, J.-R. Sack, Approximation algorithms for geometric shortest path problems, in: Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, 2000, pp. 286–295]) that converts the original continuous geometric search space into a discrete graph by placing representative points on boundary edges. However, by exploiting geometric structures that we call intervals, BUSHWHACK computes an approximate optimal path more efficiently as it accesses only a sparse subgraph of . Combined with the logarithmic discretization scheme introduced by Aleksandrov et al. [Approximation algorithms for geometric shortest path problems, in: Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, 2000, pp. 286–295], BUSHWHACK can compute an ɛ-approximation in time. By reducing complexity dependency on ɛ, this result improves on all previous results with the same discretization approach. We also provide an improvement over the discretization scheme of [L. Aleksandrov, A. Maheshwari, J.-R. Sack, Approximation algorithms for geometric shortest path problems, in: Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, 2000, pp. 286–295] so that the size of is no longer dependent on unit weight ratio, the ratio between the maximum and minimum unit weights. This leads to the first ɛ-approximation algorithm whose time complexity does not depend on unit weight ratio. [Copyright &y& Elsevier]

Abstract: In TSP with neighborhoods (TSPN) we are given a collection of regions in the plane, called neighborhoods, and we seek the shortest tour that visits all neighborhoods. Until now constant-factor approximation algorithms have been known only for cases where the neighborhoods are of approximately the same size. In this paper we present the first polynomial-time constant-factor approximation algorithm for disjoint convex fat neighborhoods of arbitrary size. We also show that in the general case, where the neighborhoods can overlap and are not required to be convex or fat, TSPN is APX-hard and cannot be approximated within a factor of in polynomial time, unless . [Copyright &y& Elsevier]

Abstract: Given a simple connected graph and a set R of pairs of vertices, we consider the problem of augmenting G by a smallest set F of new edges such that the resulting graph remains simple and has at least two internally disjoint paths between u and v for each pair . The problem is known to be NP-hard, and a -approximation algorithm has been obtained so far. In this paper, we introduce new stronger lower bounds on the optimal value, and propose an time -approximation algorithm to the problem. [Copyright &y& Elsevier]

Abstract: We consider the data migration problem, which is the problem of finding an efficient schedule to migrate data in a network. The data layout of a large storage server needs to be computed based on the expected data access pattern for load balancing. As the data access pattern changes over time, we need to recompute a new layout and then migrate the data from its current layout to the new layout. Our objective is to minimize the total completion time over all storage devices. We develop a 3-approximation algorithm when each migration job needs the same amount of time and a 9-approximation when the times required for different migration jobs are different. Another interesting objective is to minimize the total completion time over all data migration jobs. We present a 10-approximation algorithm when each migration job has a different processing time. We extend our results to the resource constrained scheduling problem where each job requires at most m resources. We obtain -approximations for both resource and job completion times. [Copyright &y& Elsevier]

We study a generalization of covering problems called partial covering. Here we wish to cover only a desired number of elements, rather than covering all elements as in standard covering problems. For example, in k-partial set cover, we wish to choose a minimum number of sets to cover at least k elements. For k-partial set cover, if each element occurs in at most f sets, then we derive a primal-dual f-approximation algorithm (thus implying a 2-approximation for k-partial vertex cover) in polynomial time. Without making any assumption about the number of sets an element is in, for instances where each set has cardinality at most three, we obtain an approximation of 4/3. We also present better-than-2-approximation algorithms for k-partial vertex cover on bounded degree graphs, and for vertex cover on expanders of bounded average degree. We obtain a polynomial-time approximation scheme for k-partial vertex cover on planar graphs, and for covering k points in Rd by disks. [Copyright &y& Elsevier]

A prevailing feature of mobile telephony systems is that the cell where a mobile user is located may be unknown. Therefore, when the system is to establish a call between users, it may need to search, or page, all the cells that it suspects the users are located in, to find the cells where the users currently reside. The search consumes expensive wireless links and so it is desirable to develop search techniques that page as few cells as possible.We consider cellular systems with c cells and m mobile users roaming among the cells. The location of the users is uncertain as given by m probability distribution vectors. Whenever the system needs to find specific users, it conducts a search operation lasting some number of rounds (the delay constraint). In each round, the system may check an arbitrary subset of cells to see which users are located there. In this setting the problem of finding one user with minimum expected number of cells searched is known to be solved optimally in polynomial time.In this paper we address the problem of finding several users with the same optimization goal. This task is motivated by the problem of establishing a conference call between mobile users. We first show that the problem is NP-hard. Then we prove that a natural heuristic is an e/(e−1)-approximation solution. [Copyright &y& Elsevier]

Motivated by a scheduling problem in multicast environments, we consider the problem of arranging a weighted graph around a circle so as to minimize the total weighted arc length. We describe the first polynomial-time approximation algorithms for this problem, and specifically an O(logn)-approximation algorithm for undirected circular arrangements and an O(√ of n)-approximation algorithm for directed circular arrangements. We will also conduct an experimental evaluation of our algorithms and show that a simple heuristic has the best performance in simulations based on busy Web server logs. [Copyright &y& Elsevier]

Let P be an optimization problem, and let A be an approximation algorithm for P. The domination ratio domr(A,n) is the maximum real q such that the solution x(I) obtained by A for any instance I of P of size n is not worse than at least a fraction q of the feasible solutions of I. We describe a deterministic, polynomial-time algorithm with domination ratio 1−o(1) for the partition problem, and a deterministic, polynomial-time algorithm with domination ratio Ω(1) for the MaxCut problem and for some far-reaching extensions of it, including Max-r-Sat, for each fixed r. The techniques combine combinatorial and probabilistic methods with tools from harmonic analysis. [Copyright &y& Elsevier]

The Multicut problem can be defined as: given a graph G and a collection of pairs of distinct vertices {si,ti} of G, find a minimum set of edges of G whose removal disconnects each si from the corresponding ti. Multicut is known to be NP-hard and Max SNP-hard even when the input graph is restricted to being a tree. The main result of the paper is a polynomial-time approximation scheme (PTAS) for Multicut in unweighted graphs with bounded degree and bounded tree-width. That is, for any ∊>0, we present a polynomial-time (1+∊)-approximation algorithm. In the particular case when the input is a bounded-degree tree, we have a linear-time implementation of the algorithm. We also provide some hardness results: we prove that Multicut is still NP-hard for binary trees and that it is Max SNP-hard if we drop any of the three conditions (unweighted, bounded-degree, bounded tree-width). Finally we show that some of these results extend to the vertex version of Multicut and to a directed version of Multicut. [Copyright &y& Elsevier]

A tree of rings is an undirected graph obtained from a tree by replacing each node of the tree with a cycle (called tree-node-cycle) and then contracting the edges of the tree so that two cycles corresponding to the two end-nodes of any edge have precisely one node in common and no three tree-node-cycles share a same node. (A more general definition may allow them to share the same node.) Given a set of paths on a tree of rings, the path coloring problem is to color these paths with the smallest number of colors so that any two paths sharing an edge are assigned different colors. In this paper, we present a 2-approximation algorithm for this problem. [Copyright &y& Elsevier]

We consider two problems: given a collection of n fat objects in a fixed dimension, (1) ( packing) find the maximum subcollection of pairwise disjoint objects, and (2) ( piercing) find the minimum point set that intersects every object. Recently, Erlebach, Jansen, and Seidel gave a polynomial-time approximation scheme (PTAS) for the packing problem, based on a shifted hierarchical subdivision method. Using shifted quadtrees, we describe a similar algorithm for packing but with a smaller time bound. Erlebach et al.''s algorithm requires polynomial space. We describe a different algorithm, based on geometric separators, that requires only linear space. This algorithm can also be applied to piercing, yielding the first PTAS for that problem. [Copyright &y& Elsevier]

Wavelength rerouting has been suggested as a viable and cost-effective method to improve the blocking performance of wavelength-routed wavelength-division multiplexing (WDM) networks. This method leads to the following combinatorial optimization problem, dubbed Venetian Routing. Given a directed multigraph G along with two vertices s and t and a collection of pairwise arc-disjoint paths, we wish to find an st-path which arc-intersects the smallest possible number of the given paths. In this paper we prove the computational hardness of this problem even in various special cases, and present several approximation algorithms for its solution. In particular we show a non-trivial connection between Venetian Routing and Label Cover. [Copyright &y& Elsevier]

We obtain the following new coloring results:

Let G be an undirected graph with nonnegative edge lengths. Given two vertices as sources and all vertices as destinations, we investigated the problem how to construct a spanning tree of G such that the sum of distances from sources to destinations is minimum. In the paper, we show the NP-hardness of the problem and present a polynomial time approximation scheme. For any ϵ>0, the approximation scheme finds a (1+ϵ)-approximation solution in O(n⌈1/ϵ+1⌉) time. We also generalize the approximation algorithm to the weighted case for distances that form a metric space. [Copyright &y& Elsevier]

We provide constant ratio approximation algorithms for two NP-hard problems, the rectangle stabbing problem and the rectilinear partitioning problem. In the rectangle stabbing problem, we are given a set of rectangles in two-dimensional space, with the objective of stabbing all rectangles with the minimum number of lines parallel to the x and y axes. We provide a 2-approximation algorithm, while the best known approximation ratio for this problem is O(logn). This algorithm is then extended to a 4-approximation algorithm for the rectilinear partitioning problem, which, given an mx × my array of nonnegative integers and positive integers v, h, asks to find a set of v vertical and h horizontal lines such that the maximum load of a subrectangle (i.e., the sum of the numbers in it) is minimized. The best known approximation ratio for this problem is 27. Our approximation ratio 4 is close to the best possible, as it is known to be NP-hard to approximate within any factor less than 2. The results are then extended to the d-dimensional space for d ≥ 2, where a d-approximation algorithm for the stabbing problem and a dd-approximation algorithm for the partitioning problem are developed. [Copyright &y& Elsevier]

The main result of this paper is a (2 + ϵ)-approximation scheme for the minimum dominating set problem on circle graphs. We first present an O(n2) time 8-approximation algorithm for this problem and then extend it to an O(n3 + 6/ϵn⌈+ 1⌉m) time (2 + ϵ)-approximation scheme for this problem. Here n and m are the number of vertices and the number of edges of the circle graph. We then present simple modifications to this algorithm that yield (3 + ϵ)-approximation schemes for the minimum connected and the minimum total dominating set problems on circle graphs. Keil (1993, Discrete Appl. Math.42, 51–63) shows that these problems are NP-complete for circle graphs and leaves open the problem of devising approximation algorithms for them. These are the first O(1)-approximation algorithms for domination problems on circle graphs. [Copyright &y& Elsevier]