We give a polynomial time approximation scheme for the weighted traveling repairman problem (TRP) in the Euclidean plane, on trees, and on planar graphs. This improves upon the quasi-polynomial time approximation schemes for the unweighted TRP in the Euclidean plane and trees and on the 3.59-approximation for planar graphs. The algorithms are based on a new decomposition technique that reduces the approximation of weighted TRP to instances for which we may restrict ourselves to solutions that are the concatenation of only a constant number of traveling salesman problem paths. A similar reduction applies to many other problems with an average completion time objective. To illustrate the strength of this approach, we apply the same technique to the well-studied scheduling problem of minimizing total weighted completion time under precedence constraints, $1|prec|\sum w_{j}C_{j}$, and present a polynomial time approximation scheme for the case of interval order precedence constraints. This improves on the known 3/2-approximation for this problem.

The field of a priori optimization is an interesting subfield of stochastic combinatorial optimization that is well suited for routing problems. In this setting, there is a probability distribution over active sets, vertices that have to be visited. For a fixed tour, the solution on an active set is obtained by restricting the solution on the active set. In the well-studied a priori traveling salesman problem, the goal is to find a tour that minimizes the expected length. In the a priori traveling repairman problem (TRP), the goal is to find a tour that minimizes the expected sum of latencies. In this paper, we study the uniform model, where a vertex is in the active set with probability p independently of the other vertices, and give the first constant-factor approximation for a priori TRP.

We give a polynomial time, (1 + ∊)-approximation algorithm for the traveling repairman problem (TRP) in the Euclidean plane, on weighted planar graphs, and on weighted trees. This improves on the known quasi-polynomial time approximation schemes for these problems. The algorithm is based on a simple technique that reduces the TRP to what we call the segmented TSP. Here, we are given numbers l1, …, lK and n1, …, nK and we need to find a path that visits at least nh points within path distance lh from the starting point for all h ∊ {1, …, K}. A solution is α-approximate if at least nh points are visited within distance αlh. It is shown that any algorithm that is α-approximate for every constant K in some metric space, gives an α(1 + ∊)-approximation for the TRP in the same metric space. Subsequently, approximation schemes are given for this segmented TSP problem in different metric spaces. The segmented TSP with only one segment (K = 1) is equivalent to the k-TSP for which a (2 + ∊)-approximation is known for a general metric space. Hence, this approach through the segmented TSP gives new impulse for improving on the 3.59-approximation for TRP in a general metric space. A similar reduction applies to many other minimum latency problems. To illustrate the strength of this approach we apply it to the well-studied scheduling problem of minimizing total weighted completion time under precedence constraints, 1|prec|Σ wjCj, and present a polynomial time approximation scheme for the case of interval order precedence constraints. This improves on the known 3/2-approximation for this problem. Both approximation schemes apply as well if release dates are added to the problem.

The generalized 2-server problem is an online optimization problem where a sequence of requests has to be served at minimal cost. Requests arrive one by one and need to be served instantly by at least one of two servers. We consider the general model where the cost function of the two servers may be different. Formally, each server moves in its own metric space and a request consists of one point in each metric space. It is served by moving one of the two servers to its request point. Requests have to be served without knowledge of future requests. The objective is to minimize the total traveled distance. The special case where both servers move on the real line is known as the CNN problem. We show that the generalized work function algorithm, $\mathrm{WFA}_{\lambda}$, is constant competitive for the generalized 2-server problem. Further, we give an outline for a possible extension to $k\geqslant2$ servers and discuss the applicability of our techniques and of the work function algorithm in general. We conclude with a discussion on several open problems in online optimization.

Let $P$ be a set of points in $\Reals^d$, and let $\alpha \ge 1$ be a real number. We define the distance between two points $p,q\in P$ as $|pq|^{\alpha}$, where $|pq|$ denotes the standard Euclidean distance between $p$ and $q$. We denote the traveling salesman problem under this distance function by \tsp($d,\alpha$). We design a 5-approximation algorithm for \tsp(2,2) and generalize this result to obtain an approximation factor of $3^{\alpha-1}+\sqrt{6}^{\,\alpha}\!/3$ for $d=2$ and all $\alpha\ge2$. We also study the variant Rev-\tsp\ of the problem where the traveling salesman is allowed to revisit points. We present a polynomial-time approximation scheme for Rev-\tsp$(2,\alpha)$ with $\alpha\ge2$, and we show that Rev-\tsp$(d, \alpha)$ is \apx-hard if $d\ge3$ and $\alpha>1$. The \apx-hardness proof carries over to \tsp$(d, \alpha)$ for the same parameter ranges.

Given a system of constraints , where ai ∊ {0, 1}n, and ℓi, ui ∊ ℝ+, for i = 1, …, m, we consider the problem Mrfs of finding the largest subsystem for which there exists a feasible solution x ≥ 0. We present approximation algorithms and inapproximability results for this problem, and study some important special cases. Our main contributions are: 1. In the general case, where ai ∊ {0, 1}n, a sharp separation in the approximability between the case when L = max{ℓ1, ⃛, ℓm} is bounded above by a polynomial in n and m, and the case when it is not. 2. In the case where A is an interval matrix, a sharp separation in approximability between the case where we allow a violation of the upper bounds by at most a (1 + ∊) factor, for any fixed ∊ > 0 and the case where no violations are allowed. Along the way, we prove that the induced matching problem on bipartite graphs is inapproximable beyond a factor of , for any ∊ > 0 unless NP=ZPP. Finally, we also show applications of Mrfs to some recently studied pricing problems.

Abstract We address the algorithmic complexity of a profit maximization problem in capacitated, undirected networks. We are asked to price a set ofmcapacitated network links to serve a set ofnpotential customers. Each customer is interested in purchasing a network connection that is specified by a simple path in the network and has a maximum budget that we assume to be known to the seller. The goal is to decide which customers to serve, and to determine prices for all network links in order to maximize the total profit. We address this pricing problem in different network topologies. More specifically, we derive several results on the algorithmic complexity of this profit maximization problem, given that the network is either a path, a cycle, a tree, or a grid. Our results include approximation algorithms as well as inapproximability results. © 2008 Wiley Periodicals, Inc. NETWORKS, 2009

In this paper we discuss the complexity and approximability of the minimum corridor connection problem where, given a rectilinear decomposition of a rectilinear polygon into "rooms", one has to find the minimum length tree along the edges of the decomposition such that every room is incident to a vertex of the tree. We show that the problem is strongly NP-hard and give an subexponential time exact algorithm. For the special case of k-outerplanar graphs the running time becomes O(n3). We develop a polynomial time approximation scheme for the case when all rooms are fat and have nearly the same size. When rooms are fat but are of varying size we give a polynomial time constant factor approximation algorithm.

We consider the generalized on-line two-server problem in which each server moves in its own metric space. Requests for service arrive one-by-one and every request is represented by two points: one in each metric space. The problem is to move, at every request, one of the two servers to its request-point such that the total distance travelled by the two servers is minimized.The special case in which both metric spaces are the real line is known as the CNN-problem. It has been a well-known open question in on-line optimization if an algorithm with a constant-competitive ratio exists for this problem. We answer this question in the affirmative by providing a constant-competitive algorithm for the generalized two-server problem on any metric space.The basic result in this article is a characterization of competitiveness for metrical service systems that seems much easier to use when looking for a competitive algorithm. The existence of a competitive algorithm for the generalized two-server problem follows rather easily from this result.

In dial-a-ride problems, items have to be transported from a source to a destination. The characteristics of the servers involved as well as the specific requirements of the rides may vary. Problems are defined on some metric space, and the goal is to find a feasible solution that minimizes a certain objective function. The structure of these problems allows for a notation similar to the standard notation for scheduling and queueing problems. We introduce such a notation and show how a class of 7,930 dial-a-ride problem types arises from this approach. In examining their computational complexity, we define a partial ordering on the problem class and incorporate it in the computer program DARCLASS. As input DARCLASS uses lists of problems whose complexity is known. The output is a classification of all problems into one of three complexity classes: solvable in polynomial time, NP-hard, or open. For a selection of the problems that form the input for DARCLASS, we exhibit a proof of polynomial-time solvability or NP-hardness.