René Sitters

The Minimum Latency Problem Is NP-Hard for Weighted Trees

In the minimum latency problem (MLP) we are given n points v1,..., vn and a distance d(vi, vj) between any pair of points. We have to find a tour, starting at v1 and visiting all points, for which the sum of arrival times is minimal. The arrival time at a point vi is the traveled distance from v1 to vi in the tour. The minimum latency problem is MAX-SNP-hard for general metric spaces, but the complexity for the problem where the metric is given by an edge-weighted tree has been a long-standing open problem. We show that the minimum latency problem is NP-hard for trees even with weights in {0, 1}.

A quasi-PTAS for profit-maximizing pricing on line graphs

We consider the problem of pricing items so as to maximize the profit made from selling these items. An instance is given by a set E of n items and a set of m clients, where each client is specified by one subset of E (the bundle of items he/she wants to buy), and a budget (valuation), which is the maximum price he is willing to pay for that subset. We restrict our attention to the model where the subsets can be arranged such that they form intervals of a line graph. Assuming an unlimited supply of any item, this problem is known as the highway problem and so far only an O(logn)-approximation algorithm is known. We show that a PTAS is likely to exist by presenting a quasi-polynomial time approximation scheme. We also combine our ideas with a recently developed quasi-PTAS for the unsplittable flow problem on line graphs to extend this approximation scheme to the limited supply version of the pricing problem.

Computer-Aided Complexity Classification of Dial-a-Ride Problems

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.

APPROXIMATION ALGORITHMS FOR THE EUCLIDEAN TRAVELING SALESMAN PROBLEM WITH DISCRETE AND CONTINUOUS NEIGHBORHOODS

In the Euclidean traveling salesman problem with discrete neighborhoods, we are given a set of points P in the plane and a set of n connected regions (neighborhoods), each containing at least one point of P. We seek to find a tour of minimum length which visits at least one point in each region. We give (i) an O(α)-approximation algorithm for the case when the regions are disjoint and α-fat, with possibly varying size; (ii) an O(α3)-approximation algorithm for intersecting α-fat regions with comparable diameters. These results also apply to the case with continuous neighborhoods, where the sought TSP tour can hit each region at any point. We also give (iii) a simple O(log n)-approximation algorithm for continuous non-fat neighborhoods. The most distinguishing features of these algorithms are their simplicity and low running-time complexities.

The traveling salesman problem on cubic and subcubic graphs

We study the traveling salesman problem (TSP) on the metric completion of cubic and subcubic graphs, which is known to be NP-hard. The problem is of interest because of its relation to the famous 4/3-conjecture for metric TSP, which says that the integrality gap, i.e., the worst case ratio between the optimal value of a TSP instance and that of its linear programming relaxation (the subtour elimination relaxation), is 4/3. We present the first algorithm for cubic graphs with approximation ratio 4/3. The proof uses polyhedral techniques in a surprising way, which is of independent interest. In fact we prove constructively that for any cubic graph on

On Profit-Maximizing Pricing for the Highway and Tollbooth Problems

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Polynomial time approximation schemes for the traveling repairman and other minimum latency problems

vertices a tour of length