Refael Hassin

Approximation schemes for the restricted shortest path problem

This note contains two fully polynomial approximation schemes for the shortest path problem with an additional constraint. The main difficulty in constructing such algorithms arises since no trivial lower and upper bounds on the solution value, whose ratio is polynomially bounded, are known. In spite of this difficulty, one of the algorithms presented here is strongly polynomial. Applications to other problems are also discussed.

To queue or not to queue : equilibrium behavior in queueing systems

Preface. 1. Introduction. 2. Observable Queues. 3. Unobservable Queues. 4. Priorities. 5. Reneging and Jockeying. 6. Schedules and Retrials. 7. Competition Among Servers. 8. Service Rate Decisions. Index.

Approximation algorithms for maximum dispersion

We describe approximation algorithms with bounded performance guarantees for the following problem: A graph is given with edge weights satisfying the triangle inequality, together with two numbers k and p. Find k disjoint subsets of p vertices each, so that the total weight of edges within subsets is maximized

Scheduling Arrivals to Queues: A Single-Server Model with No-Shows

Queueing systems with scheduled arrivals, i.e., appointment systems, are typical for frontal service systems, e.g., health clinics. An aspect of customer behavior that influences the overall efficiency of such systems is the phenomenon of no-shows. The consequences of no-shows cannot be underestimated; e.g., British surveys reveal that in the United Kingdom alone more than 12 million general practitioner (GP) appointments are missed every year, costing the British health service an estimated ₤250 million annually. In this study we answer the following key questions: How should the schedule be computed when there are no-shows? Is it sufficiently accurate to use a schedule designed for the same expected number of customers without no-shows? How important is it to invest in efforts that reduce no-shows---i.e., given that we apply a schedule that takes no-shows into consideration, is the existence of no-shows still costly to the server and customers?

The scheduling of maintenance service

Abstract We study a discrete problem of scheduling activities of several types under the constraint that at most a single activity can be scheduled to any one period. Applications of such a model are the scheduling of maintenance service to machines and multi-item replenishment of stock. In this paper we assume that the cost associated with any given type of activity increases linearly with the number of periods since the last execution of this type. The problem is to find an optimal schedule specifying at which periods to execute each of the activity types in order to minimize the long-run average cost per period. We investigate properties of an optimal solution and show that there is always a cyclic optimal policy. We propose a greedy algorithm and report on computational comparison with the optimal. We also provide a heuristic, based on regular cycles for all but one activity type, with a guaranteed worse case bound.

Improved complexity bounds for location problems on the real line

In this note we apply recent results in dynamic programming to improve the complexity bounds of several median and coverage location models on the real line.

Approximations for minimum and min-max vehicle routing problems

We consider a variety of vehicle routing problems. The input to a problem consists of a graph G = (N, E) and edge lengths l(e), e ∈ E. 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.

On Local Search for Weighted K -Set Packing

Given a collection of sets of cardinality at most k, with weights for each set, the maximum weighted packing problem is that of finding a collection of disjoint sets of maximum total weight. We study the worst case behavior of the t-local search heuristic for this problem proving a tight bound of k - 1 + 1/t . As a consequence, for any given r < 1/(k -1) we can compute in polynomial time a solution whose weight is at least r times the optimal.,

On the minimum diameter spanning tree problem

We point out a relation between the minimum diameter spanning tree of a graph and its absolute 1-center. We use this relation to solve the diameter problem and an extension of it efficiently.

The swapping problem

Each vertex of a graph initially may contain an object of a known type. A final state, specifying the type of object desired at each vertex, is also given. A single vehicle of unit capacity is available for shipping objects among the vertices. The swapping problem is to compute a shortest route such that a vehicle can accomplish the rearrangement of the objects while following this route. We exhibit several structural properties of shortest routes and develop polynomial approximation algorithms that are variations of a well-known “patching” algorithm for the traveling salesman problem. We prove tight constant performance guarantees for these algorithms and note as a side product that these bounds hold and are tight also for the latter problem.