Diana Poensgen

News from the Online Traveling Repairman

In the traveling repairman problem (TRP), a tour must be found through every one of a set of points (cities) in some metric space such that the weighted sum of completion times of the cities is minimized. Given a tour, the completion time of a city is the time traveled on the tour before the city is reached. In the online traveling repairman problem OLTRP requests for visits to cities arrive online while the repairman is traveling. We analyze the performance of algorithms for the online problem using competitive analysis, where the cost of an online algorithm is compared to that of an optimal offline algorithm. Feuerstein and Stougie [8] present a 9-competitive algorithm for the OlTrp on the real line. In this paper we show how to use techniques from online-scheduling to obtain a 6-competitive deterministic algorithm for the OlTrp on any metric space. We also present a randomized algorithm with competitive ratio of 3/ln 2 2.1282 for the L-OLDARP on the line, 4e-5/2e-3 > 2.41041 for the L-OLDARP on general metric spaces, 2 for the OLTRP on the line, and 7/3 for the OLTRP on general metric spaces.

Non-abusiveness Helps: An % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC% vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbuLwBLnhiov2DGi1BTfMBaeHb% d9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb% L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe% pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaadeWaaq% aadaqbaaGcbaGaaGOmamaaCaaaleqabaGagiiBaWMaei4Ba8Maei4z% aCgaaOWaaWbaaSqabeaadaahaaadbeqaamaaBaaabaWaaWbaaeqaba% GaaGymaiabgkHiTiabgIGiodaaaeqaaaaaaaGcdaahaaWcbeqaaiab% d6gaUbaaaaa!4546!\[2^{\log } ^{^{_{^{1 - \in } } } } ^n \](1)-Competitive Algorithm for Minimizing the Maximum Flow Time in the Online Traveling Salesman Problem

In the online traveling salesman problem OLTSP requests for visits to cities arrive online while the salesman is traveling. We study the Fmax-OLTSP where the objective is to minimize the maximum flow time. This objective is particularly interesting for applications. Unfortunately, there can be no competitive algorithm, neither deterministic nor randomized. Hence, competitive analysis fails to distinguish online algorithms. Not even resource augmentation which is helpful in scheduling works as a remedy. This unsatisfactory situation motivates the search for alternative analysis methods.We introduce a natural restriction on the adversary for the Fmax-OLTSP on the real line. A non-abusive adversary may only move in a direction if there are yet unserved requests on this side. Our main result is an algorithm which achieves a constant competitive ratio against the nonabusive adversary.In the online traveling salesman problem OlTsp requests for visits to cities arrive online while the salesman is traveling. We study the F max-OlTsp where the objective is to minimize the maximum flow time. This objective is particularly interesting for applications. Unfortunately, there can be no competitive algorithm, neither deterministic nor randomized. Hence, competitive analysis fails to distinguish online algorithms. Not even resource augmentation which is helpful in scheduling works as a remedy. This unsatisfactory situation motivates the search for alternative analysis methods.

Budgeted Maximum Graph Coverage

An instance of the maximum coverage problem is given by a set of weighted ground elements and a cost weighted family of subsets of the ground element set. The goal is to select a subfamily of total cost of at most that of a given budget maximizing the weight of the covered elements.We formulate the problem on graphs: In this situation the set of ground elements is specified by the nodes of a graph, while the family of covering sets is restricted to connected subgraphs. We show that on general graphs the problem is polynomial time solvable if restricted to sets of size at most 2, but becomes NP-hard if sets of size 3 are permitted. On trees, we prove polynomial time solvability if each node appears in a fixed number of sets. In contrast, if vertices are allowed to appear an unbounded number of times, the problem is NP-hard even on stars. We finally give a polynomial time algorithm for the special case where a star is covered by paths.

On minimizing the maximum flow time in the online dial-a-ride problem

In the online dial-a-ride problem (OlDarp), objects must be transported by a server between points in a metric space. Transportation requests (“rides”) arrive online, specifying the objects to be transported and the corresponding source and destination. We investigate the OlDarp for the objective of minimizing the maximum flow time. It has been well known that there can be no strictly competitive online algorithm for this objective and no competitive algorithm at all on unbounded metric spaces. However, the question whether on metric spaces with bounded diameter there are competitive algorithms if one allows an additive constant in the definition competitive ratio, had been open for quite a while. We provide a negative answer to this question already on the uniform metric space with three points. Our negative result is complemented by a strictly 2-competitive algorithm for the Online Traveling Salesman Problem on the uniform metric space, a special case of the problem.

Online Call Admission in Optical Networks with Larger Demands

In the problem of Online Call Admission in Optical Networks, briefly called Oca, we are given a graph G = (V, E) together with a set of wavelengths W and a finite sequence ? = r1, r2, . . . of calls which arrive in an online fashion. Each call rj specifies a pair of nodes to be connected and an integral demand indicating the number of required lightpaths. A lightpath is a path in G together with a wavelength ? ? W. Upon arrival of a call, an online algorithm must decide immediately and irrevocably whether to accept or to reject the call without any knowledge of calls which appear later in the sequence. If the call is accepted, the algorithm must provide the requested number of lightpaths to connect the specified nodes. The essential restriction is the wavelength conflict constraint: each wavelength is available only once per edge, which implies that two lightpaths sharing an edge must have different wavelengths. Each accepted call contributes a benefit equal to its demand to the overall profit. The objective in Oca is to maximize the overall profit.Competitive algorithms for Oca have been known for the special case where every call requests just a single lightpath. In this paper we present the first competitive online algorithms for the more general case in which the demand of a call may be as large as |W|.