Luigi Laura

Algorithms for the on-line quota traveling salesman problem

The Quota Traveling Salesman Problem is a generalization of the well-known Traveling Salesman Problem. The goal of the traveling salesman is, in this case, to reach a given quota of sales, minimizing the amount of time. In this paper we address the on-line version of the problem, where requests are given over time. We present algorithms for various metric spaces, and analyze their performance in the usual framework of competitive analysis. In particular we present a 2-competitive algorithm that matches the lower bound for general metric spaces. In the case of the halfline metric space, we show that it is helpful not to move at full speed, and this approach is also used to derive the best on-line polynomial time algorithm known so far for the On-Line TSP (in the homing version).

Finding strong bridges and strong articulation points in linear time

Given a directed graph G, an edge is a strong bridge if its removal increases the number of strongly connected components of G. Similarly, we say that a vertex is a strong articulation point if its removal increases the number of strongly connected components of G. In this paper, we present linear-time algorithms for computing all the strong bridges and all the strong articulation points of directed graphs, solving an open problem posed in Beldiceanu et al. (2005) [2].

The Web as a graph: How far we are

In this article we present an experimental study of the properties of webgraphs. We study a large crawl from 2001 of 200M pages and about 1.4 billion edges, made available by the WebBase project at Stanford, as well as several synthetic ones generated according to various models proposed recently. We investigate several topological properties of such graphs, including the number of bipartite cores and strongly connected components, the distribution of degrees and PageRank values and some correlations; we present a comparison study of the models against these measures.Our findings are that (i) the WebBase sample differs slightly from the (older) samples studied in the literature, and (ii) despite the fact that these models do not catch all of its properties, they do exhibit some peculiar behaviors not found, for example, in the models from classical random graph theory.Moreover we developed a software library able to generate and measure massive graphs in secondary memory; this library is publicy available under the GPL licence. We discuss its implementation and some computational issues related to secondary memory graph algorithms.

The online Prize-Collecting Traveling Salesman Problem

We study the online version of the Prize-Collecting Traveling Salesman Problem (PCTSP), a generalization of the Traveling Salesman Problem (TSP). In the TSP, the salesman has to visit a set of cities while minimizing the length of the overall tour. In the PCTSP, each city has a given weight and penalty, and the goal is to collect a given quota of the weights of the cities while minimizing the length of the tour plus the penalties of the cities not in the tour. In the online version, cities are disclosed over time. We give a 7/3-competitive algorithm for the problem, which compares with a lower bound of 2 on the competitive ratio of any deterministic algorithm. We also show how our approach can be combined with an approximation algorithm in order to obtain an O(1)-competitive algorithm that runs in polynomial time.

Algorithms and experiments for the webgraph

In this paper we present an experimental study of the properties of web graphs. We study a large crawl from 2001 of 200M pages and about 1.4 billion edges made available by the WebBase project at Stanford [19], and synthetic graphs obtained by the large scale simulation of stochastic graph models for the Webgraph. This work has required the development and the use of external and semi-external algorithms for computing properties of massive graphs, and for the large scale simulation of stochastic graph models. We report our experimental findings on the topological properties of such graphs, describe the algorithmic tools developed within this project and report the experiments on their time performance.

2-Edge Connectivity in Directed Graphs

Edge and vertex connectivity are fundamental concepts in graph theory. While they have been thoroughly studied in the case of undirected graphs, surprisingly, not much has been investigated for directed graphs. In this article, we study 2-edge connectivity problems in directed graphs and, in particular, we consider the computation of the following natural relation: We say that two vertices v and w are 2-edge-connected if there are two edge-disjoint paths from v to w and two edge-disjoint paths from w to v. This relation partitions the vertices into blocks such that all vertices in the same block are 2-edge-connected. Differently from the undirected case, those blocks do not correspond to the 2-edge-connected components of the graph. The main result of this article is an algorithm for computing the 2-edge-connected blocks of a directed graph in linear time. Besides being asymptotically optimal, our algorithm improves significantly over previous bounds. Once the 2-edge-connected blocks are available, we can test in constant time if two vertices are 2-edge-connected. Additionally, when two query vertices v and w are not 2-edge-connected, we can produce in constant time a “witness” of this property by exhibiting an edge that is contained in all paths from v to w or in all paths from w to v. We are also able to compute in linear time a sparse certificate for this relation, i.e., a subgraph of the input graph that has O(n) edges and maintains the same 2-edge-connected blocks as the input graph, where n is the number of vertices.

On the power of lookahead in on-line vehicle routing problems

Vehicle Routing Problems are generalizations of the well known Traveling Salesman Problem; we focus on the on-line version of these problems, where requests are not known in advance and arrive over time. We introduce a model of lookeahead for this class of problems, the time lookaheadΔ, which allows an on-line algorithm to foresee all the requests that will be released during next Δ time units. We present lower and upper bounds on the competitive ratio of known and studied variants of the OlTsp; we compare these results with the ones from the literature. Our results show that the effectiveness of lookahead varies significantly as we consider different problems.

Algorithms for the on-line Quota Traveling Salesman Problem

The Quota Traveling Salesman Problem is a generalization of the well known Traveling Salesman Problem. The goal of the traveling salesman is, in this case, to reach a given quota of the sales, minimizing the amount of time. In this paper we address the on-line version of the problem, where requests are given over time. We present algorithms for various metric spaces, and analyze their performance in the usual framework of the competitive analysis. In particular we present a 2-competitive algorithm that matches the lower bound for general metric spaces. In the case of the half-line metric space, we show that it is helpful not to move at full speed, and this approach is also used to derive the best on-line polynomial time algorithm known so far for the more general On-Line TSP problem (in the homing version).

Computing strong articulation points and strong bridges in large scale graphs

Let G=(V,E) be a directed graph. A vertex v∈V (respectively an edge e∈E) is a strong articulation point (respectively a strong bridge) if its removal increases the number of strongly connected components of G. We implement and engineer the linear-time algorithms in [9] for computing all the strong articulation points and all the strong bridges of a directed graph. Our implementations are tested against real-world graphs taken from several application domains, including social networks, communication graphs, web graphs, peer2peer networks and product co-purchase graphs. The algorithms implemented turn out to be very efficient in practice, and are able to run on large scale graphs, i.e., on graphs with ten million vertices and half billion edges. Our experiments on such graphs highlight some properties of strong articulation points, which might be of independent interest.

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.