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Abstract  We analyze the performance of evolutionary algorithms on various matroid optimization problems that encompass a vast number of efficiently solvable as well as NP-hard combinatorial optimization problems (including many well-known examples such as minimum spanning tree and maximum bipartite matching). We obtain very promising bounds on the expected running time and quality of the computed solution. Our results establish a better theoretical understanding of why randomized search heuristics yield empirically good results for many real-world optimization problems. [ABSTRACT FROM AUTHOR]

Abstract  Since the seminal work of Ford and Fulkerson in the 1950s, network flow theory is one of the most important and most active areas of research in combinatorial optimization. Coming from the classical maximum flow problem, we introduce and study an apparently basic but new flow problem that features a couple of interesting peculiarities. We derive several results on the complexity and approximability of the new problem. On the way we also discover two closely related basic covering and packing problems that are of independent interest. Starting from an LP formulation of the maximum s-t-flow problem in path variables, we introduce unit upper bounds on the amount of flow being sent along each path. The resulting (fractional) flow problem is NP-hard; its integral version is strongly NP-hard already on very simple classes of graphs. For the fractional problem we present an FPTAS that is based on solving the k shortest paths problem iteratively. We show that the integral problem is hard to approximate and give an interesting O(log m)-approximation algorithm, where m is the number of arcs in the considered graph. For the multicommodity version of the problem there is an -approximation algorithm. We argue that this performance guarantee is best possible, unless P=NP. [ABSTRACT FROM AUTHOR]

Abstract??Given a network with capacities and transit times on the arcs, the quickest flow problem asks for a "flow over time" that satisfies given demands within minimal time. In the setting of flows over time, flow on arcs may vary over time and the transit time of an arc is the time it takes for flow to travel through this arc. In most real-world applications (such as, e.g., road traffic, communication networks, production systems, etc.), transit times are not fixed but depend on the current flow situation in the network. We consider the model where the transit time of an arc is given as a non-decreasing function of the rate of inflow into the arc. We prove that the quickest s-t-flow problem is NP-hard in this setting and give various approximation results, including a fully polynomial time approximation scheme (FPTAS) for the quickest multicommodity flow problem with bounded cost. [ABSTRACT FROM AUTHOR]

Abstract  An optimization problem that naturally arises in the study of swarm robotics is the Freeze-Tag Problem (FTP) of how to awaken a set of "asleep" robots, by having an awakened robot move to their locations. Once a robot is awake, it can assist in awakening other slumbering robots. The objective is to have all robots awake as early as possible. While the FTP bears some resemblance to problems from areas in combinatorial optimization such as routing, broadcasting, scheduling, and covering, its algorithmic characteristics are surprisingly different. We consider both scenarios on graphs and in geometric environments. In graphs, robots sleep at vertices and there is a length function on the edges. Awake robots travel along edges, with time depending on edge length. For most scenarios, we consider the offline version of the problem, in which each awake robot knows the position of all other robots. We prove that the problem is NP-hard, even for the special case of star graphs. We also establish hardness of approximation, showing that it is NP-hard to obtain an approximation factor better than 5/3, even for graphs of bounded degree. These lower bounds are complemented with several positive algorithmic results, including: · We show that the natural greedy strategy on star graphs has a tight worst-case performance of 7/3 and give a polynomial-time approximation scheme (PTAS) for star graphs. · We give a simple O(log δ)-competitive online algorithm for graphs with maximum degree δ and locally bounded edge weights. · We give a PTAS, running in nearly linear time, for geometrically embedded instances. [ABSTRACT FROM AUTHOR]

Abstract In traditional multi-commodity flow theory, the task is to send a certain amount of each commodity from its start to its target node, subject to capacity constraints on the edges. However, no restriction is imposed on the number of paths used for delivering each commodity; it is thus feasible to spread the flow over a large number of different paths. Motivated by routing problems arising in real-life applications, e.g., telecommunication, unsplittable flows have moved into the focus of research. Here, the demand of each commodity may not be split but has to be sent along a single path. In this paper a generalization of this problem is studied. In the considered flow model, a commodity can be split into a bounded number of chunks which can then be routed on different paths. In contrast to classical (splittable) flows and unsplittable flows, the single-commodity case of this problem is already NP-hard and even hard to approximate. We present approximation algorithms for the single- and multi-commodity case and point out strong connections to unsplittable flows. Moreover, results on the hardness of approximation are presented. In particular, we show that some of our approximation results are in fact best possible, unless P = NP. [ABSTRACT FROM AUTHOR]