Parametric min-cuts analysis in a network

AbstractThe all pairs minimum cuts problem in a capacitated undirected network is well known. Gomory and Hu showed that the all pairs minimum cuts are revealed by a min-cut tree that can be obtained by solving exactly (n−1) maximum flow problems, where n is the number of nodes in the network.In this paper we consider first the problem of finding parametric min-cuts for a specified pair of nodes when the capacity of an arc i is given by min{bi,λ}, where λ is the parameter, ranging from 0 to ∞. Next we seek the parametric min-cuts for all pairs of nodes, and achieve this by constructing min-cut trees for at most 2m different values of λ, where m is the number of edges in the network

A robust branch-and-cut approach for the minimum-energy symmetric network connectivity problem

This paper considers the minimum-energy symmetric network connectivity problem (MESNC) in wireless sensor networks. The aim of the MESNC is to assign transmission power to each sensor node such that the resulting network, using only bidirectional links, is connected and the total energy consumption is minimized. We first present two new models of this problem and then propose new branch-and-cut algorithms. Based on an existing formulation, we present the first model by introducing additional constraints. These additional constraints allow us to relax certain binary variables to continuous ones and thus to reduce significantly the number of binary variables. Our second model strengthens the first one by adding an exponential number of lifted directed-connectivity constraints. We present two branch-and-cut procedures based on these proposed improvements. The computational results are reported and show that our approaches, using the proposed formulations, can efficiently solve instances with up to 120 nodes, which significantly improve our ability to solve much larger instances in comparison with other exact algorithms in the literature.Branch and bound Branch and cut Integer programming Minimum-energy topology

An adaptive memory programming metaheuristic for the heterogeneous fixed fleet vehicle routing problem

This paper studies the heterogeneous fixed fleet vehicle routing problem (HFFVRP), in which the fleet is composed of a fixed number of vehicles with different capacities, fixed costs, and variable costs. Given the fleet composition, the HFFVRP is to determine a vehicle scheduling strategy with the objective of minimizing the total transportation cost. We propose a multistart adaptive memory programming (MAMP) and path relinking algorithm to solve this problem. Through the search memory, MAMP at each iteration constructs multiple provisional solutions, which are further improved by a modified tabu search. As an intensification strategy, path relinking is integrated to enhance the performance of MAMP. We conduct a series of experiments to evaluate and demonstrate the effectiveness of the proposed algorithm.Vehicle routing Heterogeneous fixed fleet Adaptive memory programming Path relinking Metaheuristic

A branch-and-cut algorithm for the strong minimum energy topology in wireless sensor networks

This paper studies the strong minimum energy topology design problem in wireless sensor networks. The objective is to assign transmission power to each sensor node in a directed wireless sensor network such that the induced directed graph topology is strongly connected and the total energy consumption is minimized. A topology is defined to be strongly connected if there exists a communication path between each ordered pair of sensor nodes. This topology design problem with sensor nodes defined on a plane is an NP-Complete problem. We first establish a lower bound on the optimal power consumption. We then provide three formulations for a more general problem defined on a general directed graph. All these formulations involve an exponential number of constraints. Second formulation is stronger than the first one. Further, using the second formulation, we lift the connectivity constraints to generate stronger set of constraints that yield the third formulation. These lifted cuts turn out to be extremely helpful in developing an effective branch-and-cut algorithm. A series of experiments are carried out to investigate the performance of the proposed branch-and-cut algorithm. These computational results over 580 instances demonstrate the effectiveness of our approach.OR in energy Wireless sensor network Minimum energy topology Branch and bound Cutting

From primal-dual to cost shares and back: A stronger LP relaxation for the Steiner forest problem

In this paper we consider a game theoretical variant of the Steiner forest problem. An instance of this game consists of an undirected graph G = (V,E), non-negative costs c(e) for all edges e in E, and k players. Each player i has an associated pair of terminals si and ti. Consider a forest F in G. We say that player i is serviced if si and ti are connected in F. Player i derives a private utility ui for receiving service. In a recent paper, Könemann, Leonardi, and Schäfer [12] showed that a natural primal-dual algorithm, KLS, gives rise to a 2-approximate budget balanced and group-strategyproof cost sharing method for the above game. In this paper we show that the techniques used in [12] yield a new linear programming relaxation for the Steiner forest problem: the lifted-cut relaxation. First, we give an alternate proof of the approximate budget-balance result in [12] by showing that the cost shares computed by algorithm KLS are feasible for the dual of this relaxation. Second, we are able to show that this new undirected relaxation for Steiner forests is strictly stronger than the well-studied undirected cut relaxation. We conclude the paper with a negative result, arguing that no cross-monotonic cost sharing method can achieve a budget balance factor of less than 2 for the Steiner tree and Steiner forest games. This shows that the results of [11, 12] are essentially tight

From primal-dual to cost shares and back: a stronger LP relaxation for the Steiner forest problem

We consider a game-theoretical variant of the Steiner forest problem, in which each of k users i strives to connect his terminal pair (s(i), t(i)) of vertices in an undirected, edge-weighted graph G. In [1] a natural primal-dual algorithm was shown which achieved a 2-approximate budget balanced cross-monotonic cost sharing method for this game. We derive a new linear programming relaxation from the techniques of [1] which allows for a simpler proof of the budget balancedness of the algorithm from [1]. Furthermore we show that this new relaxation is strictly stronger than the well-known undirected cut relaxation for the Steiner forest problem. We conclude the paper with a negative result, arguing that no cross-monotonic cost sharing method can achieve a budget balance factor of less than 2 for the Steiner tree and Steiner forest games. This shows that the results of [1, 2] are essentially tight