Negative Cost Cycle Research Articles

A Zero-Space algorithm for Negative Cost Cycle Detection in networks

This paper is concerned with the problem of checking whether a network with positive and negative costs on its arcs contains a negative cost cycle. The Negative Cost Cycle Detection (NCCD) problem is one of the more fundamental problems in network design and finds applications in a number of domains ranging from Network Optimization and Operations Research to Constraint Programming and System Verification. As per the literature, approaches to this problem have been either Relaxation-based or Contraction-based. We introduce a fundamentally new approach for negative cost cycle detection; our approach, which we term as the Stressing Algorithm, is based on exploiting the connections between the NCCD problem and the problem of checking whether a system of difference constraints is feasible. The Stressing Algorithm is an incremental, comparison-based procedure which is as efficient as the fastest known comparison-based algorithm for this problem. In particular, on a network with n vertices and m edges, the Stressing Algorithm takes O ( m ⋅ n ) time to detect the presence of a negative cost cycle or to report that none exists. A very important feature of the Stressing Algorithm is that it uses zero extra space; this is in marked contrast to all known algorithms that require Ω ( n ) extra space. It is well known that the NCCD problem is closely related to the Single Source Shortest Paths (SSSP) problem, i.e., the problem of determining the shortest path distances of all the vertices in a network, from a specified source; indeed most algorithms in the literature for the NCCD problem are modifications of approaches to the SSSP problem. At this juncture, it is not clear whether the Stressing Algorithm could be extended to solve the SSSP problem, even if O ( n ) extra space is available.

Local optimality conditions for multicommodity flow problems with separable piecewise convex costs

We consider here a multicommodity flow network optimization problem with non-convex but piecewise convex arc cost functions. We derive complete optimality conditions for local minima based on negative-cost cycles associated with each commodity. These conditions do not extend to the convex non-smooth case.

On contrasting vertex contraction with relaxation-based approaches for negative cost cycle detection

In this paper, we present a comprehensive empirical analysis of the Vertex Contraction (VC) algorithm for the problem of checking whether a directed graph with positive and negative costs on its edges has a negative cost cycle (NCCD). VC was first presented in [K. Subramani, L. Kovalchick, Contraction versus relaxation: a comparison of two approaches for the negative cost cycle detection problem, in: P.M.A. Sloot et al. (Ed.), Proceedings of the third international conference on computational science (ICCS), Lecture notes in computer science, vol. 2659, Springer-Verlag, 2003, pp. 377–387] and is the only known purely greedy strategy for the NCCD problem. In [K. Subramani, L. Kovalchick, Contraction versus relaxation: a comparison of two approaches for the negative cost cycle detection problem, in: P.M.A. Sloot et al. (Ed.), Proceedings of the third international conference on computational science (ICCS), Lecture notes in computer science, vol. 2659, Springer-Verlag, 2003, pp. 377–387], we compared the performance profiles of a naive implementation of VC and the “standard” Bellman–Ford (BF) algorithm for negative cost cycle detection; we observed that VC performed an order of magnitude better than the BF algorithm. Our experiments were conducted on a range of randomly generated inputs over multiple classes of networks and the results conclusively demonstrated the superiority of VC over a naive implementation of BF. Our observations are surprising given that VC is asymptotically inferior to BF on sparse networks. This paper continues the study of contrasting greedy and dynamic programming (relaxation-based) approaches for the NCCD problem, by comparing sophisticated variants of VC with sophisticated implementations of the BF algorithm. Variants of VC are obtained by employing different heuristics to determine the vertex contraction sequence, in precisely the same manner as variants of BF are obtained by employing different heuristics to determine the edge relaxation sequence. Our experiments indicate that on most classes of networks, there is at least one VC variant with a performance profile that is comparable to the more sophisticated BF implementations. Our results are most surprising for the case of planar networks, where a variant of VC performs almost as well as the best relaxation-based algorithm. One of the principal conclusions of our work is that a simple, greedy approach involving vertex contraction is a good choice on sparse networks. On a more fundamental note, we establish that the selection of an algorithm for the NCCD problem should involve an analysis of the input and the architecture of the computing machine, in order to maximize performance.

A greedy strategy for detecting negative cost cycles in networks

In this paper, we develop a greedy strategy for the problem of checking whether a network (directed graph) with positive and negative costs on its edges has a negative cost cycle. We call our approach the Vertex Contraction algorithm; it is the first known greedy strategy for this problem. As per the literature, all known approaches to the problem of detecting negative cost cycles are based on dynamic programming or scaling. It is well known that the negative cost cycle detection problem is equivalent to the problem of checking whether a system of linear difference constraints is feasible. Our algorithm exploits this equivalence and uses polyhedral projection on the polyhedron of linear difference constraints, corresponding to the input network, to detect the presence of negative cost cycles. We use a variant of the Fourier–Motzkin elimination procedure to effect polyhedral projection and contrast the performance of our algorithm with the “standard” Bellman–Ford algorithm for the same problem. We observed that the Vertex Contraction algorithm performed an order of magnitude better than the Bellman–Ford algorithm on a range of randomly generated inputs, thereby conclusively demonstrating the superiority of our approach. From the perspective of asymptotic analysis, the Bellman–Ford algorithm is superior to our algorithm, in that it runs in time O ( m ⋅ n ) in the worst case, on a network with m edges and n vertices, whereas the worst-case complexity of our algorithm is O ( n 3 ) .

High-level synthesis for low power based on network flow method

We propose an effective algorithm for power optimization in behavioral synthesis. In previous work, it has been shown that several hardware allocation/binding problems for power optimization can be formulated as network flow problems and cand be solved optimally. However, in these formulations, a fixed schedule was assumed. In such a context, one key problem is that given an optimal network flow solution to a hardware allocation/binding problem for a given schedule, how to generate a new optimal network-flow solution rapidly for a local change of the given schedule. To this end, from a comprehensive analysis of the relation between network structure and flow computation, we devise a two-step procedure: Step 1) a max-flow computation step which finds a valid (maximum) flow solution while retaining the previous (maximum flow of minimum cost) solution as much as possible and Step 2) a min-cost computation step which incrementally refines the flow solution obtained in Step 1, using the concept of finding a negative cost cycle in the residual graph for the flow. The proposed algorithm can be applied effectively to several important high-level optimization problems (e.g., allocations/bindings of functional units, registers, buses, and memory ports) when we have the freedom to choose a schedule that will minimize power consumption. Experimental results (for bus synthesis) on benchmark problems show that our designs are 4%-40% more power-efficient over the designs produced by a random-move based solution and a clock-step based optimal solution, which is due to a) exploitation of the effect of scheduling and b) optimal binding for every schedule instance. Furthermore, our algorithm is about 2.6 times faster in run time over the full network flow based (optimal) algorithm, which is due to c) our novel (two-step) mechanism which utilizes the previous flow solution to reduce redundant flow computations.

A Scaling Algorithm for Multicommodity Flow Problems

We present a penalty-based algorithm that solves the multicommodity flow problem as a sequence of a finite number of scaling phases. The basis of the algorithm is simple and consists of iteratively detecting and sending flow around negative cost cycles. Two parameters control the algorithm's behavior: the penalty parameter and the scaling parameter. In the ε-scaling phase, where ε is a function of the penalty and scaling parameters, the algorithm determines an ε-optimal solution; a solution in which complementary slackness conditions are satisfied to within ε. We analyze the performance of the algorithm from both the theoretical and practical perspectives. The computational results support the theoretical behavior of the algorithm. They also demonstrate the efficiency of the algorithm for solving problem instances of different structure and size.

Constructing Good Solutions for the Spanish School Timetabling Problem

In the school timetabling problem a set of lessons (combinations of classes, teachers, subjects and rooms) has to be scheduled within the school week. Considering classes, teachers and rooms as resources for the lessons, the problem may be viewed as the scheduling of a project subject to resource constraints. We have developed an algorithm with three phases. In Phase I an initial solution is built by using the scheme of parallel heuristic algorithm with priority rules, but imbedding at each period the construction of a maximum cardinality independent set on a resource graph. In Phase II a tabu search procedure starts from the solution of Phase I and obtains a feasible solution to the problem. The solution obtained is improved in Phase III. Several procedures based on the calculation of negative cost cycles and shortest paths in a solution graph are used to get more compact timetables. The algorithms have been imbedded in a package designed to solve the problem for Spanish secondary schools. The computational results show its performance on a set of real problems. Nevertheless, it can be applied to more general problems and results on a set of large random problems are also provided.

A special case of the dynamization problem for least cost paths

Given a digraph G=( V, E) and a cost function C: E→ R , which does not imply negative cost cycles, let us denote by G(δ) the graph obtained from G by adding to the cost of each edge the positive constant δ; then we want to compute the cost of the least cost path from a given origin r to each node v in the graph G(δ) for different choices of δ⩾0, without having to run a least cost path algorithm everytime with a new cost function. Through a preprocessing of the given digraph based on the Bellmann–Ford algorithm, we will be able to obtain in time O(λ(| E|+| V|) the necessary information to generate a structure that will allow us to answer each query in time O(log λ), where λ is defined to be the length of the longest least cost path in the digraph. In particular, we will see that the only possible candidates to become least cost paths in a graph G(δ) are the least cost paths of bounded length in the original graph G; further, we will show that finding which of these candidates will become least cost path in a G(δ) and for which δ is equivalent to finding the lower envelope of at most λ lines, which, in turn, can be reduced to the lower convex hull problem for a set of ordered points.