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Abstract: The negative cost cycle detection (NCCD) problem in weighted directed graphs is a fundamental problems in theoretical computer science with applications in a wide range of domains ranging from maximum flows to image segmentation. From the perspective of program verification, this problem is identical to the problem of checking the satisfiability of a conjunction of difference constraints. There exist a number of approaches in the literature for NCCD with each approach having its own set of advantages. Recently, a greedy, space-efficient algorithm called the stressing algorithm was proposed for this problem. In this paper, we present a novel proof of the Stressing algorithm and its verification using the Prototype Verification System (PVS) theorem prover. This example is part of a larger research program to verify the soundness and completeness of a core set of decision procedures. [Copyright &y& Elsevier]

The single source shortest paths problem with positive edge weights (SSSPP) is one of the more widely studied problems in operations research and theoretical computer science, on account of its wide applicability to practical situations. This problem was first solved in polynomial time by Dijkstra, who showed that by extracting vertices with the smallest distance from the source and relaxing its outgoing edges, the shortest path to each vertex is obtained. Variations of this general theme have led to a number of algorithms which work well in practice. At the heart of a Dijkstra implementation is the technique used to implement a priority queue. It is well known that using Dijkstra's approach requires Ω( n log n) steps on a graph having n vertices, since it essentially sorts vertices based on their distances from the source. Accordingly, the fastest implementation of Dijkstra's algorithm on a graph with n vertices and m edges should take Ω( m + n · log n) time, and consequently, the Dijkstra procedure for SSSPP using Fibonacci Heaps is optimal in the comparison-based model. In this paper, we introduce a new data structure to implement priority queues called two-level heap (TLH) and a new variant of Dijkstra's algorithm called Phased Dijkstra. We contrast the performance of Dijkstra's algorithm (both the simple and the phased variants) using a number of data structures to implement the priority queue and empirically establish that TLH are far superior to Fibonacci heaps on every graph family considered. It is to be noted that our profiling includes both sparse and dense graphs. [ABSTRACT FROM AUTHOR]

Abstract: This paper discusses space–time tradeoffs associated with algorithms for the problem of detecting negative cost cycles in networks (NCCD). NCCD is one of the more ubiquitous problems in computer science and operations research, with applications ranging from program verification and real-time scheduling to image segmentation and shortest path identification. Typical algorithmic analysis, whether theoretical or empirical, focuses almost exclusively on the running time of the algorithm. However, there exist applications in which space is just as important a parameter as time is. This is especially true when the problem instances are very large, as is the case in program verification. Consequently, an algorithm that minimizes running time while ignoring the space overhead may be impractical. In this paper, we analyze a number of the more common algorithms for NCCD from the perspectives of both time and space, with a view towards providing a space–time tradeoff for the practitioner. All the algorithms discussed in this paper (with the exception of Network Simplex) run in time on a network with m arcs and n vertices; however, their space requirements range from (Stressing Algorithm) to (all the Bellman–Ford and Network Simplex variants). Our empirical results demonstrate that in the cases where space is paramount, the Stressing Algorithm is a useful alternative to the Bellman–Ford variants. [Copyright &y& Elsevier]

We contrast the performance of three algorithms for the problem of deciding whether a Partially Clairvoyant real-time system with relative timing constraints, as specified in the E-T-C scheduling framework, has a feasible schedule. In the E-T-C scheduling model, real-time scheduling problems are specified through a specialized class of constraint logic programs (CLPs) called Quantified Linear Programs (QLPs) [Subramani, K., 2003, An analysis of quantified linear programs. In: C.S. Calude (Ed.) Proceedings of the 4th International Conference on Discrete Mathematics and Theoretical Computer Science (DMTCS), volume 2731 of Lecture Notes in Computer Science, July (Springer-Verlag), pp. 265- 277]; thus algorithms for determining the schedulability of instances are procedures to determine the satisfiability of CLPs. Two of these algorithms, viz., the primal algorithm and the dual algorithm have already been discussed in the literature, while a third algorithm called the randomized dual algorithm has been recently proposed [Subramani, K. and Desovski, D. 2005, A new verification procedure for partially Clairvoyant scheduling. Proceedings of the 3rd International Conference on Formal Modelling and Analysis of Timed Systems (FORMATS), October; Subramani, K. and Desovski, D., 2005, Out of order quantifier elimination for standard quantified linear programs, Journal of Symbolic Computation, 40, 1383-1396]. Our experiments demonstrate that the dual-based algorithms (i.e. the dual and the randomized dual) are more effective from an implementational perspective; this is surprising since all three algorithms have the same worst case asymptotic complexity. [ABSTRACT FROM AUTHOR]

The article presents a study on chain programming over difference constraints. Chain programming is a restricted form of linear programming, and a chain program is characterized by a total ordering on the program variables. The study has shown that for difference constraint systems, the total ordering of the program variables results in an elegant divide and conquer algorithm for the problem of feasibility testing.

Abstract: In this paper, we present an out of order quantifier elimination algorithm for a class of Quantified Linear Programs (QLPs) called Standard Quantified Linear Programs (SQLPs). QLPs in general and SQLPs in particular are extremely useful constraint logic programming structures that find wide applicability in the modeling of real-time schedulability specifications; see Subramani [Subramani, K., 2005a. A comprehensive framework for specifying clairvoyance, constraints and periodicity in real-time scheduling. The Computer Journal 48 (3), 259–272]. Consequently any algorithmic advance in their solution has a strong practical impact. Prior to this work, the only known approaches to the solution of QLPs involved sequential variable elimination; see Subramani [Subramani, K., 2003b. An analysis of quantified linear programs. In: Calude, C.S. et al. (Eds.), Proceedings of the 4th International Conference on Discrete Mathematics and Theoretical Computer Science. DMTCS. In: Lecture Notes in Computer Science, vol. 2731. Springer-Verlag, pp. 265–277]. In the sequential approach, the innermost quantified variable is eliminated first, followed by the variable which then becomes the innermost quantified variable and so on, until we are left with a single variable from which the satisfiability of the original formula is easily deduced. This approach is applicable in both discrete and continuous domains; however, it is to be noted that the logic demanding the sequential approach requires that the variables are discrete-valued. To the best of our knowledge, the necessity for sequential elimination over continuous-valued variables has not been investigated in the literature. The techniques used in the development of our elimination algorithm may find applications in domains such as classical logic and finite model theory. The final aspect of our research concerns the structure-preserving nature of the algorithm that we introduce here; in general, it is not known whether discrete domains admit such elimination procedures. [Copyright &y& Elsevier]