In this paper, we propose an efficient method for implementing Dijkstra's algorithm for the Single Source Shortest Path Problem (SSSPP) in a graph whose edges have positive length, and where there are few distinct edge lengths. The SSSPP is one of the most widely studied problems in theoretical computer science and operations research. On a graph with n vertices, m edges and K distinct edge lengths, our algorithm runs in O (m) time if nK 2m, and O (m log nK m ) time, otherwise. We tested our algorithm against some of the fastest algorithms for SSSPP on graphs with arbitrary but positive lengths. Our experiments on graphs with few edge lengths confirmed our theoretical results, as the proposed algorithm consistently dominated the other SSSPP algorithms, which did not exploit the special structure of having few distinct edge lengths.

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Graphs are often used to model risk management in various systems. Particularly, Caskurlu et al. in [6] have considered a system which essentially represents a tripartite graph. The goal in this model is to reduce the risk in the system below a predefined risk threshold level. It can be shown that the main goal in this risk management system can be formulated as a Partial Vertex Cover problem on bipartite graphs. It is well-known that the vertex cover problem is in P on bipartite graphs; however, the computational complexity of the partial vertex cover problem on bipartite graphs is open. In this paper, we show that the partial vertex cover problem is NP-hard on bipartite graphs. Then, we show that the budgeted maximum coverage problem (a problem related to partial vertex cover problem) admits an 8 9approximation algorithm in the class of bipartite graphs, which matches the integrality gap of a natural LP relaxation.

In this paper, we show how the simple structure of the linear programs encountered during symbolic minimum-cost reachability analysis of priced timed automata can be exploited in order to substantially improve the performance of the current algorithm. The idea is rooted in duality of linear programs and we show that each encountered linear program can be reduced to the dual problem of an instance of the min-cost flow problem. Thus, we only need to solve instances of the much simpler min-cost flow problem during minimum-cost reachability analysis. Experimental results using Uppaal show a 70-80 percent performance gain. As a main application area, we show how to solve energy-optimal task graph scheduling problems using the framework of priced timed automata.

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) , An analysis of quantified linear programs. In: C.S. Calude (Ed.) 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 , A new verification procedure for partially Clairvoyant scheduling. Out of order quantifier elimination for standard quantified linear programs, Journal of Symbolic Computation, 40, 1383Computation, 40, -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.