Multi-criteria approximation schemes for the resource constrained shortest path problem

In the resource constrained shortest path problem we are given a directed graph along with a source node and a destination node, and each arc has a cost and a vector of weights specifying its requirements from a set of resources with finite budget limits. A minimum cost source-destination path is sought such that the total consumption of the arcs from each resource does not exceed its budget limit. In the case of constant number of weight functions we give a fully polynomial time multi-criteria approximation scheme for the problem which returns a source-destination path of cost at most the optimum, however, the path may slightly violate the budget limits. On the negative side, we show that there does not exist polynomial time multi-criteria approximation scheme for the problem if the number of weight functions is not a constant. The latter result applies to a broad class of problem as well, including the multi-dimensional knapsack, the multi-budgeted spanning tree, the multi-budgeted matroid basis and the multi-budgeted bipartite perfect matching problems

One-Exact Approximate Pareto Sets

Papadimitriou and Yannakakis show that the polynomial-time solvability of a certain singleobjective problem determines the class of multiobjective optimization problems that admit a polynomial-time computable $(1+\varepsilon, \dots , 1+\varepsilon)$-approximate Pareto set (also called an $\varepsilon$-Pareto set). Similarly, in this article, we characterize the class of problems having a polynomial-time computable approximate $\varepsilon$-Pareto set that is exact in one objective by the efficient solvability of an appropriate singleobjective problem. This class includes important problems such as multiobjective shortest path and spanning tree, and the approximation guarantee we provide is, in general, best possible. Furthermore, for biobjective problems from this class, we provide an algorithm that computes a one-exact $\varepsilon$-Pareto set of cardinality at most twice the cardinality of a smallest such set and show that this factor of 2 is best possible. For three or more objective functions, however, we prove that no constant-factor approximation on the size of the set can be obtained efficiently