Two applications of sparsification to parametric computing are given. The first is a fast algorithm for enumerating all distinct minimum spanning trees in a graph whose edge weights vary linearly with a parameter. The second is an asymptotically optimal algorithm for the minimum ratio spanning tree problem, as well as other search problems, on dense graphs. 1 Introduction In the parametric minimum spanning tree problem, one is given an n-node, m-edge undirected graph G where each edge e has a linear weight function w e (#)=a e +#b e . Let Z(#) denote the weight of the minimum spanning tree relative to the weights w e (#). It can be shown that Z(#) is a piecewise linear concave function of # [Gus80]; the points at which the slope of Z changes are called breakpoints. We shall present two results regarding parametric minimum spanning trees. First, we show that Z(#) can be constructed in O(min{nm log n, TMST (2n, n) # Department of Computer Science, Iowa State University, Ames, IA...

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Abstract. In financial markets high levels of risk are associated with large returns as well as large losses, whereas with lower levels of risk, the potential for either return or loss is small. Therefore, risk management is fundamentally concerned with finding an optimal tradeoff between risk and return matching an investor’s risk tolerance. Managing risk is studied mostly in a financial context; nevertheless, it is certainly relevant in any area with a significant source of uncertainty. The mean-risk tradeoff is well-studied for problems with a convex feasible set. However, this is not the case in the discrete setting, even though, in practice, portfolios are often restricted to discrete choices. In this paper we study mean-risk minimization for problems with discrete decision variables. In particular, we consider discrete optimization problems with a submodular mean-risk minimization objective. We show the connection between extended polymatroids and the convex lower envelope of this mean-risk objective. For 0-1 problems a complete linear characterization of the convex lower envelope is given. For mixed 0-1 problems we derive an exponential class of conic quadratic inequalities that are separable with the greedy algorithm.

Abstract—This paper considers a general 0-1 random fuzzy programming problem based on the degree of necessity including some previous 0-1 stochastic and fuzzy programming problems. The proposal problem is not a well-defined problem due to including random fuzzy variables. Therefore, by introducing chance constraint and fuzzy goal for objective function, and considering the maximization for the degrees of necessity that the objective function value satisfies the fuzzy goal, main problem is transformed into a deterministic equivalent problem. Furthermore, by using the assumption that each random variable is distributed according to a normal distribution, the problem is equivalently transformed into a basic 0-1 programming problem, and the efficient strict solution method to find an optimal solution is constructed. Index Terms—0-1 programming problem, Random fuzzy variables, Degree of necessity, Relaxation problem

Abstract—This paper considers two general 0-1 random fuzzy programming problems based on the degree of necessity which include some previous 0-1 stochastic and fuzzy programming problems. The proposal problems are not well defined due to including randomness and fuzziness. Therefore, by introducing chance constraint and fuzzy goal for the objective function, and considering the maximization of the aspiration level for total profit and the degree of necessity that the objective function’s value satisfies the fuzzy goal, each main problem is transformed into a deterministic equivalent problem. Furthermore, by using the assumption that each random variable is distributed according to a normal distribution, the problem is equivalently transformed into a basic 0-1 programming problem, and the efficient strict solution method to find an optimal solution is constructed. Keywords—0-1 programming problem, Random fuzzy variables, Degree of necessity, Relaxation problem I.