Unless P = NP we have to satisfy ourselves with any two out of the three goals. Most of the early undergrad algorithms like matching, shortest path etc. are exact and fast. To tackle hard problems and obtain a fast solution we use approximation algorithms, PTAS etc. FPT or fixed parameter tractable algorithms come to our rescue when we need to tackle hard problems yet obtain an optimal solution...

A natural way to deal with multiple, partially conflicting objectives is turning all the objectives but one into budget constraints. Some classical polynomial-time optimization problems, such as spanning tree and forest, shortest path, (perfect) matching, independent set (basis) in a matroid or in the intersection of two matroids, become NP-hard even with one budget constraint. Still, for most ...

In this paper, we consider an adaptive approach to address optimization problems with uncertain cost parameters. Here, the decision maker selects an initial decision, observes the realization of the uncertain cost parameters, and then is permitted to modify the initial decision. We treat the uncertainty using the framework of robust optimization in which uncertain parameters lie within a given ...

This paper introduces a new variant of the Shortest Path Problem (SPP) called Cost-Balanced (CBPP). Various real problems can either be modeled as BCPP or include sub-problem. We prove several properties related to complexity CBPP problem. In particular, we demonstrate that problem is NP-hard in its general version, but it becomes solvable polynomial time specific family instances. Moreover, ma...

This paper studies the asymptotic behavior of several continuous-time dynamical systems which are analogs of ant colony optimization algorithms that solve shortest path problems. Local asymptotic stability of the equilibrium corresponding to the shortest path is shown under mild assumptions. A complete study is given for a recently proposed model called EigenAnt: global asymptotic stability is ...