Given a set of n points on the plane colored with k ≤ n colors, the Trip Planning Problem asks for the shortest path visiting the k colors. It is a well-known NP-hard problem. We show that under some natural constraints on the path, the problem can be solved in polynomial time.

This study proposes a model of adding edges of forming a simple path to a level of depth N in a complete K-ary (K ≥ 3) tree of height H under giving priority to edges between two nodes of which the deepest common ancestor is deeper. An optimal depth N∗ is obtained by maximizing the total shortening path length which is the sum of shortening lengths of shortest paths between every pair of all no...

In this paper, we introduce the Steiner Bi-objective Shortest Path Problem. This problem is defined on a directed graph G=(V,A), with subset T?V of terminals. Arcs are labeled travel time and cost. The goal to find complete set efficient paths between every pair nodes in T. motivation behind stems from data preprocessing for vehicle routing problems. We propose solution method based labeling ap...

Abstract. Time-varying network optimization problems have tradition-ally been solved by specialized algorithms. These algorithms have NP-complement time complexity. This paper considers the time-varying short-est path problem, in which can be optimally solved in O(T(m + n)) time,where T is a given integer. For this problem with arbitrary waiting times,we propose an approximation algorithm, whic...

The enumeration of shortest paths in cubic grid is presented here. The cubic grid considers three neighborhods— 6-neighborhood (face connectivity), 18-neighborhood (edge connectivity), and 26-neighborhood (vertex connectivity). The formulation for distance metrics are given here. L1, D18, and L∞ are the three metrics for 6-neighborhood, 18-neighborhood, and 26-neighborhood. The problem is to fi...

This paper considers algorithms for counterexample generation for (bounded) probabilistic reachability properties in fully probabilistic systems. Finding the strongest evidence (i.e, the most probable path) violating a (bounded) until-formula is shown to be reducible to a single-source (hop-constrained) shortest path problem. Counterexamples of smallest size that are mostly deviating from the r...