Randomized shortest paths with net flows and capacity constraints

Network flows and shortest paths

Let G = (V,E) be a directed graph with two special vertices: a source s and a sink t. Every edge (u, v) has a capacity cap(u, v) ≥ 0. For convenience, we define cap(u, v) = 0 if (u, v) in not an edge. More specifically, we work with a modified set of edges {(u, v) : (u, v) ∈ E or (v, u) ∈ E} with cap(u, v) = 0 if (u, v) 6∈ E. A flow is a real-valued function on edges with three properties: • Sk...

Shortest Paths with Single-Point Visibility Constraints

This paper studies the problem of finding the shortest path between two points in presence of single-point visibility constraints. In this type of constraints, there should be at least one point on the output path from which a fixed viewpoint is visible. The problem is studied in various domains including simple polygons, polygonal domains, polyhedral surfaces. The method is based on partitioni...

Shortest Paths and Multicommodity Network Flows

Single Source Shortest Paths for All Flows with Integer Costs

We consider a shortest path problem for a directed graph with edges labeled with a cost and a capacity. The problem is to push an unsplittable flow f from a specified source to all other vertices with the minimum cost for all f values. Let G = (V,E) with |V | = n and |E| = m. If there are t different capacity values, we can solve the single source shortest path problem t times for all f in O(tm...

Shortest paths in simple polygons with polygon-meet constraints

We study a constrained version of the shortest path problem in simple polygons, in which the path must visit a given target polygon. We provide a worst-case optimal algorithm for this problem and also present a method to construct a subdivision of the simple polygon to efficiently answer queries to retrieve the shortest polygon-meeting paths from a single-source to the query point. The algorith...