On the Polynomial Kernelizations of Finding a Shortest Path with Positive Disjunctive Constraints

TL;DR

This paper explores kernelization and fixed-parameter tractability for a constrained shortest path problem, introducing polynomial kernels for certain parameters and graph classes, advancing parameterized complexity understanding.

Contribution

It initiates the study of kernelization for shortest path with positive disjunctive constraints, providing polynomial kernels for specific parameters and graph classes.

Findings

Kernel with O(k^5) vertices for general graphs

Kernel with O(k^3) vertices for special graph classes

Fixed-parameter tractability results for structural parameters

Abstract

We study the SHORTEST PATH problem with positive disjunctive constraints from the perspective of parameterized complexity. For positive disjunctive constraints, there are certain pair of edges such that any feasible solution must contain at least one edge from every such pair. In this paper, we initiate the study of SHORTEST PATH problem subject to some positive disjunctive constraints the classical version is known to be NP-Complete. Formally, given an undirected graph G = (V, E) with a forcing graph H = (E, F) such that the vertex set of H is same as the edge set of G. The goal is to find a set S of at most k edges from G such that S forms a vertex cover in H and there is a path from s to t in the subgraph of G induced by the edge set S. In this paper, we consider two natural parameterizations for this problem. One natural parameter is the solution size, i.e. k for which we provide a kernel with O(k^5) vertices when both G and H are general graphs. Additionally, when either G or H (but not both) belongs to some special graph classes, we provied kernelization results with O(k^3) vertices . The other natural parameter we consider is structural properties of H, i.e. the size of a vertex deletion set of H to some special graph classes. We provide some fixed-parameter tractability results for those structural parameterizations.