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.

翻译：我们研究带正析取约束的最短路径问题的参数化复杂度。在正析取约束中，存在特定的边对，使得任何可行解必须从每对边中至少选取一条。本文针对经典版本已证明为NP完全问题的带正析取约束的最短路径问题展开参数化研究。形式化地，给定无向图G=(V,E)及强制图H=(E,F)，其中H的顶点集与G的边集相同。目标是寻找G中至多k条边的集合S，使得S构成H的顶点覆盖，且边集S在G中诱导的子图包含从s到t的路径。我们考虑了该问题的两种自然参数化形式。其一为解规模参数k，当G和H均为一般图时，我们给出了包含O(k^5)个顶点的核化结果；此外，当G或H（但非二者同时）属于特殊图类时，我们得到了O(k^3)个顶点的核化结果。另一自然参数涉及H的结构性质，即H通过顶点删除转化为特殊图类所需删除的顶点规模。针对此类结构参数化问题，我们证明了若干固定参数可解性结果。