Fixed-Parameter Tractable Algorithms for Corridor Guarding Problems

TL;DR

This paper introduces fixed-parameter tractable algorithms for various corridor guarding problems in orthogonal arrangements, addressing NP-Complete cases by parameterizing key aspects like walk length and number of rooms.

Contribution

It establishes FPT algorithms for k-CMST, k-CTSP, b-MLC, and k-MCC, providing new solutions for these NP-Complete corridor guarding problems.

Findings

k-CMST/k-CTSP are FPT with respect to k.

b-MLC is FPT with respect to link-distance b.

k-MCC is FPT with respect to the number of rooms k.

Abstract

Given an orthogonal connected arrangement of line-segments, Minimum Corridor Guarding(MCG) problem asks for an optimal tree/closed walk such that, if a guard moves through the tree/closed walk, all the line-segments are visited by the guard. This problem is referred to as Corridor-MST/Corridor-TSP (CMST/CTSP) for the cases when the guarding walk is a tree/closed walk, respectively. The corresponding decision version of MCG is shown to be NP-Complete[1]. The parameterized version of CMST/CTSP referred to as k-CMST/k-CTSP, asks for an optimal tree/closed walk on at most k vertices, that visits all the line-segments. Here, vertices correspond to the endpoints and intersection points of the input line-segments. We show that k-CMST/k-CTSP is fixed-parameter tractable (FPT) with respect to the parameter k. Next, we propose a variant of CTSP referred to as Minimum Link CTSP(MLC), in which the link-distance of the closed walk has to be minimized. Here, link-distance refers to the number of links or connected line-segments with same orientation in the walk. We prove that the decision version of MLC is NP-Complete, and show that the parameterized version, namely b-MLC, is FPT with respect to the parameter b, where b corresponds to the link-distance. We also consider another related problem, the Minimum Corridor Connection (MCC). Given a rectilinear polygon partitioned into rectilinear components or rooms, MCC asks for a minimum length tree along the edges of the partitions, such that every room is incident to at least one vertex of the tree. The decision version of MCC is shown to be NP-Complete[2]. We prove the fixed parameter tractability of the parameterized version of MCC, namely k-MCC with respect to the parameter k, where k is the number of rooms.