Minimum Constraint Removal Problem for Line Segments is NP-hard

TL;DR

This paper proves that the Minimum Constraint Removal problem for line segments is NP-hard, extending the known complexity results from arbitrary shapes to the specific case of line segments.

Contribution

It demonstrates NP-hardness of MCR for line segments using a reduction from the Subset Sum problem, filling a gap in the problem's complexity classification.

Findings

MCR for line segments is NP-hard.

NP-hardness holds for both weighted and unweighted line segments.

The problem's complexity is established through a reduction from Subset Sum.

Abstract

In the minimum constraint removal ($MCR$), there is no feasible path to move from the starting point towards the goal and, the minimum constraints should be removed in order to find a collision-free path. It has been proved that $MCR$ problem is $NP-hard$ when constraints have arbitrary shapes or even they are in shape of convex polygons. However, it has a simple linear solution when constraints are lines and the problem is open for other cases yet. In this paper, using a reduction from Subset Sum problem, in three steps, we show that the problem is NP-hard for both weighted and unweighted line segments.