# Collision-Free Routing Problem with Restricted L-Path

## Abstract

We consider a variant of collision-free routing problem *CRP*. In this problem, we are given set *C* of *n* vehicles which are moving in a plane along a predefined directed rectilinear path. Our objective (*CRP*) is to find the maximum number of vehicles that can move without collision. *CRP* is shown to be NP-Hard by Ajaykumar et al. [1]. It was also shown that the approximation of this problem is as hard as Maximum Independent Set problem (*MIS*) even if the paths between a pair of vehicles intersects at most once. So we study the constrained version *CCRP* of *CRP* in which each vehicle \(c_i\) is allowed to move in a directed L-Shaped Path.

We prove *CCRP* is NP-Hard by a reduction from MIS in L-graphs, which was proved to be NP-Hard even for unit L-graph by Lahiri et al. [2]. Simultaneously, we show that any *CCRP* can be partitioned into collection \(\mathcal L\) of L-graphs such that *CCRP* reduces to a problem of finding *MIS* in L-graph for each partition in \(\mathcal L\). Thus we show that any algorithm, that can produce a \(\beta \)-approximation for L-graph, would produce a \(\beta \)-approximation for *CCRP*. We show that unit L-graphs intersected by an axis-parallel line is Co-comparable. For this problem, we propose an algorithm for finding MIS that runs in \(O(n^2)\) time and uses *O*(*n*) space. As a corollary, we get a 2-approximation algorithm for finding MIS of unit L-graph that runs in \(O(n^2)\) time and uses *O*(*n*) space.

## Keywords

Maximum Independent Set L-Graphs Approximation algorithm Collision-free Co-comparable graph## Notes

### Acknowledgments

The authors are thankful to Joydeep Mukherjee for many useful discussions.

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