# On the approximability of Time Disjoint Walks

- 4 Downloads

## Abstract

We introduce the combinatorial optimization problem Time Disjoint Walks (TDW), which has applications in collision-free routing of discrete objects (e.g., autonomous vehicles) over a network. This problem takes as input a digraph \(G\) with positive integer arc lengths, and \(k\) pairs of vertices that each represent a *trip demand* from a source to a destination. The goal is to find a walk and delay for each demand so that no two trips occupy the same vertex *at the same time*, and so that a min–max or min–sum objective over the trip durations is realized. We focus here on the min–sum variant of Time Disjoint Walks, although most of our results carry over to the min–max case. We restrict our study to various subclasses of DAGs, and observe that there is a sharp complexity boundary between Time Disjoint Walks on oriented stars and on oriented stars with the central vertex replaced by a path. In particular, we present a poly-time algorithm for min–sum and min–max TDW on the former, but show that min–sum TDW on the latter is NP-hard. Our main hardness result is that for DAGs with max degree \(\varDelta \le 3\), min–sum Time Disjoint Walks is APX-hard. We present a natural approximation algorithm for the same class, and provide a tight analysis. In particular, we prove that it achieves an approximation ratio of \(\varTheta (k/\log k)\) on bounded-degree DAGs, and \(\varTheta (k)\) on DAGs and bounded-degree digraphs.

## Keywords

Disjoint paths Scheduling Approximation Algorithms Complexity Path planning Collision avoidance Autonomous vehicles## Notes

## References

- Ausiello G, Crescenzi P, Gambosi G, Kann V, Marchetti-Spaccamela A, Protasi M (1999) Complexity and approximation: combinatorial optimization problems and their approximability properties. Springer, BerlinCrossRefGoogle Scholar
- Bayen A, Goodman J, Vinitsky E (2018) On the approximability of Time Disjoint Walks. In: Kim D, Uma RN, Zelikovsky A (eds) Proceedings of the 12th international conference on combinatorial optimization and applications (COCOA ’18), LNCS, vol 11346. Springer, Cham, pp 62–78CrossRefGoogle Scholar
- Berman P, Karpinski M (1999) On some tighter inapproximability results. In: Nielsen M, van Emde Boas P, Wiedermann J (eds) Proceedings of the 26th international colloquium on automata, languages, and programming (ICALP ’99), LNCS, vol 1644. Springer, Berlin, pp 200–209Google Scholar
- Graham RL (1966) Bounds for certain multiprocessing anomalies. Bell Syst Tech J 45(9):1563–1581CrossRefGoogle Scholar
- Groß M, Skutella M (2012) Maximum multicommodity flows over time without intermediate storage. In: Epstein L, Ferragina P (eds) Proceedings of the 20th annual European symposium on algorithms (ESA 2012), LNCS, vol 7501. Springer, Berlin, pp 539–550CrossRefGoogle Scholar
- Guruswami V, Khanna S, Rajaraman R, Shepherd B, Yannakakis M (2003) Near-optimal hardness results and approximation algorithms for edge-disjoint paths and related problems. J Comput Syst Sci 67(3):473–496MathSciNetCrossRefGoogle Scholar
- Karp RM (1975) On the computational complexity of combinatorial problems. Networks 5(1):45–68CrossRefGoogle Scholar
- Kleinberg JM (1996) Approximation algorithms for disjoint paths problems. Ph.D. thesis, Massachusetts Institute of TechnologyGoogle Scholar
- Kobayashi Y, Sommer C (2010) On shortest disjoint paths in planar graphs. Discrete Optim 7(4):234–245MathSciNetCrossRefGoogle Scholar
- Korte B, Lovász L, Promel HJ, Schrijver A (1990) Paths, flows, and VLSI-layout. Springer, BerlinzbMATHGoogle Scholar
- Lengauer T (1990) Combinatorial algorithms for integrated circuit layout. Vieweg+Teubner Verlag, WiesbadenzbMATHGoogle Scholar
- Marx D (2005) A short proof of the NP-completeness of minimum sum interval coloring. Oper Res Lett 33(4):382–384MathSciNetCrossRefGoogle Scholar
- Robertson N, Seymour PD (1995) Graph minors. XIII. The disjoint paths problem. J Comb Theory Ser B 63(1):65–110MathSciNetCrossRefGoogle Scholar
- Scheffler P (1994) A practical linear time algorithm for disjoint paths in graphs with bounded tree-width. Technical report 396, Technische Universität Berlin, Institut für MathematikGoogle Scholar
- Skutella M (2009) An introduction to network flows over time. In: Cook W, Lovász L, Vygen J (eds) Research trends in combinatorial optimization. Springer, Berlin, pp 451–482CrossRefGoogle Scholar
- Srinivas A, Modiano E (2003) Minimum energy disjoint path routing in wireless ad-hoc networks. In: Proceedings of the 9th annual international conference on mobile computing and networking (MobiCom ’03). ACM, pp 122–133Google Scholar
- Szkaliczki T (1999) Routing with minimum wire length in the dogleg-free Manhattan model is NP-complete. SIAM J Comput 29(1):274–287MathSciNetCrossRefGoogle Scholar
- Torrieri D (1992) Algorithms for finding an optimal set of short disjoint paths in a communication network. IEEE Trans Commun 40(11):1698–1702CrossRefGoogle Scholar
- Zuckerman D (2006) Linear degree extractors and the inapproximability of max clique and chromatic number. In: Proceedings of the thirty-eighth annual ACM symposium on theory of computing (STOC ’06). ACM, pp 681–690Google Scholar