A fast (2+27)-approximation algorithm for capacitated cycle covering

• Published in: Mathematical programming Vol. 192; no. 1-2; pp. 497 - 518
• Main Authors: Traub, Vera, Tröbst, Thorben
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2022; Springer; Springer Nature B.V
• Subjects: Algorithms; Analysis; Apexes; Approximation; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Full Length Paper; Graph theory; Greedy algorithms; Lower bounds; Mathematical analysis; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Mathematics of Computing; Numerical Analysis; Route planning; Theoretical; Vehicle routing; 68W20; 68W25; 90C05; 90C27; 90B06
• ISSN: 0025-5610; 1436-4646; 1436-4646
• DOI: 10.1007/s10107-021-01678-3

We consider the capacitated cycle covering problem : given an undirected, complete graph G with metric edge lengths and demands on the vertices, we want to cover the vertices with vertex-disjoint cycles, each serving a demand of at most one. The objective is to minimize a linear combination of the total length and the number of cycles. This problem is closely related to the capacitated vehicle routing problem (CVRP) and other cycle cover problems such as min-max cycle cover and bounded cycle cover. We show that a greedy algorithm followed by a post-processing step yields a ( 2 + 2 7 ) -approximation for this problem by comparing the solution to a polymatroid relaxation. We also show that the analysis of our algorithm is tight and provide a 2 + ϵ lower bound for the relaxation.