A Preemptive Algorithm for Maximizing Disjoint Paths on Trees

• Published in: Algorithmica Vol. 57; no. 3; pp. 517 - 537
• Main Authors: Azar, Yossi, Feige, Uriel, Glasner, Daniel
• Format: Journal Article Conference Proceeding
• Language: English
• Published: New York Springer-Verlag 01.07.2010; Springer
• Subjects: Algorithm Analysis and Problem Complexity; Algorithmics. Computability. Computer arithmetics; Algorithms; Applied sciences; Computer Science; Computer science; control theory; systems; Computer systems and distributed systems. User interface; Computer Systems Organization and Communication Networks; Data Structures and Information Theory; Exact sciences and technology; Mathematics of Computing; Software; Theoretical computing; Theory of Computation; Online algorithms; Admission control; Disjoint paths; Disjoint paths on trees; Call control; Access control; Preemption; Lower bound; On line; Logarithmic function; Competitiveness; Routing; Approximation algorithm; Randomized algorithm; Combinatorial optimization; Randomization; Network protocol; Capacity constraint; Deterministic algorithms
• ISSN: 0178-4617; 1432-0541
• DOI: 10.1007/s00453-009-9305-4

We consider the on-line version of the maximum vertex disjoint path problem when the underlying network is a tree. In this problem, a sequence of requests arrives in an on-line fashion, where every request is a path in the tree. The on-line algorithm may accept a request only if it does not share a vertex with a previously accepted request. The goal is to maximize the number of accepted requests. It is known that no on-line algorithm can have a competitive ratio better than Ω(log  n ) for this problem, even if the algorithm is randomized and the tree is simply a line. Obviously, it is desirable to beat the logarithmic lower bound. Adler and Azar (Proc. of the 10th ACM-SIAM Symposium on Discrete Algorithm, pp. 1–10, 1999 ) showed that if preemption is allowed (namely, previously accepted requests may be discarded, but once a request is discarded it can no longer be accepted), then there is a randomized on-line algorithm that achieves constant competitive ratio on the line. In the current work we present a randomized on-line algorithm with preemption that has constant competitive ratio on any tree. Our results carry over to the related problem of maximizing the number of accepted paths subject to a capacity constraint on vertices (in the disjoint path problem this capacity is 1). Moreover, if the available capacity is at least 4, randomization is not needed and our on-line algorithm becomes deterministic.