The Power of Recourse for Online MST and TSP

Abstract

Abstract. We consider the online MST and TSP problems with recourse. The nodes of an unknown graph with metric edge cost appear one by one and must be connected in such a way that the resulting tree or tour has low cost. In the standard online setting, with irrevocable decisions, no algorithm can guarantee a constant competitive ratio. In our model we allow recourse actions by giving a limited budget of edge rearrangements per iteration. It has been an open question for more than 20 years if an online algorithm equipped with a constant (amortized) budget can guarantee constant-approximate solutions [7].As our main result, we answer this question affirmatively in an amortized setting. We introduce an algorithm that maintains a nearly optimal tree when given constant amortized budget. In the non-amortized setting, we specify a promising proof technique and conjecture a structural property of optimal solutions that would prove a constant competitive ratio with a single recourse action. It might seem rather optimistic that such a small budget should be sufficient for a significant cost improvement. However, we can prove such power of recourse in the offline setting in which the sequence of node arrivals is known. Even this problem prohibits constant approximations if there is no recourse allowed. Surprisingly, already a smallest recourse budget significantly improves the performance guarantee from non-constant to constant.Unlike in classical TSP variants, the standard double-tree and shortcutting approach does not give constant guarantees in the online setting. However, a non-trivial robust shortcutting technique allows to translate online MST results into TSP results at the loss of small factors.

The Power of Dynamic Distance Oracles

“…Online Steiner tree There has been an increasing interest in the online Steiner tree and the related online MST problem in recent years, which started with a paper by Megow et al [30]. They showed that in the incremental case one can maintain an approximate online MST in G[S] (and consequently an approximate Steiner tree) with only a constant number of changes to the tree per terminal insertion (in amortized sense), which resolved a long standing open problem posed in [28].…”

“…The result of [30] was improved to worst-case constant by Gu, Gupta and Kumar [20]. Then, Gupta and Kumar [21] have shown that constant worst-case number of changes is sufficient in the decremental case.…”

“…This problem was first introduced in the pioneering paper by Imase and Waxman [28] and its study was later continued in [30,20,21]. However, all these papers focus on minimizing the number of changes to the tree that are necessary to maintain a good approximation, and ignore the problem of efficiently finding these changes.…”

“…approximation algorithms [7,32,5,34,9,10], through online [28,30,20,21] and stochastic models [22,19], ending with game theoretic approaches [4,8]. Taking into account the significance and the wide interest in this problem, it is somewhat surprising that no efficient dynamic algorithms for this problem have been developed so far.…”

Serve or skip: the power of rejection in online bottleneck matching

“…Megow et al (2012) study the online minimum spanning tree problem, where points arrive online and the online algorithm must connect the point to the existing tree as they arrive. They show that by allowing for a small number of re-arrangements on previously-placed edges, a nearly optimal tree can be maintained.…”