TY - GEN
T1 - The power of recourse for online MST and TSP
AU - Megow, Nicole
AU - Skutella, Martin
AU - Verschae, José
AU - Wiese, Andreas
PY - 2012
Y1 - 2012
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=84879816158&partnerID=8YFLogxK
U2 - 10.1007/978-3-642-31594-7_58
DO - 10.1007/978-3-642-31594-7_58
M3 - Conference contribution
AN - SCOPUS:84879816158
SN - 9783642315930
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 689
EP - 700
BT - Automata, Languages, and Programming - 39th International Colloquium, ICALP 2012, Proceedings
T2 - 39th International Colloquium on Automata, Languages, and Programming, ICALP 2012
Y2 - 9 July 2012 through 13 July 2012
ER -