General Bounds for Incremental Maximization

  • We propose a theoretical framework to capture incremental s olutions to cardinality con- strained maximization problems. The defining characterist ic of our framework is that the cardinality/support of the solution is bounded by a value k ∈ N that grows over time, and we allow the solution to be extended one element at a time. We i nvestigate the best-possible competitive ratio of such an incremental solution, i.e., th e worst ratio over all k between the incremental solution after k steps and an optimum solution of cardinality k . We define a large class of problems that contains many important cardin ality constrained maximization problems like maximum matching, knapsack, and packing/cov ering problems. We provide a general 2 . 618-competitive incremental algorithm for this class of pr oblems, and show that no algorithm can have competitive ratio below 2 . 18 in general. In the second part of the paper, we focus on the inherently inc remental greedy algorithm that increases the objective value as much as possible in eac h step. This algorithm is known to be 1 . 58-competitive for submodular objective functions, but it has unbounded competitive ratio for the class of incremental problems mentioned above . We define a relaxed submod- ularity condition for the objective function, capturing pr oblems like maximum (weighted) ( b -)matching and a variant of the maximum flow problem. We show t hat the greedy algo- rithm has competitive ratio (exactly) 2 . 313 for the class of problems that satisfy this relaxed submodularity condition. Note that our upper bounds on the competitive ratios transla te to approximation ratios for the underlying cardinality constrained problems.

Download full text files

Export metadata

Additional Services

Share in Twitter Search Google Scholar
Metadaten
Author:Aaron Bernstein, Yann Disser, Martin Groß
Document Type:Article
Language:English
Date of Publication (online):2017/06/30
Date of first Publication:2017/08/07
Release Date:2017/08/07
Subprojects:A07