Abstract
We propose a theoretical framework to capture incremental solutions to cardinality constrained maximization problems. The defining characteristic of our framework is that the cardinality/support of the solution is bounded by a value k in N that grows over time, and we allow the solution to be extended one element at a time. We investigate the bestpossible competitive ratio of such an incremental solution, i.e., the 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 cardinality constrained maximization problems like maximum matching, knapsack, and packing/covering problems. We provide a general 2.618competitive incremental algorithm for this class of problems, 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 incremental greedy algorithm that increases the objective value as much as possible in each step. This algorithm is known to be 1.58competitive for submodular objective functions, but it has unbounded competitive ratio for the class of incremental problems mentioned above. We define a relaxed submodularity condition for the objective function, capturing problems like maximum (weighted) (b)matching and a variant of the maximum flow problem. We show that the greedy algorithm 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 translate to approximation ratios for the underlying cardinality constrained problems.
BibTeX  Entry
@InProceedings{bernstein_et_al:LIPIcs:2017:7465,
author = {Aaron Bernstein and Yann Disser and Martin Gro{\ss}},
title = {{General Bounds for Incremental Maximization}},
booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)},
pages = {43:143:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770415},
ISSN = {18688969},
year = {2017},
volume = {80},
editor = {Ioannis Chatzigiannakis and Piotr Indyk and Fabian Kuhn and Anca Muscholl},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2017/7465},
URN = {urn:nbn:de:0030drops74650},
doi = {10.4230/LIPIcs.ICALP.2017.43},
annote = {Keywords: incremental optimization, maximization problems, greedy algorithm, competitive analysis, cardinality constraint}
}
Keywords: 

incremental optimization, maximization problems, greedy algorithm, competitive analysis, cardinality constraint 
Collection: 

44th International Colloquium on Automata, Languages, and Programming (ICALP 2017) 
Issue Date: 

2017 
Date of publication: 

07.07.2017 