General bounds for incremental maximization
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Series A
General bounds for incremental maximization Aaron Bernstein1 · Yann Disser2
· Martin Groß3 · Sandra Himburg2
Received: 16 December 2019 / Accepted: 6 October 2020 © The Author(s) 2020
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 ∈ N that grows over time, and we allow the solution to be extended one element at a time. We investigate the best-possible 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 consider 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 2.618-competitive incremental algorithm for this class of problems, and we 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 1.58-competitive 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) d-dimensional matching, maximum (weighted) (b-)matching and a variant of the maximum flow problem. We show a general bound for the competitive ratio of the greedy algorithm on the class of problems that satisfy this relaxed submodularity condition. Our bound generalizes the (tight) bound of 1.58 slightly beyond sub-modular functions and yields a tight bound of 2.313 for maximum (weighted) (b-)matching. Our bound is also tight for a more general class of functions as the relevant parameter goes to infinity. Note that our upper bounds on the competitive ratios translate to approximation ratios for the underlying cardinality constrained problems, and our bounds for the greedy algorithm carry over both.
Some results of this paper appeared in preliminary form in [1]. Supported by the German Research Foundation (DFG) within Project A07 of CRC TRR 154 and Project 413095939. Extended author information available on the last page of the article
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Keywords Incremental optimization · Maximization problems · Greedy algorithm · Competitive analysis · Cardinality constraint Mathematics Subject Classification 68W27 · 68W25 · 90C27 · 68Q25
1 Introduction Practical solutions to optimization problems are often inherently incremental in the sense that they evolve historically instead of being established in a one-shot fashion. This is especially true when solutions are expensive and need time and repeated investments to be implemented, for example when optimizing the layout of logistics and other infras
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