Abstract
Numerous combinatorial optimization problems (knapsack, maximum-weight matching, etc.) can be expressed as subset maximization problems: One is given a ground set $$N=\{1,\dots ,n\}$$ , a collection $$\mathcal {F}\subseteq 2^N$$ of subsets thereof such that $$\emptyset \in \mathcal {F}$$ , and an objective (profit) function $$p:\mathcal {F}\rightarrow \mathbb {R}_+$$ . The task is to choose a set $$S\in \mathcal {F}$$ that maximizes p(S). We consider the multistage version (Eisenstat et al., Gupta et al., both ICALP 2014) of such problems: The profit function $$p_t$$ (and possibly the set of feasible solutions $$\mathcal {F}_t$$ ) may change over time. Since in many applications changing the solution is costly, the task becomes to find a sequence of solutions that optimizes the trade-off between good per-time solutions and stable solutions taking into account an additional similarity bonus. As similarity measure for two consecutive solutions, we consider either the size of the intersection of the two solutions or the difference of n and the Hamming distance between the two characteristic vectors. We study multistage subset maximization problems in the online setting, that is, $$p_t$$ (along with possibly $$\mathcal {F}_t$$ ) only arrive one by one and, upon such an arrival, the online algorithm has to output the corresponding solution without knowledge of the future. We develop general techniques for online multistage subset maximization and thereby characterize those models (given by the type of data evolution and the type of similarity measure) that admit a constant-competitive online algorithm. When no constant competitive ratio is possible, we employ lookahead to circumvent this issue. When a constant competitive ratio is possible, we provide almost matching lower and upper bounds on the best achievable one.
Highlights
In a classical combinatorial optimization setting, given an instance of a problem one needs to find a good feasible solution
In Subsubsection 3.1.2, we show that already very natural assumptions suffice: Assuming that at each time the feasible solutions are closed under taking subsets and the objective function is subadditive, we give a (21/8 + o(1))-competitive algorithm for the model with a general evolution and Hamming bonus, improving the previous (3 + o(1))-competitive ratio
We have developed techniques for online multistage subset maximization problems and thereby settled the achievable competitive ratios in the various settings almost exactly
Summary
In a classical combinatorial optimization setting, given an instance of a problem one needs to find a good feasible solution. As it is usual in the online setting, we consider no limitations in the computational resources available This implies that at every time step t, where instance It is known, we assume the existence of an oracle able to compute the optimal solution for that time step. This is a very general class of problems, including the maximization Subset Selection problems studied by Pruhs and Woeginger in [23] (they only considered linear objective functions) It contains for instance graph problems where N is the set of. In a Multistage Subset Maximization problem P, we are given a number of steps T ∈ N, a set N of n objects; for any t ∈ T , an instance It of the optimization problem. General Evolution (GE): any modification in the input sequence is possible Both the profits and the set of feasible solutions may change over time. For knapsack, profits and weights of object (and the capacity of the bag) may change over time; for maximum independent set edges in the graph may change,
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