Online Multistage Subset Maximization Problems

Numerous combinatorial optimization problems (knapsack, maximum-weight matching, etc.) can be expressed as subset maximization problems: One is given a ground set $$N=\{1,\dots ,n\}$$ , 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 to choose $$S\in maximizes p(S). We consider the multistage version (Eisenstat et al., Gupta both ICALP 2014) profit $$p_t$$ (and possibly feasible solutions {F}_t$$ ) may change over time. Since in many applications changing solution costly, becomes find sequence optimizes trade-off between good per-time stable taking into account additional similarity bonus. As measure for two consecutive solutions, we either size intersection or difference n Hamming distance characteristic vectors. study online setting, is, (along with only arrive one by and, upon arrival, algorithm has output corresponding without knowledge future. develop general techniques thereby characterize those models (given type data evolution measure) admit constant-competitive algorithm. When no constant competitive ratio possible, employ lookahead circumvent this issue. provide almost matching lower upper bounds on best achievable one.