# Online Multistage Subset Maximization Problems

## TL;DR

This paper studies online algorithms for multistage subset maximization problems where solutions evolve over time with changing data, aiming to balance solution quality and stability, and provides bounds on algorithm performance.

## Contribution

It introduces general techniques for online multistage subset maximization and characterizes models with constant-competitive algorithms, including the use of lookahead.

## Key findings

- Characterizes models with constant-competitive online algorithms.
- Provides lower and upper bounds on algorithm performance.
- Employs lookahead to improve competitiveness when needed.

## Abstract

Numerous combinatorial optimization problems (knapsack, maximum-weight matching, etc.) can be expressed as \emph{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 \emph{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 \emph{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.