范数预算打包问题的近似算法
Approximation Algorithms for Norm-Budgeted Packing Problems
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中文总结 AI 辅助
研究范数预算打包问题，通过涉及单调、对称范数的范数预算约束建模资源，将多种经典打包问题纳入框架，开发框架获得常数因子近似保证及相关问题的PTAS，并为子模版本开发常数因子近似算法。
中文摘要 AI 辅助
近年来，对基于单调、对称范数（及其推广）的基于范数目标的优化问题研究备受关注，但几乎都集中在覆盖问题上。本文引入并研究范数预算打包问题，其资源约束通过涉及单调、对称范数的范数预算约束建模。正式地，有带相关奖励和大小的元素、可行解的向下封闭集合及预算\(B\)。目标是在范数预算约束下最大化总奖励。多种经典打包问题可纳入此框架，还为其开发了框架，能得到常数因子近似保证，背包问题和MaxGAP在相同及相关机器上有PTAS，还为一些子模版本开发了常数因子近似算法。
英文摘要
In recent years, much attention has been devoted to the study of optimization problems under norm-based objectives coming from the rich class of monotone, symmetric norms (and their generalizations). This work has however almost exclusively focused on covering problems, wherein one seeks to minimize the norm of the cost vector induced by a solution. We introduce and study the class of {\em norm-budgeted packing problems}, which are packing problems where the resource constraints underlying the packing problem are modeled via a {\em norm budget constraint} involving a {\em monotone, symmetric norm}. Formally, we have some elements with associated rewards and sizes, a downwards-closed collection of feasible solutions, and a budget $B$. Each solution induces a size vector, and the goal is to maximize the total reward subject to the norm-budget constraint $f(\text{size vector})\leq B$. The versatility of monotone, symmetric norms implies that a variety of classical packing problems can be captured under the umbrella of norm-budgeted packing problems. Moreover, the closure properties of monotone, symmetric norms, also enable one to encode multiple different norm-budget constraints via a single monotone, symmetric norm. We consider the norm-budgeted versions of a variety of canonical packing problems, including knapsack, matching, maximum-weight independent set in a $k$-set system, maximum generalized assignment problem (MaxGAP), and $k$-facility location, and develop a framework that allows us to obtain {\em constant-factor approximation guarantees} for these problems, and {\em PTASes for knapsack, and MaxGAP on identical and related machines}. We also develop constant-factor approximation algorithms for the {\em submodular} versions of some norm-budgeted packing problems, wherein the reward function is now specified by a monotone, submodular function.
