多预算优化
Optimization with More than One Budget
摘要：
处理多个部分冲突的目标的一个自然方法是将除一个目标之外的所有目标都变成预算约束。一些经典的多项式时间优化问题，如生成树与森林、最短路、（完美）匹配、拟阵中的独立集（基）和两个拟阵的交集等，即使只有一个预算约束也是NP难的。尽管如此，对于大多数这些问题的确定性和随机多项式时间近似方案是已知的。在两个或多个预算的情况下，通常只有多标准近似方案可用，其返回稍微不可行的解决方案。然而，对于严格预算限制的情况知之甚少：填补这一空白是本文的主要目标。我们表明，最短路径，完美匹配，和生成树（因此拟阵的基础和拟阵相交的基础）是不可近似的已经有两个预算约束。对于剩余的问题，其解决方案的集合形成一个独立的系统，我们提出了确定性和随机多项式时间近似计划的一个常数k的预算约束。我们的研究结果是基于各种技术：1。我们提出了一个简单而强大的机制，将多准则近似方案转化为纯近似方案。2.我们表明，在低维面的任何拟阵多面体的点几乎是完整的，一个有趣的结果本身。这给出了k-预算拟阵独立集的一个确定性逼近方案。3.我们提出了一个确定性的近似方案2预算匹配。这个结果的主干是R^2中曲线的一个纯拓扑性质。
英文摘要：
A natural way to deal with multiple, partially conflicting objectives is turning all the objectives but one into budget constraints. Some classical polynomial-time optimization problems, such as spanning tree and forest, shortest path, (perfect) matching, independent set (basis) in a matroid or in the intersection of two matroids, become NP-hard even with one budget constraint. Still, for most of these problems deterministic and randomized polynomial-time approximation schemes are known. In the case of two or more budgets, typically only multi-criteria approximation schemes are available, which return slightly infeasible solutions. Not much is known however for the case of strict budget constraints: filling this gap is the main goal of this paper. We show that shortest path, perfect matching, and spanning tree (and hence matroid basis and matroid intersection basis) are inapproximable already with two budget constraints. For the remaining problems, whose set of solutions forms an independence system, we present deterministic and randomized polynomial-time approximation schemes for a constant number k of budget constraints. Our results are based on a variety of techniques: 1. We present a simple and powerful mechanism to transform multi-criteria approximation schemes into pure approximation schemes. 2. We show that points in low dimensional faces of any matroid polytope are almost integral, an interesting result on its own. This gives a deterministic approximation scheme for k-budgeted matroid independent set. 3. We present a deterministic approximation scheme for 2-budgeted matching. The backbone of this result is a purely topological property of curves in R^2.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
