解决大规模随机优化问题的鲁棒组合算法

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"《Robust Combinatorial Optimization with Exponential Scenarios》是一篇于2006年11月10日发表的计算机科学论文，由Uriel Feige、Kamal Jain、Mohammad Mahdian和Vahab Mirrokni四位作者共同完成。该研究关注的是在具有指数数量场景的鲁棒组合优化问题，这在当时是前所未有的挑战，因为在之前的研究中，主要探讨的是具有多项式数量场景的两阶段鲁棒优化问题。论文背景源自于对概率性优化的广泛研究，如文献[14]和[17]，这些工作主要关注如何在不确定性环境下设计近似算法。然而，Dhamdhere等人在该领域的突破之前，处理的问题规模局限在了多项式的场景数量。本文作者试图拓展这一边界，引入一种新的方法来解决当场景数量呈指数级增长时的复杂情况。在两阶段鲁棒优化中，首要任务是在第一阶段预购资源，以应对未知的未来威胁。第二阶段，一旦敌手（即不确定因素）选择需要服务的客户后，需要根据这些选择调整策略，可能需要以更高的价格购买额外资源。目标是找到一个策略，使得在最糟糕的情况下，总成本最小化。作者们提出了一种基于线性规划（LP）的通用解决方案策略。这种方法揭示了一个重要的洞察：通过巧妙地利用线性规划的特性，即使在面对大量潜在场景时，也能设计出有效的近似算法。他们的方法不仅适用于资源分配问题，还可能推广到其他类型的组合优化问题，比如网络设计、调度和生产计划等领域。这篇论文的重要性在于它不仅推动了鲁棒优化理论的发展，而且为实际应用中的决策制定者提供了一种处理复杂不确定性环境的实用工具，特别是在面对大量未知变量或变数可能性时。它对于理解和解决实际问题中的不确定性和波动性具有深远的影响，并为进一步的研究提供了新的研究方向。"

optimization is that in our model, the problem the oracle needs to solve is often computationally

intractable. For example, if the set of scenarios in a robust set cover problem is given by an

upper bound on the number of active clients, the oracle needs to ﬁnd a subset of k clients

whose minimum cost of covering is maximized. We call this problem the max-min set cover

problem, and will observe that it is computationally hard. Considering the hardness of the

oracle problem, the algorithms designed for the oracle model need to be able to work with an

approximate oracle as well. Furthermore, in order to solve a robust optimization problem in

the model where the scenarios are given by an upper bound on the number of active clients,

in addition to designing an algorithm for the oracle model, we need to give an approximation

algorithm for the max-min version of the problem.

1.2 Our Contribution

In this paper, we mostly focus on the model where the scenarios are given implicitly by an

upper bound on the number of active scenarios. This is motivated by real-world situations

where a good estimate of the total number of clients who will show up is available, but we do

not exactly know where they appear. We will also give a general LP-based algorithm for the

oracle model, assuming that the oracle gives a good approximation of the worst-case scenario

with respect to the fractional solution.

A naive idea to solve the robust optimization problems is a buy-at-once algorithm: either

cover all items in the ﬁrst round in which case nothing needs to be done in the second round. Or

do nothing in the ﬁrst round and construct a solution in the second round, after the adversary

makes its choices. The choice of which of the two options to use is based on a polynomial-

time test that is problem speciﬁc. We study this algorithm in Section 4 and prove that when

the inﬂation factor is the same for all scenarios the approximation ratio of this algorithm

for robust unweighted set cover, vertex cover, and edge cover problems are max(log m, log n),

2, and 2 respectively. However, the following example shows that for the weighted version

of robust vertex cover, any buy-at-once algorithm (even with unbounded computing power)

performs poorly. The input of the robust vertex cover problem are a node-weighted graph G,

a parameter k (for number of edges to be chosen by adversary), and an inﬂation factor λ > 1.

Consider a clique on n vertices, with k = 1 and λ =

√

n. All vertices have weight 1, except for

one vertex that has weight w =

√

n. The buy-at-once algorithm will either pay at least n in

the ﬁrst round, or at least λw = n in the second round. However, an optimal algorithm can

choose only the heavy vertex in the ﬁrst round, and then pay at most w +λk = 2

√

n. Hence the

approximation ratio of the buy-at-once algorithm for weighted robust vertex cover is no better

than Ω(

√

n). This example indicates the need for more sophisticated approximation algorithms

for robust two-stage optimization problems.

In Section 2, we give a general LP-based framework for solving robust covering problems

given access to an oracle that solves the max-min problem (or the adversary’s problem) and

another oracle that rounds the LP solution for the classical (i.e., non-robust) optimization

problem. More precisely, the separation oracle for this exponential LP’s, we need to solve a

max-min variant of the fractional covering problem. For example, in the max-min fractional

set cover problem, given a collection of subsets S

, S

, . . . , S

of a universe F each with a cost

c(S

) and a parameter k, we need to ﬁnd a subset T ⊆ F of size at most k for which the cost of

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