使用Demand Oracle进行优化
Optimization with Demand Oracles
摘要：
我们研究\emph{combinatorial procurement auctions}，其中具有估价函数$v$和预算$B$的买方希望购买一组物品。每个项目$i$都有一个成本$c_i$，买方对一组$S$感兴趣，该组$S$最大化了服从$Σ_{i\in S}c_i\leq B$的$v(S)$。组合采购拍卖的特殊情况是子模块优化的经典问题。特别地，当成本都相等（\emph{cardinality constraint}）时，Nemhauser等人的经典结果表明贪婪算法提供了$\frac e {e-1}$近似。出于许多论文，利用需求查询，以引起代理在经济环境中的偏好，我们开发的算法，保证改进的近似比的存在下，需求预言。我们能够打破$\frac e {e-1}$障碍：我们提出的算法，只使用多项式许多需求查询，并有近似比$\frac 9 8+ε$的一般问题和$\frac 9 8$的最大化受到基数约束。我们还考虑了更一般的次可加赋值类。我们提出的算法，获得近似比$2+ε$的一般问题和2最大化的基数约束。即使估值是非单调的，我们也保证这些近似比。我们表明，这些比率基本上是最优的，在这个意义上，对于任何常数$ε>0$，获得近似比$2-ε$需要指数许多需求查询。
英文摘要：
We study \emph{combinatorial procurement auctions}, where a buyer with a valuation function $v$ and budget $B$ wishes to buy a set of items. Each item $i$ has a cost $c_i$ and the buyer is interested in a set $S$ that maximizes $v(S)$ subject to $Σ_{i\in S}c_i\leq B$. Special cases of combinatorial procurement auctions are classical problems from submodular optimization. In particular, when the costs are all equal (\emph{cardinality constraint}), a classic result by Nemhauser et al shows that the greedy algorithm provides an $\frac e {e-1}$ approximation. Motivated by many papers that utilize demand queries to elicit the preferences of agents in economic settings, we develop algorithms that guarantee improved approximation ratios in the presence of demand oracles. We are able to break the $\frac e {e-1}$ barrier: we present algorithms that use only polynomially many demand queries and have approximation ratios of $\frac 9 8+ε$ for the general problem and $\frac 9 8$ for maximization subject to a cardinality constraint. We also consider the more general class of subadditive valuations. We present algorithms that obtain an approximation ratio of $2+ε$ for the general problem and 2 for maximization subject to a cardinality constraint. We guarantee these approximation ratios even when the valuations are non-monotone. We show that these ratios are essentially optimal, in the sense that for any constant $ε>0$, obtaining an approximation ratio of $2-ε$ requires exponentially many demand queries.
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