We consider truthful combinatorial auctions with items $M = [m]$ for sale to $n$ bidders, where each bidder $i$ has a private monotone valuation $v_i : 2^M \to R_+$. Among truthful mechanisms, maximal-in-range (MIR) mechanisms achieve the best-known approximation guarantees among all poly-communication deterministic truthful mechanisms in all previously-studied settings. Our work settles the communication necessary to achieve any approximation guarantee via an MIR mechanism. Specifically: Let MIRsubmod$(m,k)$ denote the best approximation guarantee achievable by an MIR mechanism using $2^k$ communication between bidders with submodular valuations over $m$ items. Then for all $k = \Omega(\log(m))$, MIRsubmod$(m,k) = \Omega(\sqrt{m/(k\log(m/k))})$. When $k = \Theta(\log(m))$, this improves the previous best lower bound for poly-comm. MIR mechanisms from $\Omega(m^{1/3}/\log^{2/3}(m))$ to $\Omega(\sqrt{m}/\log(m))$. We also have MIRsubmod$(m,k) = O(\sqrt{m/k})$. Moreover, our mechanism is optimal w.r.t. the value query and succinct representation models. When $k = \Theta(\log(m))$, this improves the previous best approximation guarantee for poly-comm. MIR mechanisms from $O(\sqrt{m})$ to $O(\sqrt{m/\log(m)})$. Let also MIRgen$(m,k)$ denote the best approximation guarantee achievable by an MIR mechanism using $2^k$ communication between bidders with general valuations over $m$ items. Then for all $k = \Omega(\log(m))$, MIRgen$(m,k) = \Omega(m/k)$. When $k = \Theta(\log(m))$, this improves the previous best lower bound for poly-comm. MIR mechanisms from $\Omega(m/\log^2(m))$ to $\Omega(m/\log(m))$. We also have MIRgen$(m,k) = O(m/k)$. Moreover, our mechanism is optimal w.r.t. the value query and succinct representation models. When $k = \Theta(\log(m))$, this improves the previous best approximation guarantee for poly-comm. MIR mechanisms from $O(m/\sqrt{\log(m)})$ to $O(m/\log(m))$.

翻译：我们考虑真实组合拍卖问题，其中包含物品集合 $M = [m]$ 出售给 $n$ 个竞拍者，每个竞拍者 $i$ 拥有一个私有的单调估值函数 $v_i : 2^M \to R_+$。在所有真实机制中，最大范围（MIR）机制在所有先前研究场景下，实现了所有多项式通信确定性真实机制中已知最优的近似保证。本文工作解决了通过MIR机制实现任意近似保证所需的通信复杂度。具体而言：设MIRsubmod$(m,k)$表示在子模估值函数下，使用$2^k$通信量、针对$m$个物品可实现的MIR机制最优近似保证。则对于所有$k = \Omega(\log(m))$，有MIRsubmod$(m,k) = \Omega(\sqrt{m/(k\log(m/k))})$。当$k = \Theta(\log(m))$时，该结果将多项式通信MIR机制的先前的下界从$\Omega(m^{1/3}/\log^{2/3}(m))$改进为$\Omega(\sqrt{m}/\log(m))$。同时，我们得到MIRsubmod$(m,k) = O(\sqrt{m/k})$。此外，我们的机制在价值查询模型和简洁表示模型下均为最优。当$k = \Theta(\log(m))$时，该结果将多项式通信MIR机制先前的近似保证上界从$O(\sqrt{m})$改进为$O(\sqrt{m/\log(m)})$。设MIRgen$(m,k)$表示在一般估值函数下，使用$2^k$通信量、针对$m$个物品可实现的MIR机制最优近似保证。则对于所有$k = \Omega(\log(m))$，有MIRgen$(m,k) = \Omega(m/k)$。当$k = \Theta(\log(m))$时，该结果将多项式通信MIR机制先前的下界从$\Omega(m/\log^2(m))$改进为$\Omega(m/\log(m))$。同时，我们得到MIRgen$(m,k) = O(m/k)$。此外，我们的机制在价值查询模型和简洁表示模型下均为最优。当$k = \Theta(\log(m))$时，该结果将多项式通信MIR机制先前的近似保证上界从$O(m/\sqrt{\log(m)})$改进为$O(m/\log(m))$。