In linear combinatorial optimization, we aim to find $S^* = \arg\min_{S \in \mathcal{F}} \langle w,\mathbf{1}_S \rangle$ for a family $\mathcal{F} \subseteq 2^U$ over a ground set $U$ of $n$ elements. Traditionally, $w$ is known or accessible via a value oracle. Motivated by practical applications involving pairwise preferences, we study the weaker and more robust comparison oracle, which for any $S, T \in \mathcal{F}$ reveals only if $w(S) <, =, \text{ or } > w(T)$. We investigate the query complexity and computational efficiency of optimizing in this model. We present three main contributions. (1) Query Complexity: We establish that the query complexity over any arbitrary set system $\mathcal{F} \subseteq 2^U$ is $\tilde{O}(n^2)$. This demonstrates a fundamental separation between information and computational complexity, as the runtime may still be exponential for NP-hard problems. (2) Algorithmic Frameworks: We develop two general tools. First, a Dual Ellipsoid framework establishes an efficient reduction from optimization to certification. It shows that to optimize efficiently, it suffices to efficiently certify a candidate's optimality using only comparisons. Second, Global Subspace Learning (GSL) sorts all feasible sets using $O(nB \log(nB))$ queries for integer weights bounded by $B$. We efficiently implement GSL for linear matroids, yielding improved query complexities for problems like $k$-SUM, SUBSET-SUM, and $A+B$ sorting. (3) Combinatorial Applications: We give the first polynomial-time, low-query algorithms for classic problems, including minimum cuts, minimum weight spanning trees (and matroid bases), bipartite matching (and matroid intersection), and shortest $s$-$t$ paths. Our work provides the first general query complexity bounds and efficient algorithmic results for this fundamental model.

翻译：在线性组合优化中，我们的目标是在一个由 $n$ 个元素构成的基础集 $U$ 上，对于一个族 $\mathcal{F} \subseteq 2^U$，找到 $S^* = \arg\min_{S \in \mathcal{F}} \langle w,\mathbf{1}_S \rangle$。传统上，权重 $w$ 是已知的或可通过值预言机访问。受涉及成对偏好的实际应用启发，我们研究了更弱且更鲁棒的比较预言机模型，该模型对于任意 $S, T \in \mathcal{F}$，仅揭示 $w(S) <, =, \text{ 或 } > w(T)$。我们研究了在此模型下优化的查询复杂度和计算效率。我们提出了三个主要贡献。(1) 查询复杂度：我们证明，对于任意集合系统 $\mathcal{F} \subseteq 2^U$，查询复杂度为 $\tilde{O}(n^2)$。这揭示了信息复杂度与计算复杂度之间的根本性分离，因为对于 NP 难问题，运行时间可能仍然是指数级的。(2) 算法框架：我们开发了两个通用工具。首先，一个对偶椭球体框架建立了从优化到验证的高效归约。它表明，为了高效优化，只需能够仅使用比较来高效验证候选解的最优性。其次，全局子空间学习（GSL）方法使用 $O(nB \log(nB))$ 次查询对所有可行集进行排序，其中整数权重以 $B$ 为界。我们为线性拟阵高效实现了 GSL，从而为诸如 $k$-SUM、子集和问题以及 $A+B$ 排序等问题带来了改进的查询复杂度。(3) 组合应用：我们为经典问题给出了首个多项式时间、低查询复杂度的算法，这些问题包括最小割、最小权重生成树（及拟阵基）、二分图匹配（及拟阵交）以及最短 $s$-$t$ 路径。我们的工作为这一基础模型提供了首个通用的查询复杂度界限和高效的算法结果。