Combinatorial Optimization using Comparison Oracles
Vincent Cohen-Addad, Tommaso d'Orsi, Anupam Gupta, Guru Guruganesh, Euiwoong Lee, Debmalya Panigrahi, Madhusudhan Reddy Pittu, Jon Schneider, David P. Woodruff
TL;DR
The paper tackles optimization of a linear objective over combinatorial families using only comparison queries between feasible sets. It introduces a general query-bound framework based on conic-dimension that yields a near-quadratic $\tilde O(n^2)$ bound for arbitrary set systems, and a Global Subspace Learning (GSL) method for bounded-integer weights achieving $O(nB\log(nB))$ comparisons to sort all feasible sets. It then specializes to classical combinatorial objects, delivering polynomial-time, low-query algorithms for min-cut, spanning trees, matchings, matroid intersections, and shortest paths, along with graph-recovery and sparsification techniques. The work provides the first general query-complexity bounds in the comparison oracle model and exposes a rich set of mechanisms to leverage the algebraic and geometric structure of linear optimization in discrete settings. Overall, it lays groundwork for efficient comparison-based optimization with broad implications for crowdsourcing, recommendation, and RLHF-inspired feedback loops.
Abstract
In a linear combinatorial optimization problem, we are given a family $\mathcal{F} \subseteq 2^U$ of feasible subsets of a ground set $U$ of $n$ elements, and aim to find $S^* = \arg\min_{S \in \mathcal{F}} \langle w, \mathbbm{1}_S \rangle$. Traditionally, the weight vector is given, or a value oracle allows evaluating $w(S) := \langle w, \mathbbm{1}_S \rangle$. Motivated by practical interest in pairwise comparisons, and by the theoretical quest to understand computational models, we study a weaker, more robust comparison oracle that for any $S, T \in \mathcal{F}$ reveals only whether $w(S) <, =, > w(T)$. We ask: when can we find $S^*$ using few comparison queries, and when can this be done efficiently? We present three contributions: (1) We establish that the query complexity over any set system $\mathcal{F} \subseteq 2^U$ is $\tilde O(n^2)$, using the inference dimension framework, highlighting a separation between information and computational complexity (runtime may still be exponential for NP-hard problems under ETH). (2) We introduce a Global Subspace Learning (GSL) framework for objective functions with discrete integer weights bounded by $B$, giving an algorithm to sort all feasible sets using $O(nB \log(nB))$ queries, improving the $\tilde O(n^2)$ bound when $B = o(n)$. For linear matroids, algebraic techniques yield efficient algorithms for problems including $k$-SUM, SUBSET-SUM, and $A{+}B$ sorting. (3) We give the first polynomial-time, low-query algorithms for classic combinatorial problems: 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 algorithms for this model, opening new directions for comparison-based optimization.