Combinatorial Optimization using Comparison Oracles

2025-11-19

AI Summary

This paper studies linear combinatorial optimization in the comparison oracle model, where the optimal solution is identified solely via pairwise comparisons ( <, =, > ) of weights of feasible subsets. To address the overly strong assumptions of traditional value oracles, we propose the first global subspace learning framework, integrating inferential dimension analysis, algebraic structural modeling, and discrete integer sorting techniques. Theoretically, we achieve the first separation between information complexity and computational complexity. Algorithmically, our approach attains an $O(nB log(nB))$ query complexity for fundamental combinatorial structures—including minimum cut, shortest path, and bipartite matching—where $n$ denotes the ground set size and $B$ the bit complexity of weights. This establishes a polynomial-time, low-query paradigm for weakly supervised combinatorial optimization, significantly advancing beyond prior value-oracle–dependent methods.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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 $S^* = argmin_{S in mathcal{F}} langle w, mathbbm{1}_S angle$. Traditionally, the weight vector is given, or a value oracle allows evaluating $w(S) := langle w, mathbbm{1}_S angle$. 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 $ ilde 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 $ ilde 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.

Problem

• Studying combinatorial optimization using comparison oracles instead of value oracles
• Determining query complexity and efficient algorithms for comparison-based optimization
• Solving classic combinatorial problems with polynomial-time low-query comparison algorithms

Innovation

• Using comparison oracles instead of value oracles
• Introducing Global Subspace Learning framework
• Developing polynomial-time low-query algorithms