matching theory

This work addresses longstanding challenges in (1−ε)-approximation algorithms for maximum matching in bipartite graphs—namely, algorithmic complexity, insufficient theoretical understanding, and the absence of tight lower-bound instances. Revisiting the ALT auction algorithm, we eliminate its original vertex-freezing mechanism and introduce a novel analysis framework grounded in augmenting paths. This new perspective not only simplifies the algorithm’s structure but also provides an intuitive explanation of its convergence behavior. Moreover, we construct the first hard instance requiring Ω(1/ε²) rounds of iteration, demonstrating that this round complexity is tight even on simple path graphs and thereby establishing the fundamental theoretical limit of the algorithm.

This work addresses the limitation in hardness proofs for path-packing problems that rely on randomized weight assignments by introducing the first deterministic variant of the isolation lemma. Combining combinatorial constructions with algebraic techniques, the authors explicitly design deterministic weights and employ formal verification to guarantee their correctness. This approach successfully eliminates probabilistic assumptions from several known hardness results, replacing randomized assignments with fully deterministic ones. Consequently, it achieves complete derandomization of the corresponding complexity lower-bound proofs and significantly broadens the applicability of the isolation lemma within theoretical computer science.

High-dimensional pattern matching with wildcards suffers from poor query efficiency—existing linear-space data structures incur query time $n^c$, which is prohibitively expensive. Method: We establish a generic reduction framework linking data structure design to communication complexity, systematically transforming linear-space high-dimensional pattern matching into the analysis of unambiguous Arthur–Merlin (UAM) communication complexity under product distributions. Crucially, we identify intrinsic data sparsity as a fundamental lever for reducing both communication and query complexity, enabling dimension-independent upper bounds on UAM complexity. Leveraging a one-sided-error Set-Disjointness protocol and sublinear communication techniques, we construct a linear-space data structure. Results: Our approach achieves wildcard matching query time $n^{1-1/(clog^2 c)}$, breaking the classical lower bound from STOC’04 and marking the first asymptotic improvement for this problem under linear space.

This paper studies the problem of finding ℓ pairwise disjoint stable matchings of minimum total cost in a bipartite graph. To address this multi-matching joint optimization problem under stability constraints, we first model the structure of stable matchings as specific directed cuts and maximum antichains in a partially ordered set (poset), thereby establishing a direct connection to Dilworth’s and Mirsky’s theorems. This structural characterization yields an exact min–max formula for the minimum number of stable matchings required to cover all stable edges. Building upon this insight, we design a strongly polynomial-time network flow algorithm that uniformly solves the ℓ-disjoint minimum-cost stable matching problem. Our main contributions are: (i) the first tight min–max characterization of the minimum number of stable matchings needed to cover all stable edges; and (ii) the first strongly polynomial-time algorithm for joint optimization over multiple stable matchings.

Understanding the interplay between association schemes and Lovász–Schrijver lift-and-project relaxations—particularly SDP-based ones—in combinatorial optimization, especially for stable set polytopes and hypergraph matching. Method: We introduce the notion of *deeply vertex-transitive graphs*, construct *hypermatching pseudo-schemes* (noncommutative association structures), and develop a systematic contraction method to derive commutative sub-schemes. Our approach integrates algebraic combinatorics, spectral graph theory, and SDP analysis. Contribution/Results: We derive Delsarte–Hoffman-type tight bounds on clique and stability numbers; precisely characterize the lift-and-project rank of hypergraph matching relaxations; and establish a general contraction framework from noncommutative pseudo-schemes to commutative sub-schemes. This yields a unified algebraic–convex optimization paradigm for symmetric combinatorial optimization problems.

This work addresses the problem of testing bipartiteness in bounded-degree graphs with the goal of reducing both the number and length of random walks required. By introducing, for the first time in this context, the Goemans–Williamson semidefinite programming relaxation for Max-Cut, the authors improve upon the classic Goldreich–Ron algorithm. Their approach achieves bipartiteness testing using only $O(\sqrt{n})$ random walks, each of length $O(\log n)$, significantly lowering the query complexity. Furthermore, this technique yields a round-optimal streaming algorithm that decides bipartiteness in $O(\log n)$ passes using $O(\sqrt{n} \log n)$ space. The number of passes matches the theoretical lower bound and improves upon the previous best result, which required $O(\sqrt{n \log n})$ random walks.

This work addresses the problem of efficiently determining whether a bipartite graph admits a perfect matching within the NC complexity class. Building upon the framework introduced by CGGRT26, we propose a simpler deterministic NC algorithm that leverages algebraic and combinatorial techniques to formulate a parallel decision criterion. Our approach not only streamlines the original theoretical framework but also achieves improved bounds on key parameters, thereby enhancing the theoretical parallel efficiency of detecting perfect matchings in bipartite graphs.

This work addresses the absence of a mechanized formal verification framework for primal-dual algorithm analysis. It presents the first systematic formalization in Isabelle/HOL that supports rigorous correctness and performance verification of such algorithms, unifying a diverse range of instances—from the classical Hungarian algorithm to modern Adwords algorithms—within a single coherent framework. By achieving machine-checked proofs for multiple primal-dual algorithms, this study not only establishes their formal correctness but also develops a reusable library of verified components. The resulting infrastructure offers a novel paradigm for trustworthy verification of combinatorial optimization algorithms, enhancing both reliability and reusability in formal methods applied to algorithmic analysis.

This work investigates the communication complexity of fundamental graph problems in the deterministic two-party communication model, with a focus on maximum matching and single-source shortest paths with negative weights (including negative cycle detection). For both general and bipartite graphs, the paper introduces novel and streamlined communication protocols that eschew traditional, intricate reductions. Instead, the approach integrates vertex potentials, discretization arguments, and combinatorial optimization techniques. The main contributions include a protocol for maximum matching in general graphs with communication complexity $\tilde{O}(n^{3/2})$, and protocols requiring only $\tilde{O}(n)$ bits of communication for both negative-weight single-source shortest paths and maximum matching in bipartite graphs—substantially simplifying and improving upon existing methods.

This work addresses the fundamental challenge of efficiently maintaining a maximal matching in a graph subject to online edge insertions and deletions, particularly under an adaptive adversary. The paper introduces a novel deterministic algorithm built upon a newly devised subgraph system framework, specifically designed for verifying and maintaining maximal matchings without relying on traditional sparsifiers such as EDCS. This framework enables recursive refinement strategies that progressively optimize performance parameters. As a result, the algorithm achieves an amortized update time of $n^{1/2+o(1)}$ against an adaptive adversary, substantially improving upon the previous best-known bound of $\tilde{O}(n^{8/9})$.