combinatorial bipartite matching

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 paper addresses the maximum weight matching (MWM) problem on weighted bipartite graphs, proposing an efficient and generalizable enhancement of the Hungarian algorithm. Unlike conventional approaches, it imposes no assumptions of graph completeness or balanced vertex partitions, and avoids artificial vertex augmentation or padding. It is the first to integrate a nonlinear covering strategy into the Hungarian framework, extending its applicability to arbitrary sparse bipartite graphs. A novel weight-discretization coefficient $X$ governs label-update granularity; combined with weight-sensitive labeling and dynamic adjacency-edge scanning, this yields an improved time complexity of $O(LE + LR cdot min(L, X))$, where $L$ and $R$ denote left- and right-partition sizes, and $E$ is the number of edges. Empirical evaluation demonstrates substantial speedups over classical algorithms on sparse instances. An open-source implementation is publicly available on GitHub.

This paper studies two natural extensions of the Maximum Quadratic Assignment Problem (MaxQAP): (1) the Maximum List-Restricted MaxQAP, where each node on one side can only be assigned to a predefined candidate list; and (2) the Maximum Quadratic b-Matching Assignment Problem, requiring the solution to be a b-matching on a graph. We design the first LP-based approximation algorithms with randomized rounding for both problems, leveraging a maximum-weight b-matching solver in b independent sampling rounds. When list sizes are at least $ n - O(sqrt{n}) $ and $ b $ is constant, our algorithms achieve $ O(sqrt{n}) $ and $ O(sqrt{bn}) $ approximation ratios, respectively—matching the best-known asymptotic lower bounds. Our key contribution is the development of the first unified approximation framework applicable to both list-restricted and degree-constrained variants of MaxQAP, thereby overcoming long-standing algorithmic barriers in handling these combinatorial constraints.

This paper studies the maximum-weight $b$-matching problem on undirected bipartite graphs with vertex capacity constraints: given $G = (A cup B, E)$ and a capacity bound $b(v)$ for each vertex $v$, the goal is to select edges of maximum total weight such that each $v$ is matched to between $1$ and $b(v)$ neighbors. We present the first combinatorial algorithm with $O(n^3)$ time complexity, supporting arbitrary real-valued edge weights—unifying both maximum matching (for nonpositive weights) and minimum matching (for nonnegative weights)—thus overcoming prior restrictions to integer weights. The algorithm integrates augmenting-path techniques with potential-function-based optimization, leveraging bipartite structure to design efficient relaxation and update procedures. Under the standard assumption $|A| + |B| = O(n)$, its cubic-time complexity is tight. This result significantly improves upon generic network-flow approaches in both theoretical efficiency and practical applicability.

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 lack of a unified analytical framework and tight approximation ratios for stochastic greedy algorithms in graph matching. The authors propose a general analytical paradigm that systematically characterizes the approximation performance of vertex-iterative randomized greedy algorithms by modeling the joint distribution of vertex processing order and preference order. Leveraging probabilistic methods, coupling techniques, and structural graph theory, they establish improved approximation ratios: 0.560 for Ranking and 0.539 for FRanking on general graphs—both state-of-the-art results. Further refinements are achieved on restricted graph classes: a ratio of 0.570 for graphs excluding triangles and pentagons, and an enhanced bound of 0.615 when the shortest odd cycle has length at least 129.

This paper investigates the approximation ratio of the Ranking algorithm for online matching on general graphs, aiming to break its long-standing theoretical lower-bound stagnation. Addressing the bottleneck where the best-known lower bound (0.526, Chan et al.) and upper bound remain unseparated, we introduce the novel concept of “vertex backup” to characterize the rank distribution structure among matched vertices, thereby circumventing limitations imposed by adverse events in prior analyses. Integrating the primal-dual framework, random permutation modeling, and refined probabilistic analysis, we optimize the derivation of the approximation ratio. Our result improves the lower bound for Ranking on general graphs to 0.5469—surpassing, for the first time, the best-known approximation ratio (0.531) achievable by any non-adaptive algorithm. This advancement significantly deepens the theoretical understanding of Ranking in non-bipartite settings.

This paper investigates the theoretical capabilities and efficiency of greedy algorithms for constructing shortcut sets and hopsets in directed weighted graphs. We propose a deterministic greedy strategy that leverages path covering and transitive closure techniques to build exact β-hopsets—without relying on random sampling. Our main contributions are threefold: (1) We construct exact hopsets of size $ ilde{O}(n)$ with hopbound $O(n^{1/3})$, matching the optimal size upper bound established at SODA’22; (2) We design a deterministic algorithm achieving time complexity $O(mn^{2/3})$; (3) Under certain conditions, we attain existence-optimal exact hopsets—providing the first systematic evidence that greedy methods possess both theoretical promise and practical utility for graph sparsification and path compression.

This study addresses the lack of precise characterization regarding the efficiency gap between local and global optima in task allocation. Focusing on the matchable semi-matching problem, this work proposes an analytical framework combining explicit rational potential functions on comparison graphs with extremal construction techniques to derive closed-form expressions for the worst-case ratios of local search under varying move step sizes and load constraints. The primary contribution is the first proof that worst-case behavior can be realized even under simple tree-structured constraints, thereby surpassing traditional approximation bounds. Specifically, it establishes exact worst-case ratios of 3/2 for single-step moves and approximately 1.2945 for double-step moves in the absence of load limits, while providing rigorous theoretical guarantees under degree-constrained settings.

This work investigates the non-clashing teaching problem for concept classes defined by closed neighborhoods in graphs—a batch teaching model that satisfies anti-collusion criteria but suffers from high computational complexity and is only solvable on restricted graph classes. By leveraging parameterized complexity analysis, combinatorial graph-theoretic techniques, and reduction methods, we design the first fixed-parameter tractable (FPT) algorithm applicable to a significantly broader family of graphs. Our main contributions include an improved FPT algorithm for general graphs, tight combinatorial upper bounds, and proofs of strong inapproximability and W[1]-hardness results. These advances substantially deepen the understanding of both the algorithmic and complexity-theoretic landscape of this teaching problem.