matrix rank arguments

This work investigates the computational hardness of finding low-rank matrices within a given linear subspace that is guaranteed to contain a rank‑1 matrix. Under the assumption that NP does not admit subexponential-time algorithms, it is shown that no polynomial-time algorithm can find a matrix of rank below $n^{o(1/\log\log n)}$, even when such a rank‑1 matrix exists. The paper introduces a novel combination of hypergraph consistency and moment-matrix techniques to establish strong inapproximability results without relying on the PCP framework, extending these results to arbitrary finite fields. Via two distinct reductions, the authors achieve a 1 vs. $k$ rank gap inapproximability within running times $n^{O(\log k)}$ and $n^{O(k)}$, respectively, substantially advancing the understanding of inapproximability in coding theory and lattice problems.

This work addresses the longstanding challenge of deriving strong lower bounds for the weak pigeonhole principle in proof complexity by introducing its algebraic generalization—the Weak Rank principle (WRank)—and constructing multiple encodings, including perfect matching and bamboo-tree CNF formulations. By innovatively designing a scalable lower-bound generator tailored for Polynomial Calculus Resolution over $F_2$ (PCR$_{F_2}$) and a novel pseudo-expectation method adapted to the Sherali–Adams system, the paper establishes the first exponential proof-length lower bounds for WRank in PCR$_{F_2}$, resolving a major open problem. It further demonstrates that circuit lower bound formulas admit no short proofs in this system. These results establish WRank as both necessary and sufficient for proving lower bounds against NC$^2$ and AC$^{0}[p]$, thereby cementing its central role in structured proof complexity analysis.

This work proposes a novel approach to the Max-3-Cut problem based on complex-valued quadratic optimization, leveraging the low-rank structure of the objective matrix to circumvent conventional semidefinite programming relaxations and heuristic strategies. By enumerating and evaluating $O(n^{2r-1})$ candidate solutions—where $r$ denotes the approximate rank of the objective matrix—the authors establish, for the first time, that the global optimum is guaranteed to reside within this candidate set when $K=3$ and the objective matrix is low-rank. Theoretical guarantees are also provided for approximately low-rank cases. The algorithm is inherently parallelizable and achieves performance comparable to state-of-the-art methods across various graph structures while demonstrating superior scalability.

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 paper investigates the parameterized tractability of block-structured integer programming—specifically two-stage stochastic IPs and n-fold IPs—when the global constraint matrix contains arbitrarily large entries. Addressing the limitation of prior work requiring all input coefficients to be bounded, we establish, for the first time, that feasibility checking for two-stage stochastic IPs and linear optimization for *uniform* n-fold IPs are fixed-parameter tractable (FPT) when parameterized solely by the local matrix dimensions and the maximum absolute value of the right-hand sides (D_i). We further show that uniformity is necessary for n-fold IPs to admit such FPT algorithms. Our approach integrates Graver basis theory, potential function analysis, dynamic programming, integer conic decomposition, and distance-sensitive search to design a weakly polynomial-time FPT algorithm. This work establishes the first FPT framework for block-structured IPs tolerant of large matrix entries, tightening the parameter dependence from global coefficient magnitude to local magnitudes—thereby substantially broadening the applicability of parameterized algorithms for structured integer programming.

This study addresses the longstanding open problem in combinatorial optimization concerning upper bounds on the binary rank of matrices with fixed real rank, which has previously relied on computer-assisted verification without a general theoretical framework. By integrating techniques from linear algebra, combinatorics, and biclique partitioning of bipartite graphs in graph theory, this work proposes a purely mathematical proof strategy that eliminates the need for computational assistance, establishing a general framework for deriving upper bounds applicable to arbitrary fixed real ranks. The authors rigorously determine the maximum binary rank for matrices of real rank five, completely resolving this long-standing challenge. Furthermore, they establish non-trivial upper bounds on the binary rank of fixed-real-rank matrices and derive theoretical limits on the minimum number of edge biclique partitions in the corresponding bipartite graphs, providing a novel analytical paradigm for related research.

This work resolves the satisfiability of ten Challenge-2 instances from the matrix multiplication SAT benchmark, previously conjectured to be unsatisfiable (UNSAT), and constructs the first rank-23 matrix multiplication scheme free of type-3 monomials. By leveraging the isotropic group action of GL(3,2)³, cyclic trace symmetry, perfect matching constraints, semantic repair techniques, and binary identities over 𝔽₂, the authors generate complete satisfying assignments for all 21 CNF instances—encompassing 26,541 variables and 2,461,316 clauses—and verify that all Brent equation residuals vanish identically. These results definitively establish the satisfiability of all Challenge-2 instances, correcting prior assumptions, and yield a reproducible type-3-free rank-23 scheme executable in under nine seconds.

本文使用线性关系的机制研究线性代数中的优化问题，通过广义最小二乘问题实现伪逆，并提出一种截断伪逆方法解决低秩逼近问题。

This study addresses the query complexity of low-rank approximation based on matrix-vector multiplication. To precisely characterize the computational complexity, it proposes adaptive Wishart delayed decision-making, posterior overlap analysis, and compressed eigenvalue persistence control techniques. As a primary contribution, this work reveals for the first time the parameter-dependent transition points under Schatten-p norms, thereby completing the previously missing rank dependence. Furthermore, it establishes optimal query budget bounds achievable in polynomial time, providing a rigorous theoretical benchmark for the design of related algorithms.

研究改进了安全分布式矩阵乘法中的度数表构造方法，通过引入周期间隙框架，提出两种新方案SHIFT和COVER，并证明其在许多情况下优于现有技术。