Parallel algorithm for matroid basis computation with O(n^{1/3} log^{1/3} n) round complexity, nearly matching the KUW lower bound.
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Liu, Richard Peng, Maximilian Probst Gutenberg, and Sushant Sachdeva
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A threshold κ=Θ(1/√α) (α=m/n) separates easy collision finding from OGP-based exponential lower bounds against online algorithms in single-layer binary NNs.
First deterministic sublogarithmic-round spanner and APSP algorithms in linear, sublinear, and near-linear MPC plus Congested Clique via derandomized hitting sets.
The work proves that approximating correlation clustering to additive εn² error requires Ω(n/ε²) adjacency-matrix queries, with stronger bounds under memory constraints in random and general query models.
The first dynamic algorithms for matrix rank and related objects achieve update times scaling with rank r, specifically Õ(r^1.405) per entry update and Õ(r^1.528 + z) per column update, extending to dynamic maximum matching.
Streaming max-cut requires Ω(n) space for dense graphs but Ω(n log(ε² n)/ε²) space for graphs with Θ(n/ε²) edges when outputting the cut, with matching upper bounds for dense case and similar separations for densest subgraph.
The one-way communication complexity of reporting k-edit occurrences (including the edit sequences) is Θ(n/m · k log(m|Σ|/k)) bits for 0 < k < m < n/2.
A cut-preserving sparsifier constructed from approximate max-flow enables faster all-pairs minimum-cut algorithms in unweighted graphs across cut-query, dynamic, and streaming models.
Regularity in hypergraphs is fine-grained equivalent to the general case for clique detection, enabling a complete classification of k-sparse Boolean CSP optimization complexity by constraint degree: linear for d≤1, clique-equivalent for d=2, and exhaustive-search for d≥3 under 3-uniform hyperclique
Small symmetries create strict hierarchies in resolution with exponential separations from standard resolution, constant-depth Frege, and between SRCI and SRII.
Deterministic Õ(n^{ω(σ)}) time algorithm for multi-source reachability in digraphs with n^σ sources, improving prior randomized n^{1+2/3ω(σ)} bound.
cgFOC admits computable VC-dimension bounds on nowhere dense structures and efficient algorithms for query answering and PAC learning on locally bounded expansion classes, but a minor extension is intractable on trees.
Uniform-Ironed-Virtual-Value Item Pricing achieves a tight 3-approximation to the Duality Relaxation Benchmark in unit-demand single-buyer revenue maximization.
Any n-qubit QC Hamiltonian sparsifies to Õ(n/ε²) terms preserving all state energies within 1±ε using invariant subspace decomposition and the Alon-Kozma operator inequality.
The paper shows fine-grained hardness for approximating reachability diameter in directed graphs, gives additive approximations for unweighted cases, and constant-factor approximations for bounded treewidth and width-bounded DAGs.
First learning-augmented algorithms for online minimization problems that use stable dual LP predictions to improve theoretical guarantees on metrical task systems and laminar set cover.
Backdoors can be embedded in ResNet and ViT models as statistically indistinguishable latent directions, reducing cryptographic undetectability to an intractable hypothesis test over parameter distributions.
Sparsity helps for k-independent set only below certain density thresholds, with new algorithms achieving O(min(n^{ωk/3} + m^{k/3}, n^k)) time and conditional lower bounds showing brute-force necessity above thresholds for many binary constraint families.
GenusSink achieves near-linear time approximate generalized Sinkhorn for bounded genus graphs via separator decompositions, computational geometry, and fast distance matrix operations.
Rigorous security proofs for variable-length QKD, phase-error bounding with imperfect detectors, marginal-constrained entropy accumulation, and authentication reductions place practical QKD on firmer mathematical ground.
Semialgebraic graphs admit O(n^{1-2/(d+1)+ε})-bit adjacency labels via polynomial partitioning; semilinear graphs need only O(log n) bits.
Connectivity-preserving important separators of size at most k number 2^{O(k log k)} and can be enumerated in the same bound, yielding 2^{O(k log k)} FPT time for constant-class Node Multiway Cut-Uncut.
Incremental (1-ε)-approximate s-t max-flow algorithm achieving Õ(m + n F*/ε) total update time, first with polylog amortized updates for dense graphs.
First practical algorithm for expander hierarchies used to build a normalized-cut solver that beats state-of-the-art quality on large real-world graphs.
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Complexity of Clique-Guarded First-Order Logic with Counting
cgFOC admits computable VC-dimension bounds on nowhere dense structures and efficient algorithms for query answering and PAC learning on locally bounded expansion classes, but a minor extension is intractable on trees.