Algorithms and proofs in combinatorial optimization, algebraic complexity, dynamic graphs, and sparsification. Each problem page gives the theorem, complexity, randomness, dependencies, and verification evidence.
55 problem families · 24 supporting results · Proof index · Review and verification · Citation
Three deterministic algorithmic results. The linked problem pages give the full statements, proof dependencies, executable coverage and review status.
Deterministic exact maximum matching in general graphs.
Here m_s counts distinct nonloop edge pairs. The proof supplies the uniform cycle-search construction and names its published prerequisites. A separate executable solver returns independently verifiable optimality certificates.
Exact decision and indexed witness.
Exact rational exponent on the card.
For n integers of total bit length L, including an indexed witness. The finite startup constant C is input-independent. The complete compiler proof and exact continuous certificates support the improved third-lookup exponent.
Deterministic black-box identity testing with unknown order and unknown invertible affine coordinates, in every characteristic.
Here K(w)=2^{O(w^2)}, the full finite direction-grid minimum is displayed. Arbitrary local ranks and entry degrees; known scalar total degree D and extension-field oracle access when needed. Polynomial query count for each fixed width, with a width-dependent exponent. Complete internally reconstructed proof; external peer review and priority certification are not claimed.
Browse by problem. Each link opens its statement, proof and dependencies. Computational models and randomness contracts explain how to compare bounds.
- Deterministic general-graph maximum matching
- Degree-constrained subgraphs
- Static and threshold min-cost flow
- Incremental strongly connected components
- Incremental directed SSSP
- Fully dynamic APSP
- Dynamic spanners
- Dynamic minimum spanning forest
- Strong forests and Gomory–Hu trees
- Single-failure optimum preservers
- Flow, matching and price sensitivity
- Matroid and submodular failure summaries
- Colored and vertex-failure graph certificates
- Resource-constrained packing summaries
- Persistent-homology failure certificates
- Boolean linear optimization compiler
- Self-concordant cone sparsification
- Correlated spectral, regression and seminorm summaries
- Label and whole-block cut sparsification
- Finite groups, affine constraints and metrics
- Pauli, qudit and stabilizer blocks
- Exact-separation and non-Fano boundaries
- Arbitrary-value allocation scheme
- Recipient polymatroids
- Recipient-matroid configuration
- Explicit-item matroid union
- One-value allocation
- Intrinsic two-value allocation
- Intrinsic values near factor two
- Unrelated-value additive allocation
- Unrelated values with a joint role
- Whole-ground selection and transfers
- Selected concurrent flow
- Flow with shared setup budget
- Robust selected flow
- Robust unrelated allocation
- Robust bounded-width certificates
- Width-two robust unit allocation
- Two-scenario hardness
- Arbitrary-processing scheme
- Activated-machine scheduling
- Six-profile scheduling
- Processing-bucket tradeoff
- Two-size scheduling
- Per-machine two-time scheduling
- Many-size workload scheduling
- Commitment scheduling
- Hypergraph matching and expander routing
- Independent transversals
- Allocation and scheduling barriers
The complete index also lists 24 supporting algorithms, structural theorems, and scoped lower bounds. Shared manuscripts collect reusable proofs and backend interfaces.
Within each problem: README.md → proof.md → sources.md → verify/README.md. The review registry distinguishes manuscript review from executable checks and identifies their exact scope.
python3 tools/check_repository.pySee Contributing for the full verification mode and maintenance conventions. No external peer review or proof-assistant verification is claimed.
Yilin Liu and GPT-6 Astra Pro. Algorithm Frontier (2026). Cite the relevant problem path and commit; the citation protocol and import provenance retain the applicable licenses and notices.