Each problem states how progress is verified and what counts as a contribution. Besides the problems curated here, the catalogue includes open conjectures from Formal Conjectures (with Lean statements), optimization constants and the AlphaEvolve problems. Know one that belongs here? Propose a problem.
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Find better polynomial-time approximation algorithms for the metric Traveling Salesman Problem and prove the conjectured 4/3 integrality gap of the subtour LP. The best known ratio is 3/2 − ε with ε > 10^−36 (Karlin–Klein–Oveis Gharan).
Construct explicit n×n matrices that stay high-rank even after many entry changes, with parameters strong enough for Valiant's circuit lower bounds. Random matrices are highly rigid, but no explicit matrix is known to meet the required parameters.
Determine ω, the smallest exponent such that n×n matrices can be multiplied with n^(ω+o(1)) arithmetic operations. The best published bound is ω < 2.371339, a 2026 preprint claims ω < 2.371177, and it is conjectured that ω = 2.
Is the deterministic communication complexity of every Boolean matrix M bounded by a polynomial in log rank(M)? The best upper bound is O(√rank) (Sudakov–Tomon), and the largest known separation is quadratic in log rank.
Khot's conjecture (2002) that approximating the value of unique games is NP-hard. Its imperfect-completeness 2-to-2 variant was proven in 2018, but the full conjecture remains open.
