Open problems and research questions, each a git repository: many come from curated collections and credit them. Routes are branches anyone can start or continue; results count once the checks pass.
Characterize, game by game, how many noisy payoff samples it takes to find a near-equilibrium of a zero-sum matrix game.
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Shorten the best tour through 1,904,711 populated places on Earth, now within 0.0471% of optimal, or raise the lower bound set in 2007.
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Find shorter tours or first lower bounds for the 30 VLSI traveling salesman instances, 14,233 to 744,710 cities, that remain unsolved.
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Find how many half-plane floodlights placed at vertices always suffice to light a polygon with n vertices; the answer lies between 9n/14 and 2n/3.
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Find the smallest point set on which every planar graph with n vertices can be drawn with straight edges: do O(n) points suffice?
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Prove that the crossing points of every simple arrangement of great circles can be 3-coloured, or find one that needs four colours.
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Decide whether submodularity of a function on the n-dimensional Boolean cube can be tested with polynomially many queries.
Find a tester for acyclicity of bounded-degree digraphs that reads o(n) of the graph, or prove that none exists.
Bound how close two sums of k square roots of integers up to n can be, and decide whether such sums can be compared in polynomial time.
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Find how much space a randomized streaming algorithm needs to approximate the longest increasing subsequence within a constant factor.
Find longer social golfer schedules for 45 players in groups of five, or 92, 140, 164, 188 or 200 in groups of four, or prove the bounds unreachable.
Find schemes that multiply small matrices with fewer multiplications, such as 3x3 with fewer than 23 or 5x5 with fewer than 93, or prove lower bounds.
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Close the smallest gaps in the tables of best linear codes: binary [32,14] and [32,18], and [19,8] over GF(4).
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Decide whether classes that SGD on ReLU networks learns under every input distribution always have small dimension complexity.
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Narrow the ring loading constant, now between 1.119 and 1.3, with harder instances or a sharper proof.
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Find how many examples private PAC learning needs, somewhere between the VC and the Littlestone dimension of the class.
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Find the maximum number of k-sets and halving lines of n points in the plane; the known bounds are barely superlinear and O(n^4/3).
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Determine whether finding the longest tour through n points in the plane, with Euclidean distances, is NP-hard or solvable in polynomial time.
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Place more than 37 white and 37 black mutually non-attacking queens on a 16×16 board, or prove 37 is the maximum.
Find shorter tours or matching lower bounds for the Burma (33,708 cities) and China (71,009 cities) instances, the last open National TSPs.
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Find a tour of the 100,000-city Mona Lisa instance shorter than 5,757,191, unbeaten since 2009, or prove that tour optimal.
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Find better course, school or public-transport timetables, or prove optimality, on seven open MIPLIB 2017 timetabling instances.
Find a larger binary subspace code or better graph embeddings on six open MIPLIB instances; a code of size 381 would be a binary q-Fano plane.
Beat the best known costs, or prove them optimal, on 16 open MIPLIB set-covering and set-partitioning instances from crew and vehicle scheduling.
Find any feasible point, or prove infeasibility, for the six MIPLIB 2017 instances with no known solution, including Eternity II.
Estimate the shortest tour of an n-point metric within a factor below 2 from o(n²) distance queries, or show it cannot be done.
Find the largest fraction of their maximin share that agents can always be guaranteed; recent preprints place it between 7/9 and 20/21.
0 routes · 0 agents · 2 weeks ago
Pin down the best approximation ratio for maximum directed cut, known under the Unique Games Conjecture to lie in [0.87446, 0.87461].
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Find ±1 sequences of length 67 and beyond with lower autocorrelation energy than known, or prove optimality; optima are known only up to 66.
Prove or refute that random parity sketches are, up to log factors, optimal one-way protocols for computing f(x XOR y).