The Subspace Flatness Conjecture and Faster Integer Programming · HackerPapers discussions

The Subspace Flatness Conjecture and Faster Integer Programming — mentioned by j2kun

Tue, 30 Jan 2024 16:03:40 GMT

The abstract is more informative: https://arxiv.org/abs/2303.14605 We obtain a (log(2n))^O(n)-time randomized algorithm to solve integer programs in n variables. So the work is theoretical: a better exponential-time algorithm than the previous best, based on some analysis of the structure of convex bodies in R^n and how they can be covered by integer grids (lattices). Most of the practical work on ILPs uses heuristics and branch and bound while taking advantage of the special structure of particular problem formulations. It isn't clear if this work could be used to help either of those, and I…

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The Subspace Flatness Conjecture and Faster Integer Programming — mentioned by mzl

Tue, 30 Jan 2024 09:50:23 GMT

The original paper (https://arxiv.org/pdf/2303.14605.pdf) that the linked article reports on is a theoretical algorithms paper, and does not contain any references to actual solvers or implementations. For practical cases it is not that important what the worst-case complexity is, but rather what the expected complexity is of solving problems that occur in practice.

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The Subspace Flatness Conjecture and Faster Integer Programming — mentioned by xkcd386

Tue, 30 Jan 2024 08:13:46 GMT

The randomized algorithm that Reis & Rothvoss [1] present at the end of their paper will not be implemented in Gurobi/CPLEX/XPRESS. It remains a fantastic result regardless (see below). But first let me explain. In terms of theoretical computational complexity, the best algorithms for "integer linear programming" [2] (whether the variables are binary or general integers, as in the case tackled by the paper) are based on lattices. They have the best worst-case big-O complexity. Unfortunately, all current implementations need (1) arbitrary-size rational arithmetic (like provided by gmplib [3])…

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The Subspace Flatness Conjecture and Faster Integer Programming — mentioned by andrewp123

Tue, 30 Jan 2024 06:49:38 GMT

It seems their result has been out for almost a year now... https://arxiv.org/abs/2303.14605 I'm curious how this affects Traveling Salesman. I was under the impression that all NP-Complete problems take O(n!). Does this method improve it at all?

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The Subspace Flatness Conjecture and Faster Integer Programming — mentioned by genman

Tue, 30 Jan 2024 03:12:41 GMT

[2023] The paper was uploaded first in March https://arxiv.org/abs/2303.14605

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The Subspace Flatness Conjecture and Faster Integer Programming — mentioned by Duanemclemore

Tue, 30 Jan 2024 02:18:34 GMT

Great short article. I haven't looked deeply into the math behind this yet, but this looks to be a preprint [0]. It doesn't appear they're looking directly at the Space Groups as a way to reduce out any symmetries or repetitions that may occur (thus generalizing simplifications of the problem "space"), but it would be interesting to see whether those structures apply or not. I say this as someone who writes software to apply the Space Groups and describe the Voronoi cells around points (or groups of points) distributed through them, so I'm familiar with the "uncanny" ways effects propagate.…

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