Rapid sampling of directed Eulerian tours via flip–repair walks
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How rapidly can one sample a nearly uniform Eulerian tour in any strongly connected directed Eulerian multigraph with m arcs? Flip–repair walks aim to bypass the mn-type cost associated with earlier arborescence-based methods.
Why it matters and what counts as progress
Why it matters: Faster direct sampling would bypass an arborescence-based runtime barrier for Eulerian tours, especially on sparse directed graphs.
What counts as progress: Prove mixing and implementation bounds yielding O(m^(3/2)polylog(m/ε)) sampling time, or improve the dependence on m and error ε.
How the problem developed
- 1996 · Partial result [2]
Kandel, Matias, Unger, and Winkler Used the BEST-theorem arborescence factorization for sequence shuffling; the resulting sampler’s cost depends on random-arborescence sampling.
Boczkowski, Peres, and Sousi Established general Eulerian-digraph mixing bounds on the mn scale, which can be too costly on sparse graphs.
- 28 May 2026 · Reported claim [2]
Nima Anari and GPT-5.5 Pro Extended The preprint gives a worst-case nearly uniform sampler for any strongly connected directed Eulerian multigraph in Õ(m^{3/2}) time, using a flip–repair chain and degree reduction.
AI contribution
Sampling Directed Eulerian Tours in Õ(m³ᐟ²) Time
Anari’s Theorem 4 reports that the flip–repair walk associated with any real skew-symmetric n×n matrix has spectral gap at least 2/n. In the flat 0/1 determinant case it also gives an entropy bound and nearly linear mixing; this feeds a near-uniform sampler for strongly connected directed Eulerian multigraphs with m arcs in O(m^(3/2) polylog(m/ε)) time.
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