The Mixing of Markov Chains on Linear Extensions in Practice

Abstract

We investigate almost uniform sampling from the set of linear extensions of a given partial order. The most efficient schemes stem from Markov chains whose mixing time bounds are polynomial, yet impractically large. We show that, on instances one encounters in practice, the actual mixing times can be much smaller than the worst-case bounds, and particularly so for a novel Markov chain we put forward. We circumvent the inherent hardness of estimating standard mixing times by introducing a refined notion, which admits estimation for moderate-size partial orders. Our empirical results suggest that the Markov chain approach to sample linear extensions can be made to scale well in practice, provided that the actual mixing times can be realized by instance-sensitive upper bounds or termination rules. Examples of the latter include existing perfect simulation algorithms, whose running times in our experiments follow the actual mixing times of certain chains, albeit with significant overhead.

Counting Linear Extensions in Practice: MCMC Versus Exponential Monte Carlo

“…Telescopic Product. The basic scheme due to Brightwell and Winkler (1991), revisited by Talvitie et al (2017).…”

“…If the elements are incomparable, then we use the Monte Carlo method with a sufficient number of samples to find out which order between the elements is more probable in the linear extensions of P i , and add P i augmented with that ordering constraint to the end of the sequence, i.e., (a i+1 , b i+1 ) is either (x, y) or (y, x). An analysis shows that in total O( −2 n 2 log 2 n log δ −1 ) samples from the uniform distribution of linear extensions (of the varying posets) suffice for an ( , δ)-approximation of (P ) (Talvitie, Niinimäki, and Koivisto 2017); if the samples are from an almost uniform distribution, then the bound becomes slightly larger. Using Huber's (2006) perfect sampler that generates a random linear extension in O(n 3 log n) expected time, the whole algorithm runs in O( −2 n 5 log 3 n log δ −1 ) expected time.…”

“…It currently yields the best asymptotic running time bound for approximate counting based on the telescopic product scheme. However, Huber's (2014) later perfect sampler, which is based on Gibbs sampling, was recently shown to run significantly faster in various instances of practical size (Talvitie, Niinimäki, and Koivisto 2017), even if the sampler is currently not known to run in polynomial time in the worst case.…”

“…The later improvements were based on rapidly mixing Markov chains in the set of linear extensions (Karzanov and Khachiyan 1991;Bubley and Dyer 1999) combined with Monte Carlo counting schemes (Brightwell and Winkler 1991;Banks et al 2017). The mixing times of the Markov chains were recently studied empirically by Talvitie, Niinimäki and Koivisto (2017) and found to be often much better than the worst-case bounds. However, when approximation guarantees are desired, the degrees of the polynomial worst-case bounds can be prohibitively high.…”

Approximate Inference of Outcomes in Probabilistic Elections

“…In contrast, exact counting of the completions of a partial order is #P -complete (Brightwell and Winkler 1991). While the worst-case mixing times of the Markov chain are practically prohibitive, recent empirical results have shown that the MCMC approach can be made to scale well in practice (Talvitie, Niinimäki, and Koivisto 2017), and techniques for improving the mixing time are an active area of research (Talvitie et al 2018a;2018b).…”