{"type":"ScholarlyArticle","title":"The Mixing of Markov Chains on Linear Extensions in Practice","authors":[{"name":"Topi Talvitie","slug":"topi-talvitie","given":"Topi","family":"Talvitie","affiliation":"Dept. of Computer Science University of Helsinki"},{"name":"Teppo Niinimäki","slug":"teppo-niinimaki","given":"Teppo","family":"Niinimäki","affiliation":"Dept. of Computer Science Aalto University"},{"name":"Mikko Koivisto","slug":"mikko-koivisto","given":"Mikko","family":"Koivisto","affiliation":"Dept. of Computer Science University of Helsinki"}],"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.","doi":"10.24963/ijcai.2017/74","url":"https://ijcai.org/proceedings/2017/74","pdf":"https://ijcai.org/proceedings/2017/0074.pdf","markdown":"https://ijcai.org/proceedings/2017/74.md","bibtex":"https://ijcai.org/proceedings/2017/bibtex/74","volume":{"slug":"2017","title":"Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2017","url":"https://ijcai.org/proceedings/2017/","year":2017,"location":"Melbourne, Australia","dates":"19-25 August 2017","editors":"Carles Sierra","isbn":"978-0-9992411-0-3","publisher":"International Joint Conferences on Artificial Intelligence Organization","doi_prefix":"10.24963/ijcai.2017"},"section":"Combinatorial & Heuristic Search","track":"Main track","number":74,"pages":{"first":524,"last":530},"keywords":["Combinatorial & Heuristic Search: Evaluation and Analysis","Uncertainty in AI: Approximate Probabilistic Inference"],"videos":[],"references":[{"position":1,"raw":"[Banks et al., 2010] Jacqueline Banks, Scott Garrabrant, Mark L. Huber, and Anne Perizzolo. Using TPA to count linear extensions. arXiv preprint arXiv:1010.4981, 2010.","title":"Using TPA to count linear extensions","authors":"Jacqueline Banks, Scott Garrabrant, Mark L Huber, Anne Perizzolo","year":"2010","venue":null,"doi":null,"url":null,"internal_url":null},{"position":2,"raw":"[Bhatnagar et al., 2011] Nayantara Bhatnagar, Andrej Bog- danov, and Elchanan Mossel. The computational complex- ity of estimating MCMC convergence time. In Proc. of the 15th International Workshop on Randomization and Com- putation (RANDOM), pages 424-435, 2011.","title":"The computational complexity of estimating MCMC convergence time","authors":"Nayantara Bhatnagar, Andrej Bogdanov, Elchanan Mossel","year":"2011","venue":"Proc. of the 15th International Workshop on Randomization and Computation (RANDOM)","doi":null,"url":null,"internal_url":null},{"position":3,"raw":"[Brightwell and Winkler, 1991] Graham Brightwell and Pe- ter Winkler. Counting linear extensions. Order, 8(3):225- 242, 1991.","title":"Graham Brightwell and Peter Winkler","authors":"Winkler Brightwell","year":"1991","venue":"Order","doi":null,"url":null,"internal_url":null},{"position":4,"raw":"[Bubley and Dyer, 1999] Russ Bubley and Martin Dyer. Faster random generation of linear extensions. Discrete Mathematics, 201(1):81-88, 1999.","title":"Faster random generation of linear extensions","authors":"Bubley, ; Dyer, Russ Bubley, Martin Dyer","year":"1999","venue":"Discrete Mathematics","doi":null,"url":null,"internal_url":null},{"position":5,"raw":"[Chakraborty et al., 2015] Supratik Chakraborty, Dror Fried, Kuldeep S. Meel, and Moshe Y. Vardi. From weighted to unweighted model counting. In Proc. of the 24th Interna- tional Joint Conference on Artificial Intelligence, (IJCAI), pages 689-695, 2015.","title":"From weighted to unweighted model counting","authors":"Supratik Chakraborty, Dror Fried, Kuldeep S Meel, Moshe Y Vardi","year":"2015","venue":"Proc. of the 24th International Joint Conference on Artificial Intelligence","doi":null,"url":null,"internal_url":"/proceedings/2015/103"},{"position":6,"raw":"[Dyer and Greenhill, 2000] Martin Dyer and Catherine Greenhill. On Markov chains for independent sets. J. Algorithms, 35(1):17-49, 2000.","title":"On Markov chains for independent sets","authors":"Greenhill Dyer, Martin Dyer, Catherine Greenhill","year":"2000","venue":"J. Algorithms","doi":null,"url":null,"internal_url":null},{"position":7,"raw":"[Dyer et al., 1991] Martin Dyer, Alan Frieze, and Ravi Kan- nan. A random polynomial-time algorithm for approxi- mating the volume of convex bodies. J. ACM, 38(1):1-17, 1991.","title":"A random polynomial-time algorithm for approximating the volume of convex bodies","authors":"Martin Dyer, Alan Frieze, Ravi Kannan","year":"1991","venue":"J. ACM","doi":null,"url":null,"internal_url":null},{"position":8,"raw":"[Gomes et al., 2006] Carla P. Gomes, Ashish Sabharwal, and Bart Selman. Model counting: A new strategy for obtain- ing good bounds. In Proc. of the 21st National Conference on Artificial Intelligence (AAAI), pages 54-61, 2006.","title":"Model counting: A new strategy for obtaining good bounds","authors":"Carla P Gomes, Ashish Sabharwal, Bart Selman","year":"2006","venue":"Proc. of the 21st National Conference on Artificial Intelligence (AAAI)","doi":null,"url":null,"internal_url":null},{"position":9,"raw":"[Hoeffding, 1963] Wassily Hoeffding. Probability inequali- ties for sums of bounded random variables. J. American Statistical Association, 58(301):13-30, 1963.","title":"Wassily Hoeffding. Probability inequalities for sums of bounded random variables","authors":"Hoeffding","year":"1963","venue":"J. American Statistical Association","doi":null,"url":null,"internal_url":null},{"position":10,"raw":"[Hsu et al., 2015] Daniel Hsu, Aryeh Kontorovich, and Csaba Szepesvari. Mixing time estimation in reversible Markov chains from a single sample path. In Advances in Neural Information Processing Systems 28, pages 1459- 1467.","title":"Mixing time estimation in reversible Markov chains from a single sample path","authors":"Daniel Hsu, Aryeh Kontorovich, Csaba Szepesvari","year":null,"venue":"Advances in Neural Information Processing Systems","doi":null,"url":null,"internal_url":null},{"position":11,"raw":"Curran Associates, Inc., 2015.","title":null,"authors":"Curran Associates, Inc","year":"2015","venue":null,"doi":null,"url":null,"internal_url":null},{"position":12,"raw":"[Huber, 2006] Mark Huber. Fast perfect sampling from lin- ear extensions. Discrete Mathematics, 306(4):420-428, 2006.","title":"Fast perfect sampling from linear extensions","authors":"Mark Huber","year":"2006","venue":"Discrete Mathematics","doi":null,"url":null,"internal_url":null},{"position":13,"raw":"[Huber, 2014] Mark Huber. Near-linear time simulation of linear extensions of a height-2 poset with bounded inter- action. Chicago J. Theoretical Computer Science, 2014.","title":"Near-linear time simulation of linear extensions of a height-2 poset with bounded interaction","authors":"Mark Huber","year":"2014","venue":"Chicago J. Theoretical Computer Science","doi":null,"url":null,"internal_url":null},{"position":14,"raw":"[Jerrum et al., 2004] Mark Jerrum, Alistair Sinclair, and Eric Vigoda. A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries. J. ACM, 51(4):671-697, 2004.","title":"A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries","authors":"Mark Jerrum, Alistair Sinclair, Eric Vigoda","year":"2004","venue":"J. ACM","doi":null,"url":null,"internal_url":null},{"position":15,"raw":"[Kangas et al., 2016] Kustaa Kangas, Teemu Hankala, Teppo Niinimäki, and Mikko Koivisto. Counting linear extensions of sparse posets. In Proc. of the International Joint Conference on Artificial Intelligence (IJCAI), pages 603-609, 2016.","title":"Counting linear extensions of sparse posets","authors":"Kangas","year":"2016","venue":"Proc. of the International Joint Conference on Artificial Intelligence (IJCAI)","doi":null,"url":null,"internal_url":"/proceedings/2016/92"},{"position":16,"raw":"[Karzanov and Khachiyan, 1991] Alexander Karzanov and Leonid Khachiyan. On the conductance of order Markov chains. Order, 8(1):7-15, 1991.","title":"Alexander Karzanov and Leonid Khachiyan. On the conductance of order Markov chains","authors":"Khachiyan Karzanov","year":"1991","venue":"Order","doi":null,"url":null,"internal_url":null},{"position":17,"raw":"[Lukasiewicz et al., 2014] Thomas Lukasiewicz, Maria V. Martinez, and Gerardo I. Simari. Probabilistic preference logic networks. In Proc. of the 21st European Conference on Artificial Intelligence (ECAI), pages 561-566, 2014.","title":"Probabilistic preference logic networks","authors":"Thomas Lukasiewicz, Maria V Martinez, Gerardo I Simari","year":"2014","venue":"Proc. of the 21st European Conference on Artificial Intelligence (ECAI)","doi":null,"url":null,"internal_url":null},{"position":18,"raw":"[Mannila and Meek, 2000] Heikki Mannila and Christopher Meek. Global partial orders from sequential data. In Proc. of the Sixth International Conference on Knowledge Dis- covery and Data Mining (KDD), pages 161-168, 2000.","title":"Global partial orders from sequential data","authors":"Meek Mannila, Heikki Mannila, Christopher Meek","year":"2000","venue":"Proc. of the Sixth International Conference on Knowledge Discovery and Data Mining (KDD)","doi":null,"url":null,"internal_url":null},{"position":19,"raw":"[Mitzenmacher and Upfal, 2005] Michael Mitzenmacher and Eli Upfal. Probability and Computing: Randomized Algorithms and Probabilistic Analysis. Cambridge University Press, 2005.","title":"Probability and Computing: Randomized Algorithms and Probabilistic Analysis","authors":"Michael Mitzenmacher, Eli Upfal","year":"2005","venue":null,"doi":null,"url":null,"internal_url":null},{"position":20,"raw":"[Muise et al., 2016] Christian Muise, J. Christopher Beck, and Sheila A. McIlraith. Optimal partial-order plan re- laxation via MaxSAT. J. Artificial Intelligence Research, 57:113-149, 2016.","title":"Optimal partial-order plan relaxation via MaxSAT","authors":"Christian Muise, J Christopher Beck, Sheila A Mcilraith","year":"2016","venue":"J. Artificial Intelligence Research","doi":null,"url":null,"internal_url":null},{"position":21,"raw":"[Niinimäki et al., 2016] Teppo Niinimäki, Pekka Parviainen, and Mikko Koivisto. Structure discovery in Bayesian net- works by sampling partial orders. J. Machine Learning Research, 17(57):1-47, 2016.","title":"Structure discovery in Bayesian networks by sampling partial orders","authors":"Niinimäki","year":"2016","venue":"J. Machine Learning Research","doi":null,"url":null,"internal_url":null},{"position":22,"raw":"[Propp and Wilson, 1996] James G. Propp and David B. Wil- son. Exact sampling with coupled Markov chains and ap- plications to statistical mechanics. Random Structures & Algorithms, 9(1-2):223-252, 1996.","title":"Exact sampling with coupled Markov chains and applications to statistical mechanics","authors":"James G Propp, David B Wilson","year":"1996","venue":"Random Structures & Algorithms","doi":null,"url":null,"internal_url":null},{"position":23,"raw":"[Rabinovich et al., 2016] Maxim Rabinovich, Aaditya Ram- das, Michael I. Jordan, and Martin J. Wainwright. Function-specific mixing times and concentration away from equilibrium. CoRR, abs/1605.02077, 2016.","title":"Function-specific mixing times and concentration away from equilibrium","authors":"Rabinovich","year":"2016","venue":null,"doi":null,"url":null,"internal_url":null},{"position":24,"raw":"[Sinclair and Jerrum, 1989] Alistair Sinclair and Mark Jer- rum. Approximate counting, uniform generation and rapidly mixing Markov chains. Information and Compu- tation, 82(1):93-133, 1989.","title":"Approximate counting, uniform generation and rapidly mixing Markov chains","authors":"Alistair Sinclair, Mark Jerrum","year":"1989","venue":"Information and Computation","doi":null,"url":null,"internal_url":null},{"position":25,"raw":"[Wilson, 2004] David B. Wilson. Mixing times of lozenge tiling and card shuffling Markov chains. The Annals of Applied Probability, 14(1):274-325, 2004.","title":"Mixing times of lozenge tiling and card shuffling Markov chains","authors":"David B Wilson","year":"2004","venue":"The Annals of Applied Probability","doi":null,"url":null,"internal_url":null}],"license_note":"Copyright International Joint Conferences on Artificial Intelligence."}