Document

**Published in:** LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

We study the approximability of computing the partition functions of two-state spin systems. The problem is parameterized by a 2×2 symmetric matrix. Previous results on this problem were restricted either to the case where the matrix has non-negative entries, or to the case where the diagonal entries are equal, i.e. Ising models. In this paper, we study the generalization to arbitrary 2×2 interaction matrices with real entries. We show that in some regions of the parameter space, it’s #P-hard to even determine the sign of the partition function, while in other regions there are fully polynomial approximation schemes for the partition function. Our results reveal several new computational phase transitions.

Yumou Fei, Leslie Ann Goldberg, and Pinyan Lu. Two-State Spin Systems with Negative Interactions. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 45:1-45:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{fei_et_al:LIPIcs.ITCS.2024.45, author = {Fei, Yumou and Goldberg, Leslie Ann and Lu, Pinyan}, title = {{Two-State Spin Systems with Negative Interactions}}, booktitle = {15th Innovations in Theoretical Computer Science Conference (ITCS 2024)}, pages = {45:1--45:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-309-6}, ISSN = {1868-8969}, year = {2024}, volume = {287}, editor = {Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.45}, URN = {urn:nbn:de:0030-drops-195739}, doi = {10.4230/LIPIcs.ITCS.2024.45}, annote = {Keywords: Approximate Counting, Spin Systems, #P-Hardness, Randomized Algorithms} }

Document

**Published in:** LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

This paper proves the tight sample complexity of Second-Price Auction with Anonymous Reserve, up to a logarithmic factor, for each of all the value distribution families studied in the literature: [0,1]-bounded, [1,H]-bounded, regular, and monotone hazard rate (MHR). Remarkably, the setting-specific tight sample complexity poly(ε^{-1}) depends on the precision ε ∈ (0, 1), but not on the number of bidders n ≥ 1. Further, in the two bounded-support settings, our learning algorithm allows correlated value distributions.
In contrast, the tight sample complexity Θ̃(n) ⋅ poly(ε^{-1}) of Myerson Auction proved by Guo, Huang and Zhang (STOC 2019) has a nearly-linear dependence on n ≥ 1, and holds only for independent value distributions in every setting.
We follow a similar framework as the Guo-Huang-Zhang work, but replace their information theoretical arguments with a direct proof.

Yaonan Jin, Pinyan Lu, and Tao Xiao. Learning Reserve Prices in Second-Price Auctions. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 75:1-75:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{jin_et_al:LIPIcs.ITCS.2023.75, author = {Jin, Yaonan and Lu, Pinyan and Xiao, Tao}, title = {{Learning Reserve Prices in Second-Price Auctions}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {75:1--75:24}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-263-1}, ISSN = {1868-8969}, year = {2023}, volume = {251}, editor = {Tauman Kalai, Yael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.75}, URN = {urn:nbn:de:0030-drops-175780}, doi = {10.4230/LIPIcs.ITCS.2023.75}, annote = {Keywords: Revenue Maximization, Sample Complexity, Anonymous Reserve} }

Document

PACE Solver Description

**Published in:** LIPIcs, Volume 249, 17th International Symposium on Parameterized and Exact Computation (IPEC 2022)

A directed graph is formed by vertices and arcs from one vertex to another. The feedback vertex set problem (FVSP) consists in making a given directed graph acyclic by removing as few vertices as possible. In this write-up, we outline the core techniques used in the heuristic feedback vertex set algorithm, submitted to the heuristic track of the 2022 PACE challenge.

YuMing Du, QingYun Zhang, JunZhou Xu, ShunGen Zhang, Chao Liao, ZhiHuai Chen, ZhiBo Sun, ZhouXing Su, JunWen Ding, Chen Wu, PinYan Lu, and ZhiPeng Lv. PACE Solver Description: Hust-Solver - A Heuristic Algorithm of Directed Feedback Vertex Set Problem. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 29:1-29:3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{du_et_al:LIPIcs.IPEC.2022.29, author = {Du, YuMing and Zhang, QingYun and Xu, JunZhou and Zhang, ShunGen and Liao, Chao and Chen, ZhiHuai and Sun, ZhiBo and Su, ZhouXing and Ding, JunWen and Wu, Chen and Lu, PinYan and Lv, ZhiPeng}, title = {{PACE Solver Description: Hust-Solver - A Heuristic Algorithm of Directed Feedback Vertex Set Problem}}, booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)}, pages = {29:1--29:3}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-260-0}, ISSN = {1868-8969}, year = {2022}, volume = {249}, editor = {Dell, Holger and Nederlof, Jesper}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.29}, URN = {urn:nbn:de:0030-drops-173855}, doi = {10.4230/LIPIcs.IPEC.2022.29}, annote = {Keywords: directed feedback vertex set, local search, simulated annealing, set covering} }

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**Published in:** LIPIcs, Volume 169, 35th Computational Complexity Conference (CCC 2020)

We initiate a study of the classification of approximation complexity of the eight-vertex model defined over 4-regular graphs. The eight-vertex model, together with its special case the six-vertex model, is one of the most extensively studied models in statistical physics, and can be stated as a problem of counting weighted orientations in graph theory. Our result concerns the approximability of the partition function on all 4-regular graphs, classified according to the parameters of the model. Our complexity results conform to the phase transition phenomenon from physics.
We introduce a quantum decomposition of the eight-vertex model and prove a set of closure properties in various regions of the parameter space. Furthermore, we show that there are extra closure properties on 4-regular planar graphs. These regions of the parameter space are concordant with the phase transition threshold. Using these closure properties, we derive polynomial time approximation algorithms via Markov chain Monte Carlo. We also show that the eight-vertex model is NP-hard to approximate on the other side of the phase transition threshold.

Jin-Yi Cai, Tianyu Liu, Pinyan Lu, and Jing Yu. Approximability of the Eight-Vertex Model. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 4:1-4:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{cai_et_al:LIPIcs.CCC.2020.4, author = {Cai, Jin-Yi and Liu, Tianyu and Lu, Pinyan and Yu, Jing}, title = {{Approximability of the Eight-Vertex Model}}, booktitle = {35th Computational Complexity Conference (CCC 2020)}, pages = {4:1--4:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-156-6}, ISSN = {1868-8969}, year = {2020}, volume = {169}, editor = {Saraf, Shubhangi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.4}, URN = {urn:nbn:de:0030-drops-125564}, doi = {10.4230/LIPIcs.CCC.2020.4}, annote = {Keywords: Approximate complexity, the eight-vertex model, Markov chain Monte Carlo} }

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Complete Volume

**Published in:** LIPIcs, Volume 149, 30th International Symposium on Algorithms and Computation (ISAAC 2019)

LIPIcs, Volume 149, ISAAC'19, Complete Volume

30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@Proceedings{lu_et_al:LIPIcs.ISAAC.2019, title = {{LIPIcs, Volume 149, ISAAC'19, Complete Volume}}, booktitle = {30th International Symposium on Algorithms and Computation (ISAAC 2019)}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-130-6}, ISSN = {1868-8969}, year = {2019}, volume = {149}, editor = {Lu, Pinyan and Zhang, Guochuan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2019}, URN = {urn:nbn:de:0030-drops-116417}, doi = {10.4230/LIPIcs.ISAAC.2019}, annote = {Keywords: Theory of computation; Models of computation; Computational complexity and cryptography; Randomness, geometry and discrete structures; Theory and algorithms for application domains; Design and analysis of algorithms} }

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Front Matter

**Published in:** LIPIcs, Volume 149, 30th International Symposium on Algorithms and Computation (ISAAC 2019)

Front Matter, Table of Contents, Preface, Symposium Organization

30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 0:i-0:xvi, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{lu_et_al:LIPIcs.ISAAC.2019.0, author = {Lu, Pinyan and Zhang, Guochuan}, title = {{Front Matter, Table of Contents, Preface, Symposium Organization}}, booktitle = {30th International Symposium on Algorithms and Computation (ISAAC 2019)}, pages = {0:i--0:xvi}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-130-6}, ISSN = {1868-8969}, year = {2019}, volume = {149}, editor = {Lu, Pinyan and Zhang, Guochuan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2019.0}, URN = {urn:nbn:de:0030-drops-114967}, doi = {10.4230/LIPIcs.ISAAC.2019.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Symposium Organization} }

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RANDOM

**Published in:** LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)

We give a fully polynomial-time approximation scheme (FPTAS) to count the number of independent sets on almost every Delta-regular bipartite graph if Delta >= 53. In the weighted case, for all sufficiently large integers Delta and weight parameters lambda = Omega~ (1/(Delta)), we also obtain an FPTAS on almost every Delta-regular bipartite graph. Our technique is based on the recent work of Jenssen, Keevash and Perkins (SODA, 2019) and we also apply it to confirm an open question raised there: For all q >= 3 and sufficiently large integers Delta=Delta(q), there is an FPTAS to count the number of q-colorings on almost every Delta-regular bipartite graph.

Chao Liao, Jiabao Lin, Pinyan Lu, and Zhenyu Mao. Counting Independent Sets and Colorings on Random Regular Bipartite Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 34:1-34:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{liao_et_al:LIPIcs.APPROX-RANDOM.2019.34, author = {Liao, Chao and Lin, Jiabao and Lu, Pinyan and Mao, Zhenyu}, title = {{Counting Independent Sets and Colorings on Random Regular Bipartite Graphs}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)}, pages = {34:1--34:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-125-2}, ISSN = {1868-8969}, year = {2019}, volume = {145}, editor = {Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.34}, URN = {urn:nbn:de:0030-drops-112498}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2019.34}, annote = {Keywords: Approximate counting, Polymer model, Hardcore model, Coloring, Random bipartite graphs} }

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Brief Announcement

**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

Generating good revenue is one of the most important problems in Bayesian auction design, and many (approximately) optimal dominant-strategy incentive compatible (DSIC) Bayesian mechanisms have been constructed for various auction settings. However, most existing studies do not consider the complexity for the seller to carry out the mechanism. It is assumed that the seller knows "each single bit" of the distributions and is able to optimize perfectly based on the entire distributions. Unfortunately this is a strong assumption and may not hold in reality: for example, when the value distributions have exponentially large supports or do not have succinct representations.
In this work we consider, for the first time, the query complexity of Bayesian mechanisms. We only allow the seller to have limited oracle accesses to the players' value distributions, via quantile queries and value queries. For a large class of auction settings, we prove logarithmic lower-bounds for the query complexity for any DSIC Bayesian mechanism to be of any constant approximation to the optimal revenue. For single-item auctions and multi-item auctions with unit-demand or additive valuation functions, we prove tight upper-bounds via efficient query schemes, without requiring the distributions to be regular or have monotone hazard rate. Thus, in those auction settings the seller needs to access much less than the full distributions in order to achieve approximately optimal revenue.

Jing Chen, Bo Li, Yingkai Li, and Pinyan Lu. Brief Announcement: Bayesian Auctions with Efficient Queries. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 108:1-108:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chen_et_al:LIPIcs.ICALP.2018.108, author = {Chen, Jing and Li, Bo and Li, Yingkai and Lu, Pinyan}, title = {{Brief Announcement: Bayesian Auctions with Efficient Queries}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {108:1--108:4}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.108}, URN = {urn:nbn:de:0030-drops-91124}, doi = {10.4230/LIPIcs.ICALP.2018.108}, annote = {Keywords: The complexity of Bayesian mechanisms, quantile queries, value queries} }

Document

**Published in:** Dagstuhl Follow-Ups, Volume 7, The Constraint Satisfaction Problem: Complexity and Approximability (2017)

In this article we survey recent developments on the complexity of Holant problems. We discuss three different aspects of Holant problems: the decision version, exact counting, and approximate counting.

Heng Guo and Pinyan Lu. On the Complexity of Holant Problems. In The Constraint Satisfaction Problem: Complexity and Approximability. Dagstuhl Follow-Ups, Volume 7, pp. 159-177, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InCollection{guo_et_al:DFU.Vol7.15301.159, author = {Guo, Heng and Lu, Pinyan}, title = {{On the Complexity of Holant Problems}}, booktitle = {The Constraint Satisfaction Problem: Complexity and Approximability}, pages = {159--177}, series = {Dagstuhl Follow-Ups}, ISBN = {978-3-95977-003-3}, ISSN = {1868-8977}, year = {2017}, volume = {7}, editor = {Krokhin, Andrei and Zivny, Stanislav}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/DFU.Vol7.15301.159}, URN = {urn:nbn:de:0030-drops-69630}, doi = {10.4230/DFU.Vol7.15301.159}, annote = {Keywords: Computational complexity, Counting complexity, Dichotomy theorems, Approximate counting, Holant problems} }

Document

**Published in:** LIPIcs, Volume 60, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)

For anti-ferromagnetic 2-spin systems, a beautiful connection has been established, namely that the following three notions align perfectly: the uniqueness in infinite regular trees, the decay of correlations (also known as spatial mixing), and the approximability of the partition function. The uniqueness condition implies spatial mixing, and an FPTAS for the partition function exists based on spatial mixing. On the other hand, non-uniqueness implies some long range correlation, based on which NP-hardness reductions are built. These connections for ferromagnetic 2-spin systems are much less clear, despite their similarities to anti-ferromagnetic systems. The celebrated Jerrum-Sinclair Markov chain [JS93] works even if spatial mixing or uniqueness fails.
We provide some partial answers. We use (β,γ) to denote the (+,+) and (−,−) edge interactions and λ the external field, where βγ>1. If all fields satisfy λ<λ_c (assuming β≤γ), where λ_c=(γ/β)^{(Δ_c+1)/2} and Δ_c=(\sqrt{βγ}+1)/(\sqrt{βγ}−1), then a weaker version of spatial mixing holds in all trees. Moreover, if β≤1, then λ<λ_c is sufficient to guarantee strong spatial mixing and FPTAS. This improves the previous best algorithm, which is an FPRAS based on Markov chains and works for λ<γ/β [LLZ14a]. The bound λ_c is almost optimal. When β≤1, uniqueness holds in all infinite regular trees, if and only if λ≤λ^int_c, where λ^int_c=(γ/β)(⌈Δc⌉+1)/2. If we allow fields λ>λ^int′_c, where λ^int′_c=(γ/β)(⌊Δc⌋+2)/2, then approximating the partition function is #BIS-hard.

Heng Guo and Pinyan Lu. Uniqueness, Spatial Mixing, and Approximation for Ferromagnetic 2-Spin Systems. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 31:1-31:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{guo_et_al:LIPIcs.APPROX-RANDOM.2016.31, author = {Guo, Heng and Lu, Pinyan}, title = {{Uniqueness, Spatial Mixing, and Approximation for Ferromagnetic 2-Spin Systems}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)}, pages = {31:1--31:26}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-018-7}, ISSN = {1868-8969}, year = {2016}, volume = {60}, editor = {Jansen, Klaus and Mathieu, Claire and Rolim, Jos\'{e} D. P. and Umans, Chris}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2016.31}, URN = {urn:nbn:de:0030-drops-66547}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2016.31}, annote = {Keywords: Approximate counting; Ising model; Spin systems; Correlation decay} }

Document

**Published in:** LIPIcs, Volume 47, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)

Hardcore and Ising models are two most important families of two state spin systems in statistic physics. Partition function of spin systems is the center concept in statistic physics which connects microscopic particles and their interactions with their macroscopic and statistical properties of materials such as energy, entropy, ferromagnetism, etc. If each local interaction of the system involves only two particles, the system can be described by a graph. In this case, fully polynomial-time approximation scheme (FPTAS) for computing the partition function of both hardcore and anti-ferromagnetic Ising model was designed up to the uniqueness condition of the system. These result are the best possible since approximately computing the partition function beyond this threshold is NP-hard. In this paper, we generalize these results to general physics systems, where each local interaction may involves multiple particles. Such systems are described by hypergraphs. For hardcore model, we also provide FPTAS up to the uniqueness condition, and for anti-ferromagnetic Ising model, we obtain FPTAS under a slightly stronger condition.

Pinyan Lu, Kuan Yang, and Chihao Zhang. FPTAS for Hardcore and Ising Models on Hypergraphs. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 51:1-51:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{lu_et_al:LIPIcs.STACS.2016.51, author = {Lu, Pinyan and Yang, Kuan and Zhang, Chihao}, title = {{FPTAS for Hardcore and Ising Models on Hypergraphs}}, booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)}, pages = {51:1--51:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-001-9}, ISSN = {1868-8969}, year = {2016}, volume = {47}, editor = {Ollinger, Nicolas and Vollmer, Heribert}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.51}, URN = {urn:nbn:de:0030-drops-57526}, doi = {10.4230/LIPIcs.STACS.2016.51}, annote = {Keywords: hard-core model, ising model, hypergraph, spatial mixing, correlation decay} }

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**Published in:** LIPIcs, Volume 28, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)

We study the approximability of computing the partition function for ferromagnetic two-state spin systems. The remarkable algorithm by Jerrum and Sinclair showed that there is a fully polynomial-time randomized approximation scheme (FPRAS) for the special ferromagnetic Ising model with any given uniform external field. Later, Goldberg and Jerrum proved that it is #BIS-hard for Ising model if we allow inconsistent external fields on different nodes. In contrast to these two results, we prove that for any ferromagnetic two-state spin systems except the Ising model, there exists a threshold for external fields beyond which the problem is #BIS-hard, even if the external field is uniform.

Jingcheng Liu, Pinyan Lu, and Chihao Zhang. The Complexity of Ferromagnetic Two-spin Systems with External Fields. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 843-856, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{liu_et_al:LIPIcs.APPROX-RANDOM.2014.843, author = {Liu, Jingcheng and Lu, Pinyan and Zhang, Chihao}, title = {{The Complexity of Ferromagnetic Two-spin Systems with External Fields}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)}, pages = {843--856}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-74-3}, ISSN = {1868-8969}, year = {2014}, volume = {28}, editor = {Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.843}, URN = {urn:nbn:de:0030-drops-47428}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2014.843}, annote = {Keywords: Spin System, #BIS-hard, FPRAS} }

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**Published in:** LIPIcs, Volume 20, 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)

We study the complexity of approximation for a weighted counting constraint satisfaction problem #CSP(F). In the conservative case, where F contains all unary functions, a classification is known for the Boolean domain. We give a classification for problems with general finite domain. We define weak log-modularity and weak log-supermodularity, and show that #CSP(F) is in FP if F is weakly log-modular. Otherwise, it is at least as hard to approximate as #BIS, counting independent sets in bipartite graphs, which is believed to be intractable. We further sub-divide the #BIS-hard case. If F is weakly log-supermodular, we show that #CSP(F) is as easy as Boolean log-supermodular weighted #CSP. Otherwise, it is NP-hard to approximate. Finally, we give a trichotomy for the arity-2 case.
Then, #CSP(F) is in FP, is #BIS-equivalent, or is equivalent to #SAT, the problem of approximately counting satisfying assignments of a CNF Boolean formula.

Xi Chen, Martin Dyer, Leslie Ann Goldberg, Mark Jerrum, Pinyan Lu, Colin McQuillan, and David Richerby. The complexity of approximating conservative counting CSPs. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 20, pp. 148-159, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{chen_et_al:LIPIcs.STACS.2013.148, author = {Chen, Xi and Dyer, Martin and Goldberg, Leslie Ann and Jerrum, Mark and Lu, Pinyan and McQuillan, Colin and Richerby, David}, title = {{The complexity of approximating conservative counting CSPs}}, booktitle = {30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)}, pages = {148--159}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-50-7}, ISSN = {1868-8969}, year = {2013}, volume = {20}, editor = {Portier, Natacha and Wilke, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2013.148}, URN = {urn:nbn:de:0030-drops-39303}, doi = {10.4230/LIPIcs.STACS.2013.148}, annote = {Keywords: counting constraint satisfaction problem, approximation, complexity} }

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**Published in:** LIPIcs, Volume 9, 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)

We prove a complexity dichotomy theorem for counting weighted Boolean CSP modulo k for any positive integer $k>1$. This generalizes a theorem by Faben for the unweighted setting. In the weighted setting, there are new interesting tractable problems. We first prove a dichotomy theorem for the finite field case where k is a prime. It turns out that the dichotomy theorem for the finite field is very similar to the one for the complex weighted Boolean #CSP, found by [Cai, Lu and Xia, STOC 2009]. Then we further extend the result to an arbitrary integer k.

Heng Guo, Sangxia Huang, Pinyan Lu, and Mingji Xia. The Complexity of Weighted Boolean #CSP Modulo k. In 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 9, pp. 249-260, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{guo_et_al:LIPIcs.STACS.2011.249, author = {Guo, Heng and Huang, Sangxia and Lu, Pinyan and Xia, Mingji}, title = {{The Complexity of Weighted Boolean #CSP Modulo k}}, booktitle = {28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)}, pages = {249--260}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-25-5}, ISSN = {1868-8969}, year = {2011}, volume = {9}, editor = {Schwentick, Thomas and D\"{u}rr, Christoph}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2011.249}, URN = {urn:nbn:de:0030-drops-30158}, doi = {10.4230/LIPIcs.STACS.2011.249}, annote = {Keywords: #CSP, dichotomy theorem, counting problems, computational complexity} }

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**Published in:** LIPIcs, Volume 1, 25th International Symposium on Theoretical Aspects of Computer Science (2008)

We study the scheduling problem on unrelated machines in the
mechanism design setting. This problem was proposed and studied in
the seminal paper (Nisan and Ronen 1999), where they gave a
$1.75$-approximation randomized truthful mechanism for the case of
two machines. We improve this result by a $1.6737$-approximation
randomized truthful mechanism. We also generalize our result to a
$0.8368m$-approximation mechanism for task scheduling with $m$
machines, which improve the previous best upper bound of
$0.875m( Mu'alem and Schapira 2007)

Pinyan Lu and Changyuan Yu. An Improved Randomized Truthful Mechanism for Scheduling Unrelated Machines. In 25th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 1, pp. 527-538, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{lu_et_al:LIPIcs.STACS.2008.1314, author = {Lu, Pinyan and Yu, Changyuan}, title = {{An Improved Randomized Truthful Mechanism for Scheduling Unrelated Machines}}, booktitle = {25th International Symposium on Theoretical Aspects of Computer Science}, pages = {527--538}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-06-4}, ISSN = {1868-8969}, year = {2008}, volume = {1}, editor = {Albers, Susanne and Weil, Pascal}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2008.1314}, URN = {urn:nbn:de:0030-drops-13146}, doi = {10.4230/LIPIcs.STACS.2008.1314}, annote = {Keywords: Truthful mechanism, scheduling} }

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