research-article

Deterministic Polynomial-Time Approximation Algorithms for Partition Functions and Graph Polynomials

Authors:

Guus RegtsAuthors Info & Claims

SIAM Journal on Computing, Volume 46, Issue 6

Pages 1893 - 1919

https://doi.org/10.1137/16M1101003

Published: 01 January 2017 Publication History

Abstract

In this paper we show a new way of constructing deterministic polynomial-time approximation algorithms for computing complex-valued evaluations of a large class of graph polynomials on bounded degree graphs. In particular, our approach works for the Tutte polynomial and independence polynomial, as well as partition functions of complex-valued spin and edge-coloring models. More specifically, we define a large class of graph polynomials $\mathcal C$ and show that if $p\in \cal C$ and there is a disk $D$ centered at zero in the complex plane such that $p(G)$ does not vanish on $D$ for all bounded degree graphs $G$, then for each $z$ in the interior of $D$ there exists a deterministic polynomial-time approximation algorithm for evaluating $p(G)$ at $z$. This gives an explicit connection between absence of zeros of graph polynomials and the existence of efficient approximation algorithms, allowing us to show new relationships between well-known conjectures. Our work builds on a recent line of work initiated by Barvinok [Found. Comput. Math., 16 (2016), pp. 329--342; Theory Comput., 11 (2015), pp. 339--355; Computing the Partition Function of a Polynomial on the Boolean Cube, 2015; Discrete Anal., 2 (2017), 34pp], which provides a new algorithmic approach besides the existing Markov chain Monte Carlo method and the correlation decay method for these types of problems.

References

[1]

M. Bayati, D. Gamarnik, D. Katz, C. Nair, and P. Tetali, Simple deterministic approximation algorithms for counting matchings, in Proceedings of the 39th Annual ACM Symposium on Theory of Computing, 2007, pp. 122--127.

[2]

A. Barvinok, Computing the permanent of (some) complex matrices, Found. Comput. Math., 16 (2016), pp. 329--342.

[3]

A. Barvinok, Computing the partition function for cliques in a graph, Theory Comput., 11 (2015), pp. 339--355.

[4]

A. Barvinok, Computing the Partition Function of a Polynomial on the Boolean Cube, preprint, arXiv:1503.07463, 2015.

[5]

A. Barvinok, Approximating permanents and Hafnians, Discrete Anal., 2 (2017), 34pp.

[6]

A. Barvinok, personal communication, 2016.

[7]

A. Barvinok, Combinatorics and Complexity of Partition Functions, Algorithms Combin. 30, Springer, New York, 2017.

[8]

A. Barvinok and P. Soberón, Computing the partition function for graph homomorphisms, Combinatorica, 37 (2017), pp. 633--650, https://doi.org/10.1007/s00493-016-3357-2.

[9]

A. Barvinok and P. Soberón, Computing the partition function for graph homomorphisms with multiplicities, J. Combin. Theory Ser. A, 137 (2016), pp. 1--26.

[10]

C. Borgs, J. Chayes, J. Kahn, and L. Lovász, Left and right convergence of graphs with bounded degree, Random Structures Algorithms, 42 (2013), pp. 1--28.

[11]

R. Bubley, M. Dyer, C. Greenhill, and M. Jerrum, On approximately counting colorings of small degree graphs, SIAM J. Comput., 29 (1999), pp. 387--400.

[12]

A. Bulatov and M. Grohe, The complexity of partition functions, Theoret. Comput. Sci., 348 (2005), pp. 148--186.

[13]

J. Cai, X. Chen, and P. Lu, Graph homomorphisms with complex values: A dichotomy theorem, SIAM J. Comput., 42 (2013), pp. 924--1029.

[14]

J. Cai, H. Guo, and T. Williams, A complete dichotomy rises from the capture of vanishing signatures, in Proceedings of the 45th Annual ACM Symposium on Theory of Computing, 2013, pp. 635--644.

[15]

J. Cai, S. Huang, and P. Lu, From Holant to \#CSP and Back: Dichotomy for Holant$^c$ Problems, in Proceedings of ISAAC, 2010, pp. 253--265.

[16]

J. Cai, P. Lu, and M. Xia, Computational complexity of Holant problems, SIAM J. Comput., 40 (2011), pp. 1101--1132.

[17]

M. Chudnovsky and P. Seymour, The roots of the independence polynomial of a clawfree graph, J. Combin. Theory Ser. B, 97 (2007), pp. 350--357.

[18]

P. Csikvári and P. E. Frenkel, Benjamini--Schramm continuity of root moments of graph polynomials, European J. Combin., 52 (2016), pp. 302--320.

[19]

P. de la Harpe and V. F. R. Jones, Graph invariants related to statistical mechanical models: Examples and problems, J. Combin. Theory Ser. B, 57 (1993), pp. 207--227.

[20]

M. Dyer and C. Greenhill, On Markov chains for independent sets, J. Algorithms, 35 (2000), pp. 17--49.

[21]

M. Dyer and C. Greenhill, The complexity of counting graph homomorphisms, Random Structures Algorithms, 17 (2000), pp. 260--289.

[22]

A. Galanis, L. A. Goldberg, and D. Štefankovič, Inapproximability of the Independent Set Polynomial Below the Shearer Threshold, preprint, arXiv:1612.05832, 2016.

[23]

A. Galanis, D. Štefankovič, E. Vigoda, and L. Yang, Ferromagnetic Potts model, Refined \#BIS-hardness and related results, in RANDOM 2014, Lecture Notes in Computer Sci. 6845, 2014, pp. 677--691; arXiv:1311.4839.

[24]

D. Gamarnik and D. Katz, Correlation decay and deterministic FPTAS for counting list-colorings of a graph, J. Discrete Algorithms, 12 (2012), pp. 29--47.

[25]

D. Garijo, A. Goodall, and J. Nešetřil, Jaroslav, On the number of $B$-flows of a graph, European J. Combin., 35 (2014), pp. 273--285.

[26]

L. A. Goldberg and H. Guo, The Complexity of Approximating Complex-Valued Ising and Tutte Partition Functions, preprint, arXiv:1409.5627, 2014.

[27]

L. A. Goldberg and M. Jerrum, Approximating the partition function of the ferromagnetic Potts model, J. ACM, 59 (2012), pp. 1--25.

[28]

L. A. Goldberg and M. Jerrum, The Complexity of Computing the Sign of the Tutte Polynomial (and Consequent \#P-Hardness of Approximation), in Automata, Languages, and Programming, Springer, Berlin, 2012, pp. 399--410.

[29]

N. J. Harvey, P. Srivastava, and J. Vondrák, Computing the Independence Polynomial in Shearer's Region for the LLL, preprint, arXiv:1608.02282, 2016.

[30]

B. Jackson, Zeros of chromatic and flow polynomials of graphs, J. Geometry, 76 (2003), pp. 76--95.

[31]

B. Jackson, A. Procacci, and A. D. Sokal, Complex zero-free regions at large $|q|$ for multivariate Tutte polynomials (alias Potts-model partition functions) with general complex edge weights, J. Combin. Theory Ser. B, 103 (2013), pp. 21--45.

[32]

M. Jerrum, A very simple algorithm for estimating the number of $k$-colorings of a low-degree graph, Random Structures Algorithms, 7 (1995), pp. 157--165.

[33]

M. Jerrum and A. Sinclair, Polynomial-time approximation algorithms for the Ising model, SIAM J. Comput., 22 (1993), pp. 1087--1116.

[34]

T. Lee and T. Yang, Statistical theory of equations of state and phase transitions. I. Theory of condensation, Phys. Rev., 87 (1952), 404.

[35]

C. Lin, J. Liu, and P. Lu, A simple FPTAS for counting edge covers, in Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, 2014, pp. 341--348.

[36]

P. Lu and Y. Yin, Improved FPTAS for multi-spin systems, in Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques, Springer, Berlin, 2013, pp. 639--654.

[37]

V. Patel and G. Regts, Computing the Number of Induced Copies of a Fixed Graph in a Bounded Degree Graph, preprint, arXiv:1707.05186, 2017.

[38]

H. Peters and G. Regts, On a Conjecture of Sokal Concerning Roots of the Independence Polynomial, preprint, arXiv:1701.08049, 2017.

[39]

G. Regts, Graph Parameters and Invariants of the Orthogonal Group, Ph.D. thesis, University of Amsterdam, the Netherlands, 2013.

[40]

G. Regts, Zero-free regions of partition functions with applications to algorithms and graph limits, Combinatorica, to appear, https://doi.org/10.1007/s00493-016-3506-7.

[41]

A. D. Scott and A. D. Sokal, The repulsive lattice gas, the independent-set polynomial, and the Lovász local lemma, J. Statistical Physics, 118 (2005), pp. 1151--1261.

[42]

J. B. Shearer, On a problem of Spencer, Combinatorica, 5 (1998), pp. 241--245.

[43]

A. Sinclair, P. Srivastava, and M. Thurley, Approximation algorithms for two-state anti-ferromagnetic spin systems on bounded degree graphs, J. Stat. Phys., 155 (2014), pp. 666--686.

[44]

A. Sly and N. Sun, The computational hardness of counting in two-spin models on d-regular graphs, in Proceedings of the 53rd Annual Symposium on Foundations of Computer Science, 2012, IEEE, 2012, pp. 361--369.

[45]

A. Sokal, A personal list of unsolved problems concerning lattice gases and antiferromagnetic Potts models, Markov Process. Related Fields, 7 (2001), pp. 21--38.

[46]

P. Srivastava, Approximating the Hard Core Partition Function with Negative Activities, http://www.its.caltech.edu/~piyushs/docs/approx.pdf.

[47]

B. Szegedy, Edge-coloring models and reflection positivity, J. Amer. Math. Soc., 20 (2007), pp. 969--988.

[48]

B. Szegedy, Edge coloring models as singular vertex-coloring models, in Fete of Combinatorics and Computer Science, G.O.H. Katona, A. Schrijver, and T. Szönyi, eds., Springer, Berlin, 2010, pp. 327--336.

[49]

E. Vigoda, Improved bounds for sampling colorings, J. Math. Phys., 41 (2000), pp. 1555--1569.

[50]

D. Weitz, Counting independent sets up to the tree threshold, in Proceedings of the 38th Annual ACM Symposium on Theory of Computing, 2006, pp. 140--149.

Cited By

Bezáková IGalanis AGoldberg LŠtefankovič D(2024)Fast Sampling via Spectral Independence Beyond Bounded-degree GraphsACM Transactions on Algorithms10.1145/363135420:1(1-26)Online publication date: 22-Jan-2024
https://dl.acm.org/doi/10.1145/3631354
Anari NBurgess CTian KVuong TAgrawal KShun J(2023)Quadratic Speedups in Parallel Sampling from Determinantal DistributionsProceedings of the 35th ACM Symposium on Parallelism in Algorithms and Architectures10.1145/3558481.3591104(367-377)Online publication date: 17-Jun-2023
https://dl.acm.org/doi/10.1145/3558481.3591104
Hsieh JMohanty SXu J(2022)Certifying solution geometry in random CSPsProceedings of the 37th Computational Complexity Conference10.4230/LIPIcs.CCC.2022.11(1-18)Online publication date: 20-Jul-2022
https://dl.acm.org/doi/10.4230/LIPIcs.CCC.2022.11
Show More Cited By

Index Terms

Deterministic Polynomial-Time Approximation Algorithms for Partition Functions and Graph Polynomials
1. Mathematics of computing
  1. Discrete mathematics
  2. Mathematical analysis
    1. Functional analysis
      1. Approximation
    2. Numerical analysis
      1. Computations on polynomials
2. Theory of computation
  1. Design and analysis of algorithms
    1. Approximation algorithms analysis

Index terms have been assigned to the content through auto-classification.

Recommendations

Partition functions on k-regular graphs with {0,1}-vertex assignments and real edge functions

We prove a complexity dichotomy theorem for a class of partition functions over k-regular graphs, for any fixed k. These problems can be viewed as graph homomorphisms from an arbitrary k-regular input graph G to the weighted two vertex graph on {0,1} ...
The Complexity of Computing the Sign of the Tutte Polynomial

We study the complexity of computing the sign of the Tutte polynomial of a graph. As there are only three possible outcomes (positive, negative, and zero), this seems at first sight more like a decision problem than a counting problem. Surprisingly, however, ...
Zero-free regions for multivariate Tutte polynomials (alias Potts-model partition functions) of graphs and matroids

The chromatic polynomial P"G(q) of a loopless graph G is known to be non-zero (with explicitly known sign) on the intervals (-~,0), (0,1) and (1,32/27]. Analogous theorems hold for the flow polynomial of bridgeless graphs and for the characteristic ...

Comments

Information & Contributors

Information

Published In

cover image SIAM Journal on Computing

SIAM Journal on Computing Volume 46, Issue 6

DOI:10.1137/smjcat.46.6

Issue’s Table of Contents

© 2017, Society for Industrial and Applied Mathematics.

Publisher

Society for Industrial and Applied Mathematics

United States

Publication History

Published: 01 January 2017

Author Tags

Author Tags

Qualifiers

Research-article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

7
Total Citations
View Citations
0
Total Downloads

Downloads (Last 12 months)0
Downloads (Last 6 weeks)0

Reflects downloads up to 01 Jan 2025

Other Metrics

View Author Metrics

Citations

Cited By

Bezáková IGalanis AGoldberg LŠtefankovič D(2024)Fast Sampling via Spectral Independence Beyond Bounded-degree GraphsACM Transactions on Algorithms10.1145/363135420:1(1-26)Online publication date: 22-Jan-2024
https://dl.acm.org/doi/10.1145/3631354
Anari NBurgess CTian KVuong TAgrawal KShun J(2023)Quadratic Speedups in Parallel Sampling from Determinantal DistributionsProceedings of the 35th ACM Symposium on Parallelism in Algorithms and Architectures10.1145/3558481.3591104(367-377)Online publication date: 17-Jun-2023
https://dl.acm.org/doi/10.1145/3558481.3591104
Hsieh JMohanty SXu J(2022)Certifying solution geometry in random CSPsProceedings of the 37th Computational Complexity Conference10.4230/LIPIcs.CCC.2022.11(1-18)Online publication date: 20-Jul-2022
https://dl.acm.org/doi/10.4230/LIPIcs.CCC.2022.11
Bakali EChalki AGöbel APagourtzis AZachos S(2022)Guest column: A panorama of counting problems the decision version of which is in P3ACM SIGACT News10.1145/3561064.356107253:3(46-68)Online publication date: 30-Aug-2022
https://dl.acm.org/doi/10.1145/3561064.3561072
Jain VPerkins WSah ASawhney MLeonardi SGupta A(2022)Approximate counting and sampling via local central limit theoremsProceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing10.1145/3519935.3519957(1473-1486)Online publication date: 9-Jun-2022
https://dl.acm.org/doi/10.1145/3519935.3519957
Friedrich TGöbel AKatzmann MKrejca MPappik M(2022)Algorithms for Hard-Constraint Point Processes via DiscretizationComputing and Combinatorics10.1007/978-3-031-22105-7_22(242-254)Online publication date: 22-Oct-2022
https://dl.acm.org/doi/10.1007/978-3-031-22105-7_22
Bezáková IGalanis AGoldberg LŠtefankovič D(2021)The Complexity of Approximating the Matching Polynomial in the Complex PlaneACM Transactions on Computation Theory10.1145/344864513:2(1-37)Online publication date: 19-Apr-2021
https://dl.acm.org/doi/10.1145/3448645

View Options

View options

Media

Figures

Other

Tables

View Issue’s Table of Contents