Abstract
We study the problem of approximating the value of the matching polynomial on graphs with edge parameter γ, where γ takes arbitrary values in the complex plane.
When γ is a positive real, Jerrum and Sinclair showed that the problem admits an FPRAS on general graphs. For general complex values of γ, Patel and Regts, building on methods developed by Barvinok, showed that the problem admits an FPTAS on graphs of maximum degree Δ as long as γ is not a negative real number less than or equal to −1/(4(Δ −1)). Our first main result completes the picture for the approximability of the matching polynomial on bounded degree graphs. We show that for all Δ ≥ 3 and all real γ less than −1/(4(Δ −1)), the problem of approximating the value of the matching polynomial on graphs of maximum degree Δ with edge parameter γ is #P-hard.
We then explore whether the maximum degree parameter can be replaced by the connective constant. Sinclair et al. showed that for positive real γ, it is possible to approximate the value of the matching polynomial using a correlation decay algorithm on graphs with bounded connective constant (and potentially unbounded maximum degree). We first show that this result does not extend in general in the complex plane; in particular, the problem is #P-hard on graphs with bounded connective constant for a dense set of γ values on the negative real axis. Nevertheless, we show that the result does extend for any complex value γ that does not lie on the negative real axis. Our analysis accounts for complex values of γ using geodesic distances in the complex plane in the metric defined by an appropriate density function.
- A. Barvinok. 2016. Computing the permanent of (some) complex matrices. Foundations of Computational Mathematics 16, 2 (2016), 329--342.Google Scholar
Digital Library
- A. Barvinok. 2017. Combinatorics and Complexity of Partition Functions. Springer International.Google Scholar
Digital Library
- M. Bayati, D. Gamarnik, D. A. Katz, C. Nair, and P. Tetali. 2007. Simple deterministic approximation algorithms for counting matchings. In Proceedings of the 39th Annual ACM Symposium on Theory of Computing (STOC’07). 122--127.Google Scholar
- I. Bezáková, A. Galanis, L. A. Goldberg, and D. Štefankovič. 2020. Inapproximability of the independent set polynomial in the complex plane. SIAM Journal on Computing 49, 5 (2020), 395--448.Google Scholar
Cross Ref
- P. Buys, A. Galanis, V. Patel, and G. Regts. 2020. Lee-Yang zeros and the complexity of the ferromagnetic Ising Model on bounded-degree graphs. arXiv:2006.14828Google Scholar
- J.-Y. Cai, S. Huang, and P. Lu. 2012. From Holant to #CSP and back: Dichotomy for Holantc problems. Algorithmica 64, 3 (2012), 511--533.Google Scholar
Digital Library
- A. Galanis, L. A. Goldberg, and D. Štefankovič. 2017. Inapproximability of the independent set polynomial below the Shearer threshold. In Proceedings of the 44th International Colloquium on Automata, Languages, and Programming (ICALP’17). Article 28, 13 pages. arXiv:1612.05832Google Scholar
- C. D. Godsil. 1981. Matchings and walks in graphs. Journal of Graph Theory 5, 3 (1981), 285--297.Google Scholar
Cross Ref
- L. A. Goldberg and H. Guo. 2017. The complexity of approximating complex-valued Ising and Tutte partition functions. Computational Complexity 26, 4 (2017), 765--833.Google Scholar
Digital Library
- L. A. Goldberg and M. Jerrum. 2014. The complexity of computing the sign of the Tutte polynomial. SIAM Journal on Computing 43, 6 (2014), 1921--1952.Google Scholar
Digital Library
- N. J. A. Harvey, P. Srivastava, and J. Vondrák. 2018. Computing the independence polynomial: From the tree threshold down to the roots. In Proceedings of the 29th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’18). 1557--1576.Google Scholar
- O. J. Heilmann and E. H. Lieb. 1972. Theory of monomer-dimer systems. Communications in Mathematical Physics 25, 3 (1972), 190--232.Google Scholar
Cross Ref
- M. Jerrum and A. Sinclair. 1989. Approximating the permanent. SIAM Journal on Computing 18, 6 (1989), 1149--1178.Google Scholar
Digital Library
- D. Kraus and O. Roth. 2008. Conformal metrics. arXiv:0805.2235Google Scholar
- V. Patel and G. Regts. 2017. Deterministic polynomial-time approximation algorithms for partition functions and graph polynomials. SIAM Journal on Computing 46, 6 (2017), 1893--1919.Google Scholar
Digital Library
- H. Peters and G. Regts. 2017. On a conjecture of Sokal concerning roots of the independence polynomial. arxiv:1701.08049Google Scholar
- A. Sinclair, P. Srivastava, D. Štefankovič, and Y. Yin. 2017. Spatial mixing and the connective constant: Optimal bounds. Probability Theory and Related Fields 168, 1 (2017), 153--197.Google Scholar
Cross Ref
- A. Sinclair, P. Srivastava, and Y. Yin. 2013. Spatial mixing and approximation algorithms for graphs with bounded connective constant. In Proceedings of the 2013 IEEE 54th Annual Symposium on Foundations of Computer Science (FOCS’13). 300--309.Google Scholar
- B.-B. Wei, S.-W. Chen, H.-C. Po, and R.-B. Liu. 2014. Phase transitions in the complex plane of physical parameters. Nature Scientific Reports 4 (2014), Article 5202.Google Scholar
- D. Weitz. 2006. Counting independent sets up to the tree threshold. In Proceedings of the 38th Annual ACM Symposium on Theory of Computing (STOC’06). ACM, New York, NY, 140--149.Google Scholar
Digital Library
Index Terms
The Complexity of Approximating the Matching Polynomial in the Complex Plane
Recommendations
A dichotomy theorem for the approximate counting of complex-weighted bounded-degree Boolean CSPs
We determine the computational complexity of approximately counting the total weight of variable assignments for every complex-weighted Boolean constraint satisfaction problem (or CSP) with any number of additional unary (i.e., arity 1) constraints, ...
Zeros and approximations of Holant polynomials on the complex plane
AbstractWe present fully polynomial time approximation schemes for a broad class of Holant problems with complex edge weights, which we call Holant polynomials. We transform these problems into partition functions of abstract combinatorial structures known ...
A Complexity Trichotomy for Approximately Counting List H-Colorings
We examine the computational complexity of approximately counting the list H-colorings of a graph. We discover a natural graph-theoretic trichotomy based on the structure of the graph H. If H is an irreflexive bipartite graph or a reflexive complete ...






Comments