668 research outputs found
Counting Independent Sets and Colorings on Random Regular Bipartite Graphs
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
Sidorenko's conjecture, colorings and independent sets
Let denote the number of homomorphisms from a graph to a
graph . Sidorenko's conjecture asserts that for any bipartite graph , and
a graph we have where
and denote the number of vertices and edges of the graph and
, respectively. In this paper we prove Sidorenko's conjecture for certain
special graphs : for the complete graph on vertices, for a
with a loop added at one of the end vertices, and for a path on vertices
with a loop added at each vertex. These cases correspond to counting colorings,
independent sets and Widom-Rowlinson colorings of a graph . For instance,
for a bipartite graph the number of -colorings
satisfies
In fact, we will prove that in the last two cases (independent sets and
Widom-Rowlinson colorings) the graph does not need to be bipartite. In all
cases, we first prove a certain correlation inequality which implies
Sidorenko's conjecture in a stronger form.Comment: Two references added and Remark 2.1 is expande
Counting dominating sets and related structures in graphs
We consider some problems concerning the maximum number of (strong)
dominating sets in a regular graph, and their weighted analogues. Our primary
tool is Shearer's entropy lemma. These techniques extend to a reasonably broad
class of graph parameters enumerating vertex colorings satisfying conditions on
the multiset of colors appearing in (closed) neighborhoods. We also generalize
further to enumeration problems for what we call existence homomorphisms. Here
our results are substantially less complete, though we do solve some natural
problems
Fast Algorithms at Low Temperatures via Markov Chains
For spin systems, such as the hard-core model on independent sets weighted by fugacity lambda>0, efficient algorithms for the associated approximate counting/sampling problems typically apply in the high-temperature region, corresponding to low fugacity. Recent work of Jenssen, Keevash and Perkins (2019) yields an FPTAS for approximating the partition function (and an efficient sampling algorithm) on bounded-degree (bipartite) expander graphs for the hard-core model at sufficiently high fugacity, and also the ferromagnetic Potts model at sufficiently low temperatures. Their method is based on using the cluster expansion to obtain a complex zero-free region for the partition function of a polymer model, and then approximating this partition function using the polynomial interpolation method of Barvinok. We present a simple discrete-time Markov chain for abstract polymer models, and present an elementary proof of rapid mixing of this new chain under sufficient decay of the polymer weights. Applying these general polymer results to the hard-core and ferromagnetic Potts models on bounded-degree (bipartite) expander graphs yields fast algorithms with running time O(n log n) for the Potts model and O(n^2 log n) for the hard-core model, in contrast to typical running times of n^{O(log Delta)} for algorithms based on Barvinok\u27s polynomial interpolation method on graphs of maximum degree Delta. In addition, our approach via our polymer model Markov chain is conceptually simpler as it circumvents the zero-free analysis and the generalization to complex parameters. Finally, we combine our results for the hard-core and ferromagnetic Potts models with standard Markov chain comparison tools to obtain polynomial mixing time for the usual spin system Glauber dynamics restricted to even and odd or "red" dominant portions of the respective state spaces
A reverse Sidorenko inequality
Let be a graph allowing loops as well as vertex and edge weights. We
prove that, for every triangle-free graph without isolated vertices, the
weighted number of graph homomorphisms satisfies the inequality
where denotes the degree of vertex in . In particular, one has for every -regular
triangle-free . The triangle-free hypothesis on is best possible. More
generally, we prove a graphical Brascamp-Lieb type inequality, where every edge
of is assigned some two-variable function. These inequalities imply tight
upper bounds on the partition function of various statistical models such as
the Ising and Potts models, which includes independent sets and graph
colorings.
For graph colorings, corresponding to , we show that the
triangle-free hypothesis on may be dropped; this is also valid if some of
the vertices of are looped. A corollary is that among -regular graphs,
maximizes the quantity for every and ,
where counts proper -colorings of .
Finally, we show that if the edge-weight matrix of is positive
semidefinite, then This implies that among -regular graphs,
maximizes . For 2-spin Ising models, our results give a
complete characterization of extremal graphs: complete bipartite graphs
maximize the partition function of 2-spin antiferromagnetic models and cliques
maximize the partition function of ferromagnetic models.
These results settle a number of conjectures by Galvin-Tetali, Galvin, and
Cohen-Csikv\'ari-Perkins-Tetali, and provide an alternate proof to a conjecture
by Kahn.Comment: 30 page
The Bipartite Swapping Trick on Graph Homomorphisms
We provide an upper bound to the number of graph homomorphisms from to
, where is a fixed graph with certain properties, and varies over
all -vertex, -regular graphs. This result generalizes a recently resolved
conjecture of Alon and Kahn on the number of independent sets. We build on the
work of Galvin and Tetali, who studied the number of graph homomorphisms from
to when is bipartite. We also apply our techniques to graph
colorings and stable set polytopes.Comment: 22 pages. To appear in SIAM J. Discrete Mat
Extremes of the internal energy of the Potts model on cubic graphs
We prove tight upper and lower bounds on the internal energy per particle
(expected number of monochromatic edges per vertex) in the anti-ferromagnetic
Potts model on cubic graphs at every temperature and for all . This
immediately implies corresponding tight bounds on the anti-ferromagnetic Potts
partition function.
Taking the zero-temperature limit gives new results in extremal
combinatorics: the number of -colorings of a -regular graph, for any , is maximized by a union of 's. This proves the case of a
conjecture of Galvin and Tetali
Ferromagnetic Potts Model: Refined #BIS-hardness and Related Results
Recent results establish for 2-spin antiferromagnetic systems that the
computational complexity of approximating the partition function on graphs of
maximum degree D undergoes a phase transition that coincides with the
uniqueness phase transition on the infinite D-regular tree. For the
ferromagnetic Potts model we investigate whether analogous hardness results
hold. Goldberg and Jerrum showed that approximating the partition function of
the ferromagnetic Potts model is at least as hard as approximating the number
of independent sets in bipartite graphs (#BIS-hardness). We improve this
hardness result by establishing it for bipartite graphs of maximum degree D. We
first present a detailed picture for the phase diagram for the infinite
D-regular tree, giving a refined picture of its first-order phase transition
and establishing the critical temperature for the coexistence of the disordered
and ordered phases. We then prove for all temperatures below this critical
temperature that it is #BIS-hard to approximate the partition function on
bipartite graphs of maximum degree D. As a corollary, it is #BIS-hard to
approximate the number of k-colorings on bipartite graphs of maximum degree D
when k <= D/(2 ln D).
The #BIS-hardness result for the ferromagnetic Potts model uses random
bipartite regular graphs as a gadget in the reduction. The analysis of these
random graphs relies on recent connections between the maxima of the
expectation of their partition function, attractive fixpoints of the associated
tree recursions, and induced matrix norms. We extend these connections to
random regular graphs for all ferromagnetic models and establish the Bethe
prediction for every ferromagnetic spin system on random regular graphs. We
also prove for the ferromagnetic Potts model that the Swendsen-Wang algorithm
is torpidly mixing on random D-regular graphs at the critical temperature for
large q.Comment: To appear in SIAM J. Computin
- …