The number of independent sets in an irregular graph

Sah, Ashwin; Sawhney, Mehtaab; Stoner, David; Zhao, Yufei

doi:10.1016/j.jctb.2019.01.007

Cited by 15 publications

(30 citation statements)

References 32 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A decade later (2010), it was extended in [23] to regular graphs (that are not necessarily bipartite); a year later (2011), it was proved in [24] for graphs with a small maximal degree (up to 5). Finally, this conjecture was recently (2019) proved in general [25], by utilizing a new approach. The reader is referred to [26] for an announcement on the solution of this conjecture as a frustrating combinatorial problem for two decades, along with the history and ramifications of this problem, and some reflections of the authors on their work in [25].…”

Section: Introductionmentioning

confidence: 87%

“…Finally, this conjecture was recently (2019) proved in general [25], by utilizing a new approach. The reader is referred to [26] for an announcement on the solution of this conjecture as a frustrating combinatorial problem for two decades, along with the history and ramifications of this problem, and some reflections of the authors on their work in [25].…”

Section: Introductionmentioning

confidence: 87%

“…The recently introduced proof of the conjecture for general undirected graphs [25] uses an induction on the number of vertices in a graph, and it obtains a recurrence inequality whose derivation involves some judicious applications of Hölder's inequality (see Sections 2 and 4 in [25]). The work in [25] proved, for the first time, Conjecture 4.2 in [11] by using an interesting approach which is unrelated to information theory. The possibility of generalizing the information-theoretic proof in [11] to irregular bipartite graphs was left in [25] as an open issue.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A Generalized Information-Theoretic Approach for Bounding the Number of Independent Sets in Bipartite Graphs

Sason¹

2021

Entropy

View full text Add to dashboard Cite

This paper studies the problem of upper bounding the number of independent sets in a graph, expressed in terms of its degree distribution. For bipartite regular graphs, Kahn (2001) established a tight upper bound using an information-theoretic approach, and he also conjectured an upper bound for general graphs. His conjectured bound was recently proved by Sah et al. (2019), using different techniques not involving information theory. The main contribution of this work is the extension of Kahn’s information-theoretic proof technique to handle irregular bipartite graphs. In particular, when the bipartite graph is regular on one side, but may be irregular on the other, the extended entropy-based proof technique yields the same bound as was conjectured by Kahn (2001) and proved by Sah et al. (2019).

show abstract

Section: Introductionmentioning

confidence: 87%

Section: Introductionmentioning

confidence: 87%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A Generalized Information-Theoretic Approach for Bounding the Number of Independent Sets in Bipartite Graphs

Sason¹

2021

Entropy

View full text Add to dashboard Cite

show abstract

“…In our case, the number of vertices in graph ‫ܩ‬ ሖ corresponds to the number of independent sets in the collision graph Q. In general, finding an upper bound for the number of independent sets in different types of graphs is a problem in its own right (e.g., see [35]). Pessimistically, the trivial upper bound of 2 |E| (for a topology graph with |E| links) is as tight as one can get, since the extremely sparse collision graph that has no edges does indeed have 2 |E| independent sets.…”

Section: Fig 5 Corresponding Bipartite Graphmentioning

confidence: 99%

IEEE 802.15.4.e TSCH-Based Scheduling for Throughput Optimization: A Combinatorial Multi-Armed Bandit Approach

2020

View full text Add to dashboard Cite

In TSCH, which is a MAC mechanism set of the IEEE 802.15.4e amendment, calculation, construction, and maintenance of the packet transmission schedules are not defined. Moreover, to ensure optimal throughput, most of the existing scheduling methods are based on the assumption that instantaneous and accurate Channel State Information (CSI) is available. However, due to the inevitable errors in the channel estimation process, this assumption cannot be materialized in many practical scenarios. In this paper, we propose two alternative and realistic approaches. In our first approach, we assume that only the statistical knowledge of CSI is available a priori. Armed with this knowledge, the average packet rate on each link is computed and then, using the results, the throughput-optimal schedule for the assignment of (slotframe) cells to links can be formulated as a max-weight bipartite matching problem, which can be solved efficiently using the well-known Hungarian algorithm. In the second approach, we assume that no CSI knowledge (even statistical) is available at the design stage. For this zeroknowledge setting, we introduce a machine learning-based algorithm by formally modeling the scheduling problem in terms of a combinatorial multi-armed bandit (CMAB) process. Our CMAB-based scheme is widely applicable to many real operational environments, thanks to its reduced reliance on design-time knowledge. Simulation results show that the average throughput obtained by the statistical CSIbased method is within the margin of 15% from the theoretical upper bound associated with perfect instantaneous CSI. The aforesaid margin is around 18% for our learning-theoretic solution.Index Terms-IEEE 802.15.4e, TSCH, Scheduling, CSI, CMAB.

show abstract

“…Relation to previous work. This work builds on our earlier work [34] proving Kahn's conjecture on independent sets, Theorem 1.5, but requires several significantly new ideas. Our proof of Theorem 1.11 in Section 3 actually gives a new and more streamlined proof of Theorem 1.5.…”

Section: Graphical Brascamp-lieb Inequalitiesmentioning

confidence: 99%

A reverse Sidorenko inequality

et al. 2020

Self Cite

View full text Add to dashboard Cite

Let H be a graph allowing loops as well as vertex-and edge-weights. We prove that, for every triangle-free graph G without isolated vertices, the weighted number of graph homomorphisms hom(G, H) satisfies the inequalitywhere du denotes the degree of vertex u in G. In particular, one has hom(G, H) 1/|E(G)| ≤ hom (K d,d , H) 1/d 2 for every d-regular triangle-free G. The triangle-free hypothesis on G is best possible. More generally, we prove a graphical Brascamp-Lieb type inequality, where every edge of G 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 H = Kq (also valid if some of the vertices of Kq are looped), we show that the triangle-free hypothesis on G may be dropped. A corollary is that among d-regular graphs, G = K d,d maximizes the quantity cq(G) 1/|V (G)| for every q and d, where cq(G) counts proper q-colorings of G.Finally, we show that if the edge-weight matrix of H is positive semidefinite, thenThis implies that among d-regular graphs, G = K d+1 maximizes hom(G, H) 1/|V (G)| . 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-Csikvári-Perkins-Tetali, and provide an alternate proof to a conjecture by Kahn. arXiv:1809.09462v2 [math.CO] , where denotes a disjoint union. Question 1.1 was initially raised by Granville in 1988 in connection with the Cameron-Erdős conjecture on the number of sum-free sets. Alon [1] and Kahn [27] conjectured that G = K d,d is the exact maximizer. Alon [1] proved an asymptotic version as d → ∞, Kahn [27] proved the exact version under the additional hypothesis that G is bipartite, and Zhao [38] later removed this bipartite assumption. The results of Kahn [27] and Zhao [38] together answer Question 1.1: the maximizer is K d,d (unique up to taking disjoint unions of copies of itself). Galvin and Tetali [22] initiated the study of Questions 1.2 and 1.3 and extended Kahn's entropy method [27] to prove that, under the additional hypothesis that G is bipartite, G = K d,d is also the maximizer for hom(G, H) 1/|V (G)| . See Lubetzky and Zhao [32, Section 6] for a different proof using Hölder/Brascamp-Lieb type inequalities. Can the bipartite hypothesis on G also be dropped in this case? Not for all H: e.g., for H = , G = K d+1 is the maximizer instead of K d,d . Extending the technique for independent sets, Zhao [39] showed that the bipartite hypothesis can be dropped for certain classes of H, but the techniques failed for H = K q , corresponding to colorings (Question 1.2). It remained a tantalizing conjecture to remove the bipartite hypothesis for colorings.Recently, Davies, Jenssen, Perki...

show abstract

The number of independent sets in an irregular graph

Cited by 15 publications

References 32 publications

A Generalized Information-Theoretic Approach for Bounding the Number of Independent Sets in Bipartite Graphs

A Generalized Information-Theoretic Approach for Bounding the Number of Independent Sets in Bipartite Graphs

IEEE 802.15.4.e TSCH-Based Scheduling for Throughput Optimization: A Combinatorial Multi-Armed Bandit Approach

A reverse Sidorenko inequality

Contact Info

Product

Resources

About