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1. Near-Linear Time Homomorphism Counting in Bounded Degeneracy Graphs: The Barrier of Long Induced Cycles

   Suman K. Bera, Noujan Pashanasangi (January 2021, Symposium on Discrete Algorithms (SODA))

   null (Ed.)

   Counting homomorphisms of a constant sized pattern graph H in an input graph G is a fundamental computational problem. There is a rich history of studying the complexity of this problem, under various constraints on the input G and the pattern H. Given the significance of this problem and the large sizes of modern inputs, we investigate when near-linear time algorithms are possible. We focus on the case when the input graph has bounded degeneracy, a commonly studied and practically relevant class for homomorphism counting. It is known from previous work that for certain classes of H, H-homomorphisms can be counted exactly in near-linear time in bounded degeneracy graphs. Can we precisely characterize the patterns H for which near-linear time algorithms are possible? We completely resolve this problem, discovering a clean dichotomy using fine-grained complexity. Let m denote the number of edges in G. We prove the following: if the largest induced cycle in H has length at most 5, then there is an O(mlogm) algorithm for counting H-homomorphisms in bounded degeneracy graphs. If the largest induced cycle in H has length at least 6, then (assuming standard fine-grained complexity conjectures) there is a constant γ>0, such that there is no o(m1+γ) time algorithm for counting H-homomorphisms. 

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2. Near-Linear Time Homomorphism Counting in Bounded Degeneracy Graphs: The Barrier of Long Induced Cycles

   https://doi.org/10.1137/1.9781611976465.138  

   Bera, Suman K.; Pashanasangi, Noujan; Seshadhri, C. (January 2021, Proceedings of the Annual ACMSIAM Symposium on Discrete Algorithms)

   null (Ed.)

   Counting homomorphisms of a constant sized pattern graph H in an input graph G is a fundamental computational problem. There is a rich history of studying the complexity of this problem, under various constraints on the input G and the pattern H. Given the significance of this problem and the large sizes of modern inputs, we investigate when near-linear time algorithms are possible. We focus on the case when the input graph has bounded degeneracy, a commonly studied and practically relevant class for homomorphism counting. It is known from previous work that for certain classes of H, H-homomorphisms can be counted exactly in near-linear time in bounded degeneracy graphs. Can we precisely characterize the patterns H for which near-linear time algorithms are possible? We completely resolve this problem, discovering a clean dichotomy using fine-grained complexity. Let m denote the number of edges in G. We prove the following: if the largest induced cycle in H has length at most 5, then there is an O(m log m) algorithm for counting H-homomorphisms in bounded degeneracy graphs. If the largest induced cycle in H has length at least 6, then (assuming standard fine-grained complexity conjectures) there is a constant γ > 0, such that there is no o(m1+γ) time algorithm for counting H-homomorphisms. 

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3. A Dichotomy for Bounded Degree Graph Homomorphisms with Nonnegative Weights

   Govorov, Artem Cai (July 2020, 47th International Colloquium on Automata, Languages, and Programming)

   Czumaj, Artur Dawar (Ed.)

   We consider the complexity of counting weighted graph homomorphisms defined by a symmetric matrix A. Each symmetric matrix A defines a graph homomorphism function Z A (·), also known as the partition function. Dyer and Greenhill [10] established a complexity dichotomy of Z A (·) for symmetric {0, 1}-matrices A, and they further proved that its #P-hardness part also holds for bounded degree graphs. Bulatov and Grohe [4] extended the Dyer-Greenhill dichotomy to nonnegative symmetric matrices A. However, their hardness proof requires graphs of arbitrarily large degree, and whether the bounded degree part of the Dyer-Greenhill dichotomy can be extended has been an open problem for 15 years. We resolve this open problem and prove that for nonnegative symmetric A, either Z A (G) is in polynomial time for all graphs G, or it is #P-hard for bounded degree (and simple) graphs G. We further extend the complexity dichotomy to include nonnegative vertex weights. Additionally, we prove that the #P-hardness part of the dichotomy by Goldberg et al. [12] for Z A (·) also holds for simple graphs, where A is any real symmetric matrix. 

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4. A decidable dichotomy theorem on directed graph homomorphisms with non-negative weights

   https://doi.org/10.1109/FOCS.2010.49  

   Cai, Jin-Yi; Chen, Xi. (January 2019, Computational complexity)

   The complexity of graph homomorphism problems has been the subject of intense study for some years. In this paper, we prove a decidable complexity dichotomy theorem for the partition function of directed graph homomorphisms. Our theorem applies to all non-negative weighted forms of the problem: given any fixed matrix A with non-negative algebraic entries, the partition function ZA(G) of directed graph homomorphisms from any directed graph G is either tractable in polynomial time or #P-hard, depending on the matrix A. The proof of the dichotomy theorem is combinatorial, but involves the definition of an infinite family of graph homomorphism problems. The proof of its decidability on the other hand is algebraic and based on properties of polynomials. 

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5. 11th Innovations in Theoretical Computer Science Conference (ITCS 2020).

   Cai, Jin-Yi; Govorov, Artem (January 2020, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020).)

   Vidick, Thomas (Ed.)

   Graph homomorphism has been an important research topic since its introduction [14]. Stated in the language of binary relational structures in that paper [14], Lovász proved a fundamental theorem that the graph homomorphism function G 7→ hom(G, H) for 0-1 valued H (as the adjacency matrix of a graph) determines the isomorphism type of H. In the past 50 years various extensions have been proved by Lovász and others [15, 9, 1, 19, 17]. These extend the basic 0-1 case to admit vertex and edge weights; but always with some restrictions such as all vertex weights must be positive. In this paper we prove a general form of this theorem where H can have arbitrary vertex and edge weights. An innovative aspect is that we prove this by a surprisingly simple and unified argument. This bypasses various technical obstacles and unifies and extends all previous known versions of this theorem on graphs. The constructive proof of our theorem can be used to make various complexity dichotomy theorems for graph homomorphism effective, i.e., it provides an algorithm that for any H either outputs a P-time algorithm solving hom(·, H) or a P-time reduction from a canonical #P-hard problem to hom(·, H). 

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