A homomorphism from a graph G to a graph H is an edge-preserving mapping from V (G) to V (H). In the graph homomorphism problem, denoted by Hom(H), the graph H is fixed and we need to determine if there exists a homomorphism from an instance graph G to H. We study the complexity of the problem parameterized by the cutwidth of G, i.e., we assume that G is given along with a linear ordering v1, . . ., vn of V (G) such that, for each i ∈ {1, . . ., n − 1}, the number of edges with one endpoint in {v1, . . ., vi} and the other in {vi+1, . . ., vn} is at most k. We aim, for each H, for algorithms for Hom(H) running in time c^k_Hn^O(1) and matching lower bounds that exclude c^k_H^·o⁽¹⁾n^O(1) or c^k_H^(1−Ω(1))n^O(1) time algorithms under the (Strong) Exponential Time Hypothesis. In the paper we introduce a new parameter that we call mimsup(H). Our main contribution is strong evidence of a close connection between cH and mimsup(H): an information-theoretic argument that the number of states needed in a natural dynamic programming algorithm is at most mimsup(H)^k, lower bounds that show that for almost all graphs H indeed we have cH ≥ mimsup(H), assuming the (Strong) Exponential-Time Hypothesis, and an algorithm with running time exp(O(mimsup(H) · k log k))n^O(1). In the last result we do not need to assume that H is a fixed graph. Thus, as a consequence, we obtain that the problem of deciding whether G admits a homomorphism to H is fixed-parameter tractable, when parameterized by cutwidth of G and mimsup(H). The parameter mimsup(H) can be thought of as the p-th root of the maximum induced matching number in the graph obtained by multiplying p copies of H via a certain graph product, where p tends to infinity. It can also be defined as an asymptotic rank parameter of the adjacency matrix of H. Such parameters play a central role in, among others, algebraic complexity theory and additive combinatorics. Our results tightly link the parameterized complexity of a problem to such an asymptotic matrix parameter for the first time.

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We introduce a new model of indeterminacy in graphs: instead of specifying all the edges of the graph, the input contains all triples of vertices that form a connected subgraph. In general, different (labelled) graphs may have the same set of connected triples, making unique reconstruction of the original graph from the triples impossible. We identify some families of graphs (including triangle-free graphs) for which all graphs have a different set of connected triples. We also give algorithms that reconstruct a graph from a set of triples, and for testing if this reconstruction is unique. Finally, we study a possible extension of the model in which the subsets of size k that induce a connected graph are given for larger (fixed) values of k.@en

We study the fine-grained complexity of counting the number of colorings and connected spanning edge sets parameterized by the cutwidth and treewidth of the graph. While decompositions of small treewidth decompose the graph with small vertex separators, decompositions with small cutwidth decompose the graph with small edge separators. Let p, q ∈ N such that p is a prime and q ≥ 3. We show: If p divides q - 1, there is a (q - 1)^ctwn^O(1) time algorithm for counting list q-colorings modulo p of n-vertex graphs of cutwidth ctw. Furthermore, there is no ε > 0 for which there is a (q - 1 - ε)^ctwn^O(1) time algorithm that counts the number of list q-colorings modulo p of n-vertex graphs of cutwidth ctw, assuming the Strong Exponential Time Hypothesis (SETH). If p does not divide q - 1, there is no ε > 0 for which there exists a (q - ε)^ctwn^O(1) time algorithm that counts the number of list q-colorings modulo p of n-vertex graphs of cutwidth ctw, assuming SETH. The lower bounds are in stark contrast with the existing 2^ctwn^O(1) time algorithm to compute the chromatic number of a graph by Jansen and Nederlof [Theor. Comput. Sci.’18]. Furthermore, by building upon the above lower bounds, we obtain the following lower bound for counting connected spanning edge sets: there is no ε > 0 for which there is an algorithm that, given a graph G and a cutwidth ordering of cutwidth ctw, counts the number of spanning connected edge sets of G modulo p in time (p - ε)^ctwn^O(1), assuming SETH. We also give an algorithm with matching running time for this problem. Before our work, even for the treewidth parameterization, the best conditional lower bound by Dell et al. [ACM Trans. Algorithms’14] only excluded 2^o(tw)n^O(1) time algorithms for this problem. Both our algorithms and lower bounds employ use of the matrix rank method, by relating the complexity of the problem to the rank of a certain “compatibility matrix” in a non-trivial way.

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