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 v_1,…,v_n of V(G) such that, for each i ∈ {1,…,n-1}, the number of edges with one endpoint in {v_1,…,v_i} and the other in {v_{i+1},…,v_n} is at most k. We aim, for each H, for algorithms for Hom(H) running in time c_H^k n^𝒪(1) and matching lower bounds that exclude c_H^{k⋅o(1)} n^𝒪(1) or c_H^{k(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 c_H 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 c_H ≥ 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.

We give a fine-grained classification of evaluating the Tutte polynomial $T(G;x,y)$ on all integer points on graphs with small treewidth and cutwidth. Specifically, we show for any point $(x,y) \in \mathbb{Z}^2$ that either - can be computed in polynomial time, - can be computed in $2^{O(tw)}n^{O(1)}$ time, but not in $2^{o(ctw)}n^{O(1)}$ time assuming the Exponential Time Hypothesis (ETH), - can be computed in $2^{O(tw \log tw)}n^{O(1)}$ time, but not in $2^{o(ctw \log ctw)}n^{O(1)}$ time assuming the ETH, where we assume tree decompositions of treewidth $tw$ and cutwidth decompositions of cutwidth $ctw$ are given as input along with the input graph on $n$ vertices and point $(x,y)$. To obtain these results, we refine the existing reductions that were instrumental for the seminal dichotomy by Jaeger, Welsh and Vertigan~[Math. Proc. Cambridge Philos. Soc'90]. One of our technical contributions is a new rank bound of a matrix that indicates whether the union of two forests is a forest itself, which we use to show that the number of forests of a graph can be counted in $2^{O(tw)}n^{O(1)}$ time.

The Integer Multicommodity Flow problem has been studied extensively in the literature. However, from a parameterised perspective, mostly special cases, such as the Disjoint Path problem, have been considered. Therefore, we investigate the parameterised complexity of the general Integer Multicommodity Flow problem. We show that the decision version of this problem on directed graphs for a constant number of commodities, when the capacities are given in unary, is XNLP-complete with pathwidth as parameter and XALP-complete with treewidth as parameter. When the capacities are given in binary, the problem is NP-complete even for graphs of pathwidth at most 13. We give related results for undirected graphs. These results imply that the problem is unlikely to be fixed-parameter tractable by these parameters. In contrast, we show that the problem does become fixed-parameter tractable when weighted tree partition width (a variant of tree partition width for edge weighted graphs) is used as parameter.

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 \emph{edge} separators. Let $p,q \in \mathbb{N}$ such that $p$ is a prime and $q \geq 3$. - If $p$ divides $q-1$, there is a $(q-1)^{\text{ctw}}n^{O(1)}$ time algorithm for counting list $q$-colorings modulo $p$ of $n$-vertex graphs of cutwidth $\text{ctw}$ and for all $\varepsilon>0$ there is no algorithm running in time $(q-1-\varepsilon)^{\text{ctw}} n^{O(1)}$, assuming the Strong Exponential Time Hypothesis (SETH). - If $p$ does not divide $q-1$, there is a (folklore) $q^{\text{ctw}}n^{O(1)}$ time algorithm for counting list $q$-colorings modulo $p$ of $n$-vertex graphs of cutwidth $\text{ctw}$ and for all $\varepsilon>0$ there is no algorithm running in time $(q-\varepsilon)^{\text{ctw}} n^{O(1)}$, assuming SETH. The lower bounds are in stark contrast with the existing $2^{\text{ctw}}n^{O(1)}$ time algorithm to compute the chromatic number of a graph by Jansen and Nederlof~[Theor. Comput. Sci.'18]. 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. We extend our lower bounds to counting connected spanning edge sets modulo $p$ and give an algorithm with matching running time for both treewidth and cutwidth.