Tight Bounds for Counting Colorings and Connected Edge Sets Parameterized by Cutwidth

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 ? ? such that p is a prime and q ? 3. We show: - If p divides q-1, there is a (q-1)^{ctw}n^{O(1)} time algorithm for counting list q-colorings modulo p of n-vertex graphs of cutwidth ctw. Furthermore, there is no ? > 0 for which there is a (q-1-?)^{ctw} n^{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 ? > 0 for which there exists a (q-?)^{ctw} n^{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^{ctw}n^{O(1)} time algorithm to compute the chromatic number of a graph by Jansen and Nederlof [Theor. Comput. Sci.\u2718]. Furthermore, by building upon the above lower bounds, we obtain the following lower bound for counting connected spanning edge sets: there is no ? > 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 - ?)^{ctw} n^{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\u2714] 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

A Fine-Grained Classification of the Complexity of Evaluating the Tutte Polynomial on Integer Points Parameterized by Treewidth and Cutwidth

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) ∈ ℤ² that either - T(G; x, y) can be computed in polynomial time, - T(G; x, y) 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), - T(G; x, y) 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

Tight Bounds for Counting Colorings and Connected Edge Sets Parameterized by Cutwidth

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)^{ctw}n^{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)^{ctw} n^{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)^{ctw} n^{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^{ctw}n^{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)^{ctw} n^{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

The Parameterised Complexity of Integer Multicommodity Flow

The Integer Multicommodity Flow problem has been studied extensively in the literature. However, from a parameterised perspective, mostly special cases, such as the Disjoint Paths 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

Tight Bounds for Counting Colorings and Connected Edge Sets Parameterized by Cutwidth

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)^{ctw}n^{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)^{ctw} n^{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)^{ctw} n^{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^{ctw}n^{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)^{ctw} n^{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

On the Parameterized Complexity of the Connected Flow and Many Visits TSP Problem

We study a variant of Min Cost Flow in which the flow needs to be connected. Specifically, in the Connected Flow problem one is given a directed graph G, along with a set of demand vertices D⊆ V(G) with demands dem: D→ N, and costs and capacities for each edge. The goal is to find a minimum cost flow that satisfies the demands, respects the capacities and induces a (strongly) connected subgraph. This generalizes previously studied problems like the (Many Visits) TSP. We study the parameterized complexity of Connected Flow parameterized by |D|, the treewidth tw and by vertex cover size k of G and provide: 1.NP-completeness already for the case | D| = 2 with only unit demands and capacities and no edge costs, and fixed-parameter tractability if there are no capacities,2.a fixed-parameter tractable O⋆(kO ( k )) time algorithm for the general case, and a kernel of size polynomial in k for the special case of Many Visits TSP,3.a | V(G) |O ( t w ) time algorithm and a matching | V(G) |o ( t w ) time conditional lower bound conditioned on the Exponential Time Hypothesis. To achieve some of our results, we significantly extend an approach by Kowalik et al. [ESA’20]

On the Parameterized Complexity of the Connected Flow and Many Visits TSP Problem

We study a variant of Min Cost Flow in which the flow needs to be connected. Specifically, in the Connected Flow problem one is given a directed graph G, along with a set of demand vertices D⊆ V(G) with demands dem: D→ N, and costs and capacities for each edge. The goal is to find a minimum cost flow that satisfies the demands, respects the capacities and induces a (strongly) connected subgraph. This generalizes previously studied problems like the (Many Visits) TSP. We study the parameterized complexity of Connected Flow parameterized by |D|, the treewidth tw and by vertex cover size k of G and provide: 1.NP-completeness already for the case | D| = 2 with only unit demands and capacities and no edge costs, and fixed-parameter tractability if there are no capacities,2.a fixed-parameter tractable O⋆(kO ( k )) time algorithm for the general case, and a kernel of size polynomial in k for the special case of Many Visits TSP,3.a | V(G) |O ( t w ) time algorithm and a matching | V(G) |o ( t w ) time conditional lower bound conditioned on the Exponential Time Hypothesis. To achieve some of our results, we significantly extend an approach by Kowalik et al. [ESA’20]