Cutwidth: obstructions and algorithmic aspects

Cutwidth is one of the classic layout parameters for graphs. It measures how well one can order the vertices of a graph in a linear manner, so that the maximum number of edges between any prefix and its complement suffix is minimized. As graphs of cutwidth at most $k$ are closed under taking immersions, the results of Robertson and Seymour imply that there is a finite list of minimal immersion obstructions for admitting a cut layout of width at most $k$. We prove that every minimal immersion obstruction for cutwidth at most $k$ has size at most $2^{O(k^3\log k)}$. As an interesting algorithmic byproduct, we design a new fixed-parameter algorithm for computing the cutwidth of a graph that runs in time $2^{O(k^2\log k)}\cdot n$, where $k$ is the optimum width and $n$ is the number of vertices. While being slower by a $\log k$-factor in the exponent than the fastest known algorithm, given by Thilikos, Bodlaender, and Serna in [Cutwidth I: A linear time fixed parameter algorithm, J. Algorithms, 56(1):1--24, 2005] and [Cutwidth II: Algorithms for partial $w$-trees of bounded degree, J. Algorithms, 56(1):25--49, 2005], our algorithm has the advantage of being simpler and self-contained; arguably, it explains better the combinatorics of optimum-width layouts

Computing the Chromatic Number Using Graph Decompositions via Matrix Rank

Computing the smallest number $q$ such that the vertices of a given graph can be properly $q$-colored is one of the oldest and most fundamental problems in combinatorial optimization. The $q$-Coloring problem has been studied intensively using the framework of parameterized algorithmics, resulting in a very good understanding of the best-possible algorithms for several parameterizations based on the structure of the graph. While there is an abundance of work for parameterizations based on decompositions of the graph by vertex separators, almost nothing is known about parameterizations based on edge separators. We fill this gap by studying $q$-Coloring parameterized by cutwidth, and parameterized by pathwidth in bounded-degree graphs. Our research uncovers interesting new ways to exploit small edge separators. We present two algorithms for $q$-Coloring parameterized by cutwidth $cutw$: a deterministic one that runs in time $O^*(2^{\omega \cdot cutw})$, where $\omega$ is the matrix multiplication constant, and a randomized one with runtime $O^*(2^{cutw})$. In sharp contrast to earlier work, the running time is independent of $q$. The dependence on cutwidth is optimal: we prove that even 3-Coloring cannot be solved in $O^*((2-\varepsilon)^{cutw})$ time assuming the Strong Exponential Time Hypothesis (SETH). Our algorithms rely on a new rank bound for a matrix that describes compatible colorings. Combined with a simple communication protocol for evaluating a product of two polynomials, this also yields an $O^*((\lfloor d/2\rfloor+1)^{pw})$ time randomized algorithm for $q$-Coloring on graphs of pathwidth $pw$ and maximum degree $d$. Such a runtime was first obtained by Bj\"orklund, but only for graphs with few proper colorings. We also prove that this result is optimal in the sense that no $O^*((\lfloor d/2\rfloor+1-\varepsilon)^{pw})$-time algorithm exists assuming SETH.Comment: 29 pages. An extended abstract appears in the proceedings of the 26th Annual European Symposium on Algorithms, ESA 201

Tight Bounds for Connectivity Problems Parameterized by Cutwidth

In this work we start the investigation of tight complexity bounds for connectivity problems parameterized by cutwidth assuming the Strong Exponential-Time Hypothesis (SETH). Van Geffen et al. [Bas A. M. van Geffen et al., 2020] posed this question for Odd Cycle Transversal and Feedback Vertex Set. We answer it for these two and four further problems, namely Connected Vertex Cover, Connected Dominating Set, Steiner Tree, and Connected Odd Cycle Transversal. For the latter two problems it sufficed to prove lower bounds that match the running time inherited from parameterization by treewidth; for the others we provide faster algorithms than relative to treewidth and prove matching lower bounds. For upper bounds we first extend the idea of Groenland et al. [Carla Groenland et al., 2022] to solve what we call coloring-like problems. Such problems are defined by a symmetric matrix M over ?? indexed by a set of colors. The goal is to count the number (modulo some prime p) of colorings of a graph such that M has a 1-entry if indexed by the colors of the end-points of any edge. We show that this problem can be solved faster if M has small rank over ?_p. We apply this result to get our upper bounds for CVC and CDS. The upper bounds for OCT and FVS use a subdivision trick to get below the bounds that matrix rank would yield

Low-Stretch Spanning Trees of Graphs with Bounded Width

We study the problem of low-stretch spanning trees in graphs of bounded width: bandwidth, cutwidth, and treewidth. We show that any simple connected graph $G$ with a linear arrangement of bandwidth $b$ can be embedded into a distribution $\mathcal T$ of spanning trees such that the expected stretch of each edge of $G$ is $O(b^2)$. Our proof implies a linear time algorithm for sampling from $\mathcal T$. Therefore, we have a linear time algorithm that finds a spanning tree of $G$ with average stretch $O(b^2)$ with high probability. We also describe a deterministic linear-time algorithm for computing a spanning tree of $G$ with average stretch $O(b^3)$. For graphs of cutwidth $c$, we construct a spanning tree with stretch $O(c^2)$ in linear time. Finally, when $G$ has treewidth $k$ we provide a dynamic programming algorithm computing a minimum stretch spanning tree of $G$ that runs in polynomial time with respect to the number of vertices of $G$

Towards Exact Structural Thresholds for Parameterized Complexity

Parameterized complexity seeks to optimally use input structure to obtain faster algorithms for NP-hard problems. This has been most successful for graphs of low treewidth, i.e., graphs decomposable by small separators: Many problems admit fast algorithms relative to treewidth and many of them are optimal under the Strong Exponential-Time Hypothesis (SETH). Fewer such results are known for more general structure such as low clique-width (decomposition by large and dense but structured separators) and more restrictive structure such as low deletion distance to some sparse graph class. Despite these successes, such results remain "islands" within the realm of possible structure. Rather than adding more islands, we seek to determine the transitions between them, that is, we aim for structural thresholds where the complexity increases as input structure becomes more general. Going from deletion distance to treewidth, is a single deletion set to a graph with simple components enough to yield the same lower bound as for treewidth or does it take many disjoint separators? Going from treewidth to clique-width, how much more density entails the same complexity as clique-width? Conversely, what is the most restrictive structure that yields the same lower bound? For treewidth, we obtain both refined and new lower bounds that apply already to graphs with a single separator X such that G-X has treewidth at most r = ?(1), while G has treewidth |X|+?(1). We rule out algorithms running in time ?^*((r+1-?)^k) for Deletion to r-Colorable parameterized by k = |X|; this implies the same lower bound relative to treedepth and (hence) also to treewidth. It specializes to ?^*((3-?)^k) for Odd Cycle Transversal where tw(G-X) ? r = 2 is best possible. For clique-width, an extended version of the above reduction rules out time ?^*((4-?)^k), where X is allowed to be a possibly large separator consisting of k (true) twinclasses, while the treewidth of G - X remains r; this is proved also for the more general Deletion to r-Colorable and it implies the same lower bound relative to clique-width. Further results complement what is known for Vertex Cover, Dominating Set and Maximum Cut. All lower bounds are matched by existing and newly designed algorithms

Parameterization of Tensor Network Contraction

We present a conceptually clear and algorithmically useful framework for parameterizing the costs of tensor network contraction. Our framework is completely general, applying to tensor networks with arbitrary bond dimensions, open legs, and hyperedges. The fundamental objects of our framework are rooted and unrooted contraction trees, which represent classes of contraction orders. Properties of a contraction tree correspond directly and precisely to the time and space costs of tensor network contraction. The properties of rooted contraction trees give the costs of parallelized contraction algorithms. We show how contraction trees relate to existing tree-like objects in the graph theory literature, bringing to bear a wide range of graph algorithms and tools to tensor network contraction. Independent of tensor networks, we show that the edge congestion of a graph is almost equal to the branchwidth of its line graph