Tree Decomposition Of Width Research Articles

Tight Complexity Bounds for Counting Generalized Dominating Sets in Bounded-Treewidth Graphs Part II: Hardness Results

For a well-studied family of domination-type problems, in bounded-treewidth graphs, we investigate whether it is possible to find faster algorithms. For sets σ , ρ of non-negative integers, a ( σ , ρ )-set of a graph G is a set S of vertices such that | N ( u )∩ S | ∈ σ for every u ∈ S , and | N ( v )∩ S | ∈ ρ for every \(v\not\in S \) . The problem of finding a ( σ , ρ )-set (of a certain size) unifies common problems like Independent Set , Dominating Set , Independent Dominating Set , and many others. In an accompanying paper, it is proven that, for all pairs of finite or cofinite sets ( σ , ρ ), there is an algorithm that counts ( σ , ρ )-sets in time \((c_{\sigma,\rho })^{\sf tw}\cdot n^{{\rm O}(1)} \) (if a tree decomposition of width \({\sf tw} \) is given in the input). Here, c σ , ρ is a constant with an intricate dependency on σ and ρ . Despite this intricacy, we show that the algorithms in the accompanying paper are most likely optimal, i.e., for any pair ( σ , ρ ) of finite or cofinite sets where the problem is non-trivial, and any ε > 0, a \((c_{\sigma,\rho }-\varepsilon)^{\sf tw}\cdot n^{{\rm O}(1)} \) -algorithm counting the number of ( σ , ρ )-sets would violate the Counting Strong Exponential-Time Hypothesis (#SETH). For finite sets σ and ρ , our lower bounds also extend to the decision version, showing that those algorithms are optimal in this setting as well.

Space efficient algorithm for solving reachability using tree decomposition and separators

To solve reachability is to determine whether there is a path from one vertex to the other in a graph. Standard graph traversal algorithms such as DFS and BFS take linear time to solve reachability; however, their space complexity is also linear. On the other hand, Savitch's algorithm takes quasipolynomial time, although the space-bound is O(log2⁡n). In this paper, we study space-efficient algorithms for deciding reachability that runs in polynomial time.We show a polynomial-time algorithm that solves reachability in directed graphs using O(wlog⁡n) space. Our algorithm requires access to a tree decomposition of width w for the underlying undirected graph of the input. This requirement can be waived for graphs for which recursive balanced vertex separators can be computed space-efficiently.

Detecting Feedback Vertex Sets of Size k in O ⋆ (2.7 k ) Time

In the Feedback Vertex Set (FVS) problem, one is given an undirected graph G and an integer k , and one needs to determine whether there exists a set of k vertices that intersects all cycles of G (a so-called feedback vertex set). Feedback Vertex Set is one of the most central problems in parameterized complexity: It served as an excellent testbed for many important algorithmic techniques in the field such as Iterative Compression [Guo et al. (JCSS’06)], Randomized Branching [Becker et al. (J. Artif. Intell. Res’00)] and Cut&Count [Cygan et al. (FOCS’11)]. In particular, there has been a long race for the smallest dependence f(k) in run times of the type O ⋆ (f(k)) , where the O ⋆ notation omits factors polynomial in n . This race seemed to have reached a conclusion in 2011, when a randomized O ⋆ (3 k ) time algorithm based on Cut&Count was introduced. In this work, we show the contrary and give a O ⋆ (2.7 k ) time randomized algorithm. Our algorithm combines all mentioned techniques with substantial new ideas: First, we show that, given a feedback vertex set of size k of bounded average degree, a tree decomposition of width (1-Ω (1))k can be found in polynomial time. Second, we give a randomized branching strategy inspired by the one from [Becker et al. (J. Artif. Intell. Res’00)] to reduce to the aforementioned bounded average degree setting. Third, we obtain significant run time improvements by employing fast matrix multiplication.

Optimizing tree decompositions in MSO

The classic algorithm of Bodlaender and Kloks [J. Algorithms, 1996] solves the following problem in linear fixed-parameter time: given a tree decomposition of a graph of (possibly suboptimal) width k, compute an optimum-width tree decomposition of the graph. In this work, we prove that this problem can also be solved in mso in the following sense: for every positive integer k, there is an mso transduction from tree decompositions of width k to tree decompositions of optimum width. Together with our recent results [LICS 2016], this implies that for every k there exists an mso transduction which inputs a graph of treewidth k, and nondeterministically outputs its tree decomposition of optimum width. We also show that mso transductions can be implemented in linear fixed-parameter time, which enables us to derive the algorithmic result of Bodlaender and Kloks as a corollary of our main result.

Structural Parameterizations of Clique Coloring

A clique coloring of a graph is an assignment of colors to its vertices such that no maximal clique is monochromatic. We initiate the study of structural parameterizations of the Clique Coloring problem which asks whether a given graph has a clique coloring with q colors. For fixed q ge 2, we give an mathscr {O}^{star }(q^{{mathsf {tw}}})-time algorithm when the input graph is given together with one of its tree decompositions of width {mathsf {tw}} . We complement this result with a matching lower bound under the Strong Exponential Time Hypothesis. We furthermore show that (when the number of colors is unbounded) Clique Coloring is mathsf {XP} parameterized by clique-width.

Constructing tree decompositions of graphs with bounded gonality

In this paper, we give a constructive proof of the fact that the treewidth of a graph is at most its divisorial gonality. The proof gives a polynomial time algorithm to construct a tree decomposition of width at most k, when an effective divisor of degree k that reaches all vertices is given. We also give a similar result for two related notions: stable divisorial gonality and stable gonality.

On the intersection graph of the disks with diameters the sides of a convex [formula omitted]-gon

Given a convex n-gon, we can draw n disks (called side disks) where each disk has a different side of the polygon as diameter and the midpoint of the side as its center. The intersection graph of such disks is the undirected graph with vertices the n disks and two disks are adjacent if and only if they have a point in common. Such a graph was introduced by Huemer and Pérez-Lantero (Discrete Mathematics, 2020), proved to be planar and Hamiltonian. In this paper, we study further combinatorial properties of this graph. We prove that the treewidth is at most 3, by showing an O(n)-time algorithm that builds a tree decomposition of width at most 3, given the polygon as input. This implies that we can construct the intersection graph of the side disks in O(n) time. We further study the independence number, which is the maximum number of pairwise disjoint disks. The planarity condition implies that for every convex n-gon we can select at least ⌈n/4⌉ pairwise disjoint disks, and we prove that for every n≥3 there exist convex n-gons in which we cannot select more than this number. Finally, we show that our class of graphs includes all outerplanar Hamiltonian graphs except the cycle of length four, and that it is a proper subclass of the planar Hamiltonian graphs.

A (probably) optimal algorithm for Bisection on bounded-treewidth graphs

The maximum/minimum bisection problems are, given an edge-weighted graph, to find a bipartition of the vertex set into two sets whose sizes differ by at most one, such that the total weight of edges between the two sets is maximized/minimized. Although these two problems are known to be NP-hard, there is an efficient algorithm for bounded-treewidth graphs. In particular, Jansen et al. (2005) [5] gave an O(2tn3)-time algorithm when given a tree decomposition of width t of the input graph, where n is the number of vertices of the input graph. Eiben et al. (2021) [10] improved the dependency of n in the running time by giving an O(8tt5n2log⁡n)-time algorithm. Moreover, they showed that there is no O(n2−ε)-time algorithm for trees under some reasonable complexity assumption.In this paper, we show an O(2t(tn)2)-time algorithm for both problems, which is asymptotically tight to their conditional lower bound. We also show that the exponential dependency of the treewidth is asymptotically optimal under the Strong Exponential Time Hypothesis. Finally, we discuss the (in)tractability of both problems with respect to special graph classes.

Computing the Number of k-Component Spanning Forests of a Graph with Bounded Treewidth

Let G be a graph on n vertices with bounded treewidth. We use $$f_k(G)$$ to denote the number of spanning forests of G with k components. Given a tree decomposition of width at most p of G, we present an algorithm to compute $$f_k(G)$$ for $$k = 1,2, \cdots , n$$ . The running time of our algorithm is $$O((B(p+1))^3pn^3)$$ , where $$B(p+1)$$ is the $$(p+1)$$ -th Bell number.

On Algorithms Employing Treewidth for $L$-bounded Cut Problems

Given a graph $G=(V,E)$ with two distinguished vertices $s,t\in V$ and an integer parameter $L>0$, an $L$-bounded cut is a subset $F$ of edges (vertices) such that the every path between $s$ and $t$ in $G\setminus F$ has length more than $L$. The task is to find an $L$-bounded cut of minimum cardinality. Though the problem is very simple to state and has been studied since the beginning of the 70's, it is not much understood yet. The problem is known to be $\cal{NP}$-hard to approximate within a small constant factor even for $L\geq 4$ (for $L\geq 5$ for the vertex-deletion version). On the other hand, the best known approximation algorithm for general graphs has approximation ratio only $\mathcal{O}({n^{2/3}})$ in the edge case, and $\mathcal{O}({\sqrt{n}})$ in the vertex case, where $n$ denotes the number of vertices. We show that for planar graphs, it is possible to solve both the edge- and the vertex-deletion version of the problem optimally in $\mathcal{O}((L+2)^{3L}n)$ time. That is, the problem is fixed-parameter tractable (FPT) with respect to $L$ on planar graphs. Furthermore, we show that the problem remains FPT even for bounded genus graphs, a super class of planar graphs. Our second contribution deals with approximations of the vertex-deletion version of the problem. We describe an algorithm that for a given graph $G$, its tree decomposition of width $\tau$ and vertices $s$ and $t$ computes a $\tau$-approximation of the minimum $L$-bounded $s-t$ vertex cut; if the decomposition is not given, then the approximation ratio is $\mathcal{O}(\tau \sqrt{\log \tau})$. For graphs with treewidth bounded by $\mathcal{O}(n^{1/2-\epsilon})$ for any $\epsilon>0$, but not by a constant, this is the best approximation in terms of $n$ that we are aware of.

Bounding the Order of a Graph Using Its Diameter and Metric Dimension: A Study Through Tree Decompositions and VC Dimension

The metric dimension of a graph is the minimum size of a set of vertices such that each vertex is uniquely determined by the distances to the vertices of that set. Our aim is to upper-bound the order $n$ of a graph in terms of its diameter $d$ and metric dimension $k$. In general, the bound $n\leq d^k+k$ is known to hold. We prove a bound of the form $n=\mathcal{O}(kd^2)$ for trees and outerplanar graphs (for trees we determine the best possible bound and the corresponding extremal examples). More generally, for graphs having a tree decomposition of width $w$ and length $\ell$, we obtain a bound of the form $n=\mathcal{O}(kd^2(2\ell+1)^{3w+1})$. This implies in particular that $n=\mathcal{O}(kd^{\mathcal{O}(1)})$ for graphs of constant treewidth and $n=\mathcal{O}(f(k)d^2)$ for chordal graphs, where $f$ is a doubly-exponential function. Using the notion of distance-VC dimension (introduced in 2014 by Bousquet and Thomass\'e) as a tool, we prove the bounds $n\leq (dk+1)^{t-1}+1$ for $K_t$-minor-free graphs, and $n\leq (dk+1)^{d(3\cdot 2^{r}+2)}+1$ for graphs of rankwidth at most $r$.

A unified treatment of linked and lean tree-decompositions

There are many results asserting the existence of tree-decompositions of minimal width which still represent local connectivity properties of the underlying graph, perhaps the best known being Thomas' theorem that proves for every graph G the existence of a linked tree-decomposition of width tw(G). We prove a general theorem on the existence of linked and lean tree-decompositions, providing a unifying proof of many known results in the field, as well as implying some new results. In particular we prove that every matroid M admits a lean tree-decomposition of width tw(M), generalizing the result of Thomas.

Minimum size tree-decompositions

We study in this paper the problem of computing a tree-decomposition of a graph with width at most k and minimum number of bags. More precisely, we focus on the following problem: given a fixed k≥1, what is the complexity of computing a tree-decomposition of width at most k with minimum number of bags in the class of graphs with treewidth at most k? We prove that the problem is NP-complete for any fixed k≥4 and polynomial for k≤2; for k=3, we show that it is polynomial in the class of trees and 2-connected outerplanar graphs.

Approximate tree decompositions of planar graphs in linear time

Many algorithms have been developed for NP-hard problems on graphs with small treewidth k. For example, all problems that are expressible in linear extended monadic second order can be solved in linear time on graphs of bounded treewidth. It turns out that the bottleneck of many algorithms for NP-hard problems is the computation of a tree decomposition of width O(k). In particular, by the bidimensional theory, there are many linear extended monadic second order problems that can be solved on n-vertex planar graphs with treewidth k in a time linear in n and subexponential in k if a tree decomposition of width O(k) can be found in such a time.We present the first algorithm that, on n-vertex planar graphs with treewidth k, finds a tree decomposition of width O(k) in such a time. In more detail, our algorithm has a running time of O(nk2log⁡k). We show the result as a special case of a result concerning so-called weighted treewidth of weighted graphs.

On the vertex cover $$P_3$$ P 3 problem parameterized by treewidth

Consider a graph G. A subset of vertices, F, is called a vertex cover \(P_t\) (\(VCP_t\)) set if every path of order t contains at least one vertex in F. Finding a minimum \(VCP_t\) set in a graph is is NP-hard for any integer \(t\ge 2\) and is called the \(MVCP_3\) problem. In this paper, we study the parameterized algorithms for the \(MVCP_3\) problem when the underlying graph G is parameterized by the treewidth. Given an n-vertex graph together with its tree decomposition of width at most p, we present an algorithm running in time \(4^{p}\cdot n^{O(1)}\) for the \(MVCP_3\) problem. Moreover, we show that for the \(MVCP_3\) problem on planar graphs, there is a subexponential parameterized algorithm running in time \(2^{O(\sqrt{k})}\cdot n^{O(1)}\) where k is the size of the optimal solution.

Parameterized algorithms for load coloring problem

One way to state the Load Coloring Problem (LCP) is as follows. Let G=(V,E) be graph and let f:V→{red,blue} be a 2-coloring. An edge e∈E is called red (blue) if both end-vertices of e are red (blue). For a 2-coloring f, let rf′ and bf′ be the number of red and blue edges and let μf(G)=min{rf′,bf′}. Let μ(G) be the maximum of μf(G) over all 2-colorings.We introduce the parameterized problem k-LCP of deciding whether μ(G)⩾k, where k is the parameter. We prove that this problem admits a kernel with at most 7k. Ahuja et al. (2007) proved that one can find an optimal 2-coloring on trees in polynomial time. We generalize this by showing that an optimal 2-coloring on graphs with tree decomposition of width t can be found in time O⁎(2t). We also show that either G is a Yes-instance of k-LCP or the treewidth of G is at most 2k. Thus, k-LCP can be solved in time O⁎(4k).

Tree-width and large grid minors in planar graphs

Graphs and Algorithms We show that for a planar graph with no g-grid minor there exists a tree-decomposition of width at most 5g - 6. The proof is constructive and simple. The underlying algorithm for the tree-decomposition runs in O(n(2) log n) time.

Linked tree-decompositions of represented infinite matroids

We prove that a represented infinite matroid having finite tree-width w has a linked tree-decomposition of width at most 2 w. This result should be a key lemma in showing that any class of infinite matroids representable over a fixed finite field and having bounded tree-width is well-quasi-ordered under taking minors. We also show that for every finite w, a represented infinite matroid has tree-width at most w if and only if all its finite submatroids have tree-width at most w. Both proofs rely on the use of a notion of chordality for represented matroids.

Approximation Algorithms for Treewidth

This paper presents algorithms whose input is an undirected graph, and whose output is a tree decomposition of width that approximates the optimal, the treewidth of that graph. The algorithms differ in their computation time and their approximation guarantees. The first algorithm works in polynomial-time and finds a factor-O(log OPT) approximation, where OPT is the treewidth of the graph. This is the first polynomial-time algorithm that approximates the optimal by a factor that does not depend on n, the number of nodes in the input graph. As a result, we get an algorithm for finding pathwidth within a factor of O(log OPT⋅log n) from the optimal. We also present algorithms that approximate the treewidth of a graph by constant factors of 3.66, 4, and 4.5, respectively and take time that is exponential in the treewidth. These are more efficient than previously known algorithms by an exponential factor, and are of practical interest. Finding triangulations of minimum treewidth for graphs is central to many problems in computer science. Real-world problems in artificial intelligence, VLSI design and databases are efficiently solvable if we have an efficient approximation algorithm for them. Many of those applications rely on weighted graphs. We extend our results to weighted graphs and weighted treewidth, showing similar approximation results for this more general notion. We report on experimental results confirming the effectiveness of our algorithms for large graphs associated with real-world problems.

Improved Approximation Algorithms for Minimum Weight Vertex Separators

We develop the algorithmic theory of vertex separators and its relation to the embeddings of certain metric spaces. Unlike in the edge case, we show that embeddings into $L_1$ (and even Euclidean embeddings) are insufficient but that the additional structure provided by many embedding theorems does suffice for our purposes. We obtain an $O(\sqrt{\log n})$ approximation for minimum ratio vertex cuts in general graphs, based on a new semidefinite relaxation of the problem, and a tight analysis of the integrality gap which is shown to be $\Theta(\sqrt{\log n})$. We also prove an optimal $O(\log k)$-approximate max-flow/min-vertex-cut theorem for arbitrary vertex-capacitated multicommodity flow instances on k terminals. For uniform instances on any excluded-minor family of graphs, we improve this to $O(1)$, and this yields a constant-factor approximation for minimum ratio vertex cuts in such graphs. Previously, this was known only for planar graphs, and for general excluded-minor families the best known ratio was $O(\log n)$. These results have a number of applications. We exhibit an $O(\sqrt{\log n})$ pseudoapproximation for finding balanced vertex separators in general graphs. In fact, we achieve an approximation ratio of $O(\sqrt{\log {opt}})$, where ${opt}$ is the size of an optimal separator, improving over the previous best bound of $O(\log {opt})$. Likewise, we obtain improved approximation ratios for treewidth: in any graph of treewidth k, we show how to find a tree decomposition of width at most $O(k \sqrt{\log k})$, whereas previous algorithms yielded $O(k \log k)$. For graphs excluding a fixed graph as a minor (which includes, e.g., bounded genus graphs), we give a constant-factor approximation for the treewidth. This in turn can be used to obtain polynomial-time approximation schemes for several problems in such graphs.