We revisit the issue of low-distortion embedding of metric spaces into the line, and more generally, into the shortest path metric of trees, from the parameterized complexity perspective.Let $M=M(G)$ be the shortest path metric of an edge weighted graph $G=(V,E)$ on $n$ vertices. We describe algorithms for the problem of finding a low distortion non-contracting embedding of $M$ into line and tree metrics.
We give an $O(nd^4(2d+1)^{2d})$ time algorithm that for an unweighted graph metric $M$ and integer $d$ either constructs an embedding of $M$ into the line with distortion at most $d$, or concludes that no such embedding exists. We find the result surprising, because the considered problem bears a strong resemblance to the notoriously hard Bandwidth Minimization problem which does not admit any FPT algorithm unless an unlikely collapse of parameterized complexity classes occurs.
We show that our algorithm can also be applied to construct small distortion embeddings of weighted graph metrics. The running time of our algorithm is $O(n(dW)^4(2d+1)^{2dW})$ where $W$ is the largest edge weight of the input graph. We also show that deciding whether a weighted graph metric $M(G)$ with maximum weight $W < |V(G)|$ can be embedded into the line with distortion at most $d$ is NP-Complete for every fixed rational $d \geq 2$. This rules out any possibility of an algorithm with running time $O((nW)^{h(d)})$ where $h$ is a function of $d$ alone.
We generalize the result on embedding into the line by proving that for any tree $T$ with maximum degree $Δ$, embedding of $M$ into a shortest path metric of $T$ is FPT, parameterized by $(Δ,d)$.
The Maximum Agreement Forest problem has been extensively studied in phylogenetics. Most previous work is on two binary phylogenetic trees. In this paper, we study a generalized version of the problem: the Maximum Agreement Forest problem on multiple rooted multifurcating phylogenetic trees, from the perspective of fixed-parameter algorithms. By taking advantage of a new branch-and-bound strategy, two parameterized algorithms, with running times $O(2.42^k m^3 n^4)$ and $O(2.74^k m^3 n^5)$, respectively, are presented for the hard version and the soft version of the problem, which correspond to…
We study the Directed Feedback Vertex Set problem parameterized by the treewidth of the input graph. We prove that unless the Exponential Time Hypothesis fails, the problem cannot be solved in time $2^{o(t\log t)}\cdot n^{\mathcal{O}(1)}$ on general directed graphs, where $t$ is the treewidth of the underlying undirected graph. This is matched by a dynamic programming algorithm with running time $2^{\mathcal{O}(t\log t)}\cdot n^{\mathcal{O}(1)}$. On the other hand, we show that if the input digraph is planar, then the running time can be improved to $2^{\mathcal{O}(t)}\cdot n^{\mathcal{O}(1)}$…
The notion of $\mathcal{H}$-treewidth, where $\mathcal{H}$ is a hereditary graph class, was recently introduced as a generalization of the treewidth of an undirected graph. Roughly speaking, a graph of $\mathcal{H}$-treewidth at most $k$ can be decomposed into (arbitrarily large) $\mathcal{H}$-subgraphs which interact only through vertex sets of size $O(k)$ which can be organized in a tree-like fashion. $\mathcal{H}$-treewidth can be used as a hybrid parameterization to develop fixed-parameter tractable algorithms for $\mathcal{H}$-deletion problems, which ask to find a minimum vertex set whos…
We study the minimum diameter spanning tree problem under the reload cost model (DIAMETER-TREE for short) introduced by Wirth and Steffan (2001). In this problem, given an undirected edge-colored graph $G$, reload costs on a path arise at a node where the path uses consecutive edges of different colors. The objective is to find a spanning tree of $G$ of minimum diameter with respect to the reload costs. We initiate a systematic study of the parameterized complexity of the DIAMETER-TREE problem by considering the following parameters: the cost of a solution, and the treewidth and the maximum de…
The bandwidth of a $n$-vertex graph $G$ is the smallest integer $b$ such that there exists a bijective function $f : V(G) \rightarrow \{1,...,n\}$, called a layout of $G$, such that for every edge $uv \in E(G)$, $|f(u) - f(v)| \leq b$. In the {\sc Bandwidth} problem we are given as input a graph $G$ and integer $b$, and asked whether the bandwidth of $G$ is at most $b$. We present two results concerning the parameterized complexity of the {\sc Bandwidth} problem on trees. First we show that an algorithm for {\sc Bandwidth} with running time $f(b)n^{o(b)}$ would violate the Exponential Time Hyp…
A resolving set $S$ of a graph $G$ is a subset of its vertices such that no two vertices of $G$ have the same distance vector to $S$. The Metric Dimension problem asks for a resolving set of minimum size, and in its decision form, a resolving set of size at most some specified integer. This problem is NP-complete, and remains so in very restricted classes of graphs. It is also W[2]-complete with respect to the size of the solution. Metric Dimension has proven elusive on graphs of bounded treewidth. On the algorithmic side, a polytime algorithm is known for trees, and even for outerplanar graph…
After the number of vertices, Vertex Cover is the largest of the classical graph parameters and has more and more frequently been used as a separate parameter in parameterized problems, including problems that are not directly related to the Vertex Cover. Here we consider the TREEWIDTH and PATHWIDTH problems parameterized by k, the size of a minimum vertex cover of the input graph. We show that the PATHWIDTH and TREEWIDTH can be computed in O*(3^k) time. This complements recent polynomial kernel results for TREEWIDTH and PATHWIDTH parameterized by the Vertex Cover.
In an edge-colored graph, the cost incurred at a vertex on a path when two incident edges with different colors are traversed is called reload or changeover cost. The "Minimum Changeover Cost Arborescence" (MINCCA) problem consists in finding an arborescence with a given root vertex such that the total changeover cost of the internal vertices is minimized. It has been recently proved by Gözüpek et al. [TCS 2016] that the problem is FPT when parameterized by the treewidth and the maximum degree of the input graph. In this article we present the following results for the MINCCA problem: - the pr…
The Cut & Count technique and the rank-based approach have lead to single-exponential FPT algorithms parameterized by treewidth, that is, running in time $2^{O(tw)}n^{O(1)}$, for Feedback Vertex Set and connected versions of the classical graph problems (such as Vertex Cover and Dominating Set). We show that Subset Feedback Vertex Set, Subset Odd Cycle Transversal, Restricted Edge-Subset Feedback Edge Set, Node Multiway Cut, and Multiway Cut are unlikely to have such running times. More precisely, we match algorithms running in time $2^{O(tw \log tw)}n^{O(1)}$ with tight lower bounds under the…
Parameterized complexity seeks to use input structure to obtain faster algorithms for NP-hard problems. This has been most successful for graphs of low treewidth: Many problems admit fast algorithms relative to treewidth and many of them are optimal under SETH. Fewer such results are known for more general structure such as low clique-width and more restrictive structure such as low deletion distance to a 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 betwe…
Technische Universität Berlin（柏林工业大学）；Université Paris-Dauphine, PSL Research University, CNRS, UMR 7243, LAMSADE（巴黎第九大学，PSL研究大学，法国国家科学研究中心，UMR 7243，LAMSADE）；Orange Research（法国电信研究院）
Many hard graph problems can be solved efficiently when restricted to graphs of bounded treewidth, and more generally to graphs of bounded clique-width. But there is a price to be paid for this generality, exemplified by the four problems MaxCut, Graph Coloring, Hamiltonian Cycle and Edge Dominating Set that are all FPT parameterized by treewidth but none of which can be FPT parameterized by clique-width unless FPT = W[1], as shown by Fomin et al [7, 8]. We therefore seek a structural graph parameter that shares some of the generality of clique-width without paying this price. Based on splits,…
This paper makes two main contributions: The first is the construction of a near-minimum spanning tree with constant average distortion. The second is a general equivalence theorem relating two refined notions of distortion: scaling distortion and prioritized distortion. Minimum Spanning Trees of weighted graphs are fundamental objects in numerous applications. In particular in distributed networks, the minimum spanning tree of the network is often used to route messages between network nodes. Unfortunately, while being most efficient in the total cost of connecting all nodes, minimum spanning…
The metric dimension has been introduced independently by Harary, Melter and Slater in 1975 to identify vertices of a graph G using its distances to a subset of vertices of G. A resolving set X of a graph G is a subset of vertices such that, for every pair (u,v) of vertices of G, there is a vertex x in X such that the distance between x and u and the distance between x and v are distinct. The metric dimension of the graph is the minimum size of a resolving set. Computing the metric dimension of a graph is NP-hard even on split graphs and interval graphs. Bonnet and Purohit proved that the metr…
Many computational problems admit fast algorithms on special inputs, however, the required properties might be quite restrictive. E.g., many graph problems can be solved much faster on interval or cographs, or on graphs of small modular-width or small tree-width, than on general graphs. One challenge is to attain the greatest generality of such results, i.e., being applicable to less restrictive input classes, without losing much in terms of running time. Building on the use of algebraic expressions we present a clean and robust way of combining such homogeneous structure into more complex het…
The notion of treewidth plays an important role in theoretical and practical studies of graph problems. It has been recognized that, especially in practical environments, when computing the treewidth of a graph it is invaluable to first apply an array of preprocessing rules that simplify and shrink it. This work seeks to prove rigorous performance guarantees for such preprocessing rules, both known and new ones, by studying them in the framework of kernelization from parameterized complexity. It is known that the NP-complete problem of determining whether a given graph G has treewidth at most …
We show that the Minimal Length-Bounded L-But problem can be computed in linear time with respect to L and the tree-width of the input graph as parameters. In this problem the task is to find a set of edges of a graph such that after removal of this set, the shortest path between two prescribed vertices is at least L long. We derive an FPT algorithm for a more general multi-commodity length bounded cut problem when parameterized by the number of terminals also. For the former problem we show a W[1]-hardness result when the parameterization is done by the path-width only (instead of the tree-wi…
We extend Babai's quasipolynomial-time graph isomorphism test (STOC 2016) and develop a quasipolynomial-time algorithm for the multiple-coset isomorphism problem. The algorithm for the multiple-coset isomorphism problem allows to exploit graph decompositions of the given input graphs within Babai's group-theoretic framework. We use it to develop a graph isomorphism test that runs in time $n^{\operatorname{polylog}(k)}$ where $n$ is the number of vertices and $k$ is the minimum treewidth of the given graphs and $\operatorname{polylog}(k)$ is some polynomial in $\operatorname{log}(k)$. Our resul…
We study connectivity problems from a fine-grained parameterized perspective. Cygan et al. (TALG 2022) obtained algorithms with single-exponential running time $α^{tw} n^{O(1)}$ for connectivity problems parameterized by treewidth ($tw$) by introducing the cut-and-count-technique, which reduces connectivity problems to locally checkable counting problems. In addition, the bases $α$ were proven to be optimal assuming the Strong Exponential-Time Hypothesis (SETH). As only sparse graphs may admit small treewidth, these results do not apply to graphs with dense structure. A well-known tool to capt…
The Edge Multicut problem is a classical cut problem where given an undirected graph $G$, a set of pairs of vertices $\mathcal{P}$, and a budget $k$, the goal is to determine if there is a set $S$ of at most $k$ edges such that for each $(s,t) \in \mathcal{P}$, $G-S$ has no path from $s$ to $t$. Edge Multicut has been relatively recently shown to be fixed-parameter tractable (FPT), parameterized by $k$, by Marx and Razgon [SICOMP 2014], and independently by Bousquet et al. [SICOMP 2018]. In the weighted version of the problem, called Weighted Edge Multicut one is additionally given a weight fu…
In the Mixed Chinese Postman Problem (MCPP), given a weighted mixed graph $G$ ($G$ may have both edges and arcs), our aim is to find a minimum weight closed walk traversing each edge and arc at least once. The MCPP parameterized by the number of edges in $G$ or the number of arcs in $G$ is fixed-parameter tractable as proved by van Bevern {\em et al.} (in press) and Gutin, Jones and Sheng (ESA 2014), respectively. In this paper, we consider the unweighted version of MCPP. Solving an open question of van Bevern {\em et al.} (in press), we show that somewhat unexpectedly MCPP parameterized by th…
We present a general setting for structure-sequence comparison in a large class of RNA structures that unifies and generalizes a number of recent works on specific families on structures. Our approach is based on tree decomposition of structures and gives rises to a general parameterized algorithm, where the exponential part of the complexity depends on the family of structures. For each of the previously studied families, our algorithm has the same complexity as the specific algorithm that had been given before.
The recently introduced graph parameter tree-cut width plays a similar role with respect to immersions as the graph parameter treewidth plays with respect to minors. In this paper, we provide the first algorithmic applications of tree-cut width to hard combinatorial problems. Tree-cut width is known to be lower-bounded by a function of treewidth, but it can be much larger and hence has the potential to facilitate the efficient solution of problems that are not known to be fixed-parameter tractable (FPT) when parameterized by treewidth. We introduce the notion of nice tree-cut decompositions an…
A set $S$ of vertices of a graph is a defensive alliance if, for each element of $S$, the majority of its neighbours is in $S$. We consider the notion of local minimality in this paper. We are interested in locally minimal defensive alliance of maximum size. We also look at connected version of defensive alliance. This problem is known to be NP-hard but its parameterized complexity remains open until now. We enhance our understanding of the problem from the viewpoint of parameterized complexity. The main results of the paper are the following: (1) Locally Minimal Defensive Alliance is NP-compl…
In the Mixed Chinese Postman Problem (MCPP), given an edge-weighted mixed graph $G$ ($G$ may have both edges and arcs), our aim is to find a minimum weight closed walk traversing each edge and arc at least once. The MCPP parameterized by the number of edges was known to be fixed-parameter tractable using a simple argument. Solving an open question of van Bevern et al., we prove that the MCPP parameterized by the number of arcs is also fixed-parameter tractable. Our proof is more involved and, in particular, uses a well-known result of Marx, O'Sullivan and Razgon (2013) on the treewidth of tors…
We obtain new parameterized algorithms for the classical problem of determining whether a directed acyclic graph admits an upward planar drawing. Our results include a new fixed-parameter algorithm parameterized by the number of sources, an XP-algorithm parameterized by treewidth, and a fixed-parameter algorithm parameterized by treedepth. All three algorithms are obtained using a novel framework for the problem that combines SPQR tree-decompositions with parameterized techniques. Our approach unifies and pushes beyond previous tractability results for the problem on series-parallel digraphs, …
TU Delft（代尔夫特理工大学）；Middlesex University（密德萨斯大学）；LIGM, Université Gustave Eiffel（拉格朗日计算与建模实验室，古斯塔夫·埃菲尔大学）；Centre National de Recherche Scientifique（法国国家科学研究中心）
In this paper we show that a CNF cannot be compiled into an Ordered Binary Decision Diagram (OBDD) of fixed-parameter size parameterized by the primal graph treewidth of the CNF. Thus we provide a parameterized separation between OBDDs and Sentential Decision Diagrams (SDDs) for which such fixed-parameter compilation is possible. In fact, we demonstrate that the proposed lower bound also yields a classical (non-parameterized) separation of OBDDs and SDDs. We also show that the best existing parameterized upper bound for OBDDs in fact holds for incidence graph treewidth parameterization.
A breakthrough result of Cygan et al. (FOCS 2011) showed that connectivity problems parameterized by treewidth can be solved much faster than the previously best known time $\mathcal{O}^*(2^{\mathcal{O}(tw \log(tw))})$. Using their inspired Cut\&Count technique, they obtained $\mathcal{O}^*(α^{tw})$ time algorithms for many such problems. Moreover, they proved these running times to be optimal assuming the Strong Exponential-Time Hypothesis. Unfortunately, like other dynamic programming algorithms on tree decompositions, these algorithms also require exponential space, and this is widely belie…
We study the parameterized complexity of the directed variant of the classical {\sc Steiner Tree} problem on various classes of directed sparse graphs. While the parameterized complexity of {\sc Steiner Tree} parameterized by the number of terminals is well understood, not much is known about the parameterization by the number of non-terminals in the solution tree. All that is known for this parameterization is that both the directed and the undirected versions are W[2]-hard on general graphs, and hence unlikely to be fixed parameter tractable FPT. The undirected {\sc Steiner Tree} problem bec…
The Secluded Path problem models a situation where a sensitive information has to be transmitted between a pair of nodes along a path in a network. The measure of the quality of a selected path is its exposure, which is the total weight of vertices in its closed neighborhood. In order to minimize the risk of intercepting the information, we are interested in selecting a secluded path, i.e. a path with a small exposure. Similarly, the Secluded Steiner Tree problem is to find a tree in a graph connecting a given set of terminals such that the exposure of the tree is minimized. The problems were …
We investigate the complexity of several fundamental polynomial-time solvable problems on graphs and on matrices, when the given instance has low treewidth; in the case of matrices, we consider the treewidth of the graph formed by non-zero entries. In each of the considered cases, the best known algorithms working on general graphs run in polynomial time, however the exponent of the polynomial is large. Therefore, our main goal is to construct algorithms with running time of the form $\textrm{poly}(k)\cdot n$ or $\textrm{poly}(k)\cdot n\log n$, where $k$ is the width of the tree decomposition …
One way to state the Load Coloring Problem (LCP) is as follows. Let $G=(V,E)$ be graph and let $f:V\rightarrow \{{\rm red}, {\rm blue}\}$ be a 2-coloring. An edge $e\in E$ is called red (blue) if both end-vertices of $e$ are red (blue). For a 2-coloring $f$, let $r'_f$ and $b'_f$ be the number of red and blue edges and let $μ_f(G)=\min\{r'_f,b'_f\}$. Let $μ(G)$ be the maximum of $μ_f(G)$ over all 2-colorings. We introduce the parameterized problem $k$-LCP of deciding whether $μ(G)\ge k$, where $k$ is the parameter. We prove that this problem admits a kernel with at most $7k$. Ahuja et al. (200…
Combining the techniques of approximation algorithms and parameterized complexity has long been considered a promising research area, but relatively few results are currently known. In this paper we study the parameterized approximability of a number of problems which are known to be hard to solve exactly when parameterized by treewidth or clique-width. Our main contribution is to present a natural randomized rounding technique that extends well-known ideas and can be used for both of these widths. Applying this very generic technique we obtain approximation schemes for a number of problems, e…
A defensive alliance in an undirected graph $G=(V,E)$ is a non-empty set of vertices $S$ satisfying the condition that every vertex $v\in S$ has at least as many neighbours (including itself) in $S$ as it has in $V\setminus S$. We consider the notion of global minimality in this paper. We are interested in globally minimal defensive alliance of maximum size. This problem is known to be NP-hard but its parameterized complexity remains open until now. We enhance our understanding of the problem from the viewpoint of parameterized complexity by showing that the Globally Minimal Defensive Alliance…
For a family of graphs $\cal F$, the $\mathcal{F}$-Contraction problem takes as an input a graph $G$ and an integer $k$, and the goal is to decide if there exists $S \subseteq E(G)$ of size at most $k$ such that $G/S$ belongs to $\cal F$. Here, $G/S$ is the graph obtained from $G$ by contracting all the edges in $S$. Heggernes et al.~[Algorithmica (2014)] were the first to study edge contraction problems in the realm of Parameterized Complexity. They studied $\cal F$-Contraction when $\cal F$ is a simple family of graphs such as trees and paths. In this paper, we study the $\mathcal{F}$-Contra…
Parameterized algorithms are a very useful tool for dealing with NP-hard problems on graphs. Yet, to properly utilize parameterized algorithms it is necessary to choose the right parameter based on the type of problem and properties of the target graph class. Tree-width is an example of a very successful graph parameter, however it cannot be used on dense graph classes and there also exist problems which are hard even on graphs of bounded tree-width. Such problems can be tackled by using vertex cover as a parameter, however this places severe restrictions on admissible graph classes. Michael L…
Many fixed-parameter tractable algorithms using a bounded search tree have been repeatedly improved, often by describing a larger number of branching rules involving an increasingly complex case analysis. We introduce a novel and general search strategy that branches on the forbidden subgraphs of a graph class relaxation. By using the class of $P_4$-sparse graphs as the relaxed graph class, we obtain efficient bounded search tree algorithms for several parameterized deletion problems. We give the first non-trivial bounded search tree algorithms for the cograph edge-deletion problem and the tri…
We give two algorithms computing representative families of linear and uniform matroids and demonstrate how to use representative families for designing single-exponential parameterized and exact exponential time algorithms. The applications of our approach include - LONGEST DIRECTED CYCLE - MINIMUM EQUIVALENT GRAPH (MEG) - Algorithms on graphs of bounded treewidth -k-PATH, k-TREE, and more generally, k-SUBGRAPH ISOMORPHISM, where the k-vertex pattern graph is of constant treewidth.
For a given graph $G=(V,\, E)$ with a terminal set $S$ and a selected root $r\in S$, a positive integer cost and a delay on every edge and a delay constraint $D\in Z^{+}$, the shallow-light Steiner tree (\emph{SLST}) problem is to compute a minimum cost tree spanning the terminals of $S$, in which the delay between root and every vertex is restrained by $D$. This problem is NP-hard and very hard to approximate. According to known inapproximability results, this problem admits no approximation with ratio better than factor $(1,\, O(\log^{2}n))$ unless $NP\subseteq DTIME(n^{\log\log n})$ \cite{k…
We study the parameterized complexity of the $s$-Club Cluster Edge Deletion problem: Given a graph $G$ and two integers $s \ge 2$ and $k \ge 1$, is it possible to remove at most $k$ edges from $G$ such that each connected component of the resulting graph has diameter at most $s$? This problem is known to be NP-hard already when $s = 2$. We prove that it admits a fixed-parameter tractable algorithm when parameterized by $s$ and the treewidth of the input graph.
We show that the eccentricities, diameter, radius, and Wiener index of an undirected $n$-vertex graph with nonnegative edge lengths can be computed in time $O(n\cdot \binom{k+\lceil\log n\rceil}{k} \cdot 2^k k^2 \log n)$, where $k$ is the treewidth of the graph. For every $ε>0$, this bound is $n^{1+ε}\exp O(k)$, which matches a hardness result of Abboud, Vassilevska Williams, and Wang (SODA 2015) and closes an open problem in the multivariate analysis of polynomial-time computation. To this end, we show that the analysis of an algorithm of Cabello and Knauer (Comp. Geom., 2009) in the regime o…
The Maximum Agreement Forest (Maf) problem is a well-studied problem in evolutionary biology, which asks for a largest common subforest of a given collection of phylogenetic trees with identical leaf label-set. However, the previous work about the Maf problem are mainly on two binary phylogenetic trees or two general (i.e., binary and non-binary) phylogenetic trees. In this paper, we study the more general version of the problem: the Maf problem on multiple general phylogenetic trees. We present a parameterized algorithm of running time $O(3^k n^2m)$ and a 3-approximation algorithm for the Maf…
A commonly studied means of parameterizing graph problems is the deletion distance from triviality (Guo et al. 2004), which counts vertices that need to be deleted from a graph to place it in some class for which efficient algorithms are known. In the context of graph isomorphism, we define triviality to mean a graph with maximum degree bounded by a constant, as such graph classes admit polynomial-time isomorphism tests. We generalise deletion distance to a measure we call elimination distance to triviality, based on elimination trees or tree-depth decompositions. We establish that graph canon…
We consider the problem of finding a 1-planar drawing for a general graph, where a 1-planar drawing is a drawing in which each edge participates in at most one crossing. Since this problem is known to be NP-hard we investigate the parameterized complexity of the problem with respect to the vertex cover number, tree-depth, and cyclomatic number. For these parameters we construct fixed-parameter tractable algorithms. However, the problem remains NP-complete for graphs of bounded bandwidth, pathwidth, or treewidth.
Average-Tree Phylogenetic Diversity Parameterized by Scanwidth and Invisibility
Delft University of Technology（代尔夫特理工大学）；Middlesex University（米德尔塞克斯大学）；ISEM, Université de Montpellier（蒙彼利埃大学生物演化与多样性研究所）；LIGM, CNRS, Univsersité Gustave Eiffel（巴黎-塞纳河东高等学院计算机科学与数字科学实验室）；Univsersité Gustave Eiffel（古斯塔夫·艾菲尔大学）
