In the Disjoint Paths problem, the input consists of an $n$-vertex graph $G$ and a collection of $k$ vertex pairs, $\{(s_i,t_i)\}_{i=1}^k$, and the objective is to determine whether there exists a collection $\{P_i\}_{i=1}^k$ of $k$ pairwise vertex-disjoint paths in $G$ where the end-vertices of $P_i$ are $s_i$ and $t_i$. This problem was shown to admit an $f(k)n^3$-time algorithm by Robertson and Seymour (Graph Minors XIII, The Disjoint Paths Problem, JCTB). In modern terminology, this means that Disjoint Paths is fixed parameter tractable (FPT) with respect to $k$. Remarkably, the above algorithm for Disjoint Paths is a cornerstone of the entire Graph Minors Theory, and conceptually vital to the $g(k)n^3$-time algorithm for Minor Testing (given two undirected graphs, $G$ and $H$ on $n$ and $k$ vertices, respectively, determine whether $G$ contains $H$ as a minor).
In this semi-survey, we will first give an exposition of the Graph Minors Theory with emphasis on efficiency from the viewpoint of Parameterized Complexity. Secondly, we will review the state of the art with respect to the Disjoint Paths and Planar Disjoint Paths problems. Lastly, we will discuss the main ideas behind a new algorithm that combines treewidth reduction and an algebraic approach to solve Planar Disjoint Paths in time $2^{k^{O(1)}}n^{O(1)}$ (for undirected graphs).
Given a directed graph $D$ on $n$ vertices and a positive integer $k$, the Arc-Disjoint Cycle Packing problem is to determine whether $D$ has $k$ arc-disjoint cycles. This problem is known to be W[1]-hard in general directed graphs. In this paper, we initiate a systematic study on the parameterized complexity of the problem restricted to tournaments. We show that the problem is fixed-parameter tractable and admits a polynomial kernel when parameterized by the solution size $k$. In particular, we show that it can be solved in $2^{\mathcal{O}(k \log k)} n^{\mathcal{O}(1)}$ time and has a kernel …
In this paper, we study the \textsf{Planar Disjoint Paths} problem: Given an undirected planar graph $G$ with $n$ vertices and a set $T$ of $k$ pairs $(s_i,t_i)_{i=1}^k$ of vertices, the goal is to find a set $\mathcal P$ of $k$ pairwise vertex-disjoint paths connecting $s_i$ and $t_i$ for all indices $i\in\{1,\ldots,k\}$. We present a $2^{O(k^2)}n$-time algorithm for the \textsf{Planar Disjoint Paths} problem. This improves the two previously best-known algorithms: $2^{2^{O(k)}}n$-time algorithm [Discrete Applied Mathematics 1995] and $2^{O(k^2)}n^6$-time algorithm [STOC 2020].
In the Disjoint Paths problem, the input is an undirected graph $G$ on $n$ vertices and a set of $k$ vertex pairs, $\{s_i,t_i\}_{i=1}^k$, and the task is to find $k$ pairwise vertex-disjoint paths connecting $s_i$ to $t_i$. The problem was shown to have an $f(k)n^3$ algorithm by Robertson and Seymour. In modern terminology, this means that Disjoint Paths is fixed parameter tractable (FPT), parameterized by the number of vertex pairs. This algorithm is the cornerstone of the entire graph minor theory, and a vital ingredient in the $g(k)n^3$ algorithm for Minor Testing (given two undirected grap…
For the well-known Survivable Network Design Problem (SNDP) we are given an undirected graph $G$ with edge costs, a set $R$ of terminal vertices, and an integer demand $d_{s,t}$ for every terminal pair $s,t\in R$. The task is to compute a subgraph $H$ of $G$ of minimum cost, such that there are at least $d_{s,t}$ disjoint paths between $s$ and $t$ in $H$. If the paths are required to be edge-disjoint we obtain the edge-connectivity variant (EC-SNDP), while internally vertex-disjoint paths result in the vertex-connectivity variant (VC-SNDP). Another important case is the element-connectivity va…
On the Parameterized Tractability of Packing Vertex-Disjoint A-Paths with Length Constraints
Tata Institute of Fundamental Research（塔塔基础研究所）；National Institute of Science, Education and Research（国家科学教育与研究学院）；Indraprastha Institute of Information Technology Delhi（德里因德拉普拉斯塔信息技术学院）
This paper revisits the classical Edge Disjoint Paths (EDP) problem, where one is given an undirected graph $G$ and a set of terminal pairs $P$ and asks whether $G$ contains a set of pairwise edge-disjoint paths connecting every terminal pair in $P$. Our aim is to identify structural properties (parameters) of graphs which allow the efficient solution of EDP without restricting the placement of terminals in $P$ in any way. In this setting, EDP is known to remain NP-hard even on extremely restricted graph classes, such as graphs with a vertex cover of size $3$. We present three results which us…
A directed graph $G$ is called a pumpkin if $G$ is a union of induced paths with a common start vertex $s$ and a common end vertex $t$, and the internal vertices of every two paths are disjoint. We give an algorithm that given a directed graph $G$ and an integer $k$, decides whether a pumpkin can be obtained from $G$ by deleting at most $k$ vertices. The algorithm runs in $O^*(2^k)$ time.
We study Subgraph Isomorphism on graph classes defined by a fixed forbidden graph. Although there are several ways for forbidding a graph, we observe that it is reasonable to focus on the minor relation since other well-known relations lead to either trivial or equivalent problems. When the forbidden minor is connected, we present a near dichotomy of the complexity of Subgraph Isomorphism with respect to the forbidden minor, where the only unsettled case is $P_{5}$, the path of five vertices. We then also consider the general case of possibly disconnected forbidden minors. We show fixed-parame…
(see paper for full abstract) We show that the Edge-Disjoint Paths problem is W[1]-hard parameterized by the number $k$ of terminal pairs, even when the input graph is a planar directed acyclic graph (DAG). This answers a question of Slivkins (ESA '03, SIDMA '10). Moreover, under the Exponential Time Hypothesis (ETH), we show that there is no $f(k)\cdot n^{o(k)}$ algorithm for Edge-Disjoint Paths on planar DAGs, where $k$ is the number of terminal pairs, $n$ is the number of vertices and $f$ is any computable function. Our hardness holds even if both the maximum in-degree and maximum out-degre…
We study an "above guarantee" version of the {\sc Longest Path} problem in directed graphs: We are given a graph $G$, two vertices $s$ and $t$ of $G$, and a non-negative integer $k$, and the objective is to determine whether $G$ contains a path of length at least $dist_G(s,t) +k$ where $dist_G(s,t)$ is the length of a shortest path from $s$ to $t$ in $G$ (assuming that one exists). We show that the problem is fixed parameter tractable (FPT) parameterized by $k$ in the class of graphs where {\sc $2$-Disjoint Paths} problem is polynomial time solvable.
Let ${\cal G}$ be a minor-closed graph class. We say that a graph $G$ is a $k$-apex of ${\cal G}$ if $G$ contains a set $S$ of at most $k$ vertices such that $G\setminus S$ belongs to ${\cal G}$. We denote by ${\cal A}_k ({\cal G})$ the set of all graphs that are $k$-apices of ${\cal G}.$ In the first paper of this series we obtained upper bounds on the size of the graphs in the minor-obstruction set of ${\cal A}_k ({\cal G})$, i.e., the minor-minimal set of graphs not belonging to ${\cal A}_k ({\cal G}).$ In this article we provide an algorithm that, given a graph $G$ on $n$ vertices, runs in…
Given a graph $G = (V,E)$, $A \subseteq V$, and integers $k$ and $\ell$, the \textsc{$(A,\ell)$-Path Packing} problem asks to find $k$ vertex-disjoint paths of length $\ell$ that have endpoints in $A$ and internal points in $V \setminus A$. We study the parameterized complexity of this problem with parameters $|A|$, $\ell$, $k$, treewidth, pathwidth, and their combinations. We present sharp complexity contrasts with respect to these parameters. Among other results, we show that the problem is polynomial-time solvable when $\ell \le 3$, while it is NP-complete for constant $\ell \ge 4$. We also…
The Firefighting problem is defined as follows. At time $t=0$, a fire breaks out at a vertex of a graph. At each time step $t \geq 0$, a firefighter permanently defends (protects) an unburned vertex, and the fire then spread to all undefended neighbors from the vertices on fire. This process stops when the fire cannot spread anymore. The goal is to find a sequence of vertices for the firefighter that maximizes the number of saved (non burned) vertices. The Firefighting problem turns out to be NP-hard even when restricted to bipartite graphs or trees of maximum degree three. We study the parame…
On Structural Parameterizations of the Edge Disjoint Paths Problem
在本文中，我们重新审视经典的边缘不相交路径（EDP）的问题，其中一个是一个无向图G和一组终端对P，并要求G是否包含一组成对的边缘不相交路径连接每个终端对P.我们的重点在于结构参数化的问题，允许有效的（多项式时间或FPT）算法。作为我们的第一个结果，我们回答了Fleszar，Mnich和Spoerhase（2016）中提出的一个开放性问题，通过证明如果输入图具有大小为1的反馈顶点集，则可以在多项式时间内解决该问题。我们还表明，EDP参数的树宽和最大程度的输入图是固定参数听话。已经开发了两种新的算法EDP使用结构上的限制输入图，然后我们把我们的注意力转向增强图，即，在每个端子对之间添加一条边之后从输入图获得的图。在对输入图的扩展中，即使树宽为2，EDP也是NP难的，Zhou et al.（2000）的结果表明，如果扩展图具有恒定的树宽，则EDP可以在非均匀多项式时间内求解;我们注意到，从那时起，这个结果对FPT算法的可能改进仍然是开放的。我们表明，这是非常不可能的，通过建立W[1]-硬度的问题参数化的树宽（甚至反馈顶点集）的增强图。最后，我们开发了一个FPT算法的EDP利用一个新的结构参数的增强图。
We revisit classic string problems considered in the area of parameterized complexity, and study them through the lens of dynamic data structures. That is, instead of asking for a static algorithm that solves the given instance efficiently, our goal is to design a data structure that efficiently maintains a solution, or reports a lack thereof, upon updates in the instance. We first consider the Closest String problem, for which we design randomized dynamic data structures with amortized update times $d^{\mathcal{O}(d)}$ and $|Σ|^{\mathcal{O}(d)}$, respectively, where $Σ$ is the alphabet and $d…
We extend the notion of lossy kernelization, introduced by Lokshtanov et al. [STOC 2017], to approximate Turing kernelization. An $α$-approximate Turing kernel for a parameterized optimization problem is a polynomial-time algorithm that, when given access to an oracle that outputs $c$-approximate solutions in $O(1)$ time, obtains an $(α\cdot c)$-approximate solution to the considered problem, using calls to the oracle of size at most $f(k)$ for some function $f$ that only depends on the parameter. Using this definition, we show that Independent Set parameterized by treewidth $\ell$ has a $(1+\…
Technische Universität Berlin（柏林工业大学）；Discrete Mathematics Group, Institute for Basic Science (IBS)（基础科学研究所离散数学组）；LIRMM, Univ Montpellier, CNRS（蒙彼利埃大学CNRS LIRMM实验室）；School of Computing, KAIST（韩国科学技术院计算机学院）
We consider node-weighted survivable network design (SNDP) in planar graphs and minor-closed families of graphs. The input consists of a node-weighted undirected graph $G=(V,E)$ and integer connectivity requirements $r(uv)$ for each unordered pair of nodes $uv$. The goal is to find a minimum weighted subgraph $H$ of $G$ such that $H$ contains $r(uv)$ disjoint paths between $u$ and $v$ for each node pair $uv$. Three versions of the problem are edge-connectivity SNDP (EC-SNDP), element-connectivity SNDP (Elem-SNDP) and vertex-connectivity SNDP (VC-SNDP) depending on whether the paths are require…
Bidimensionality Theory was introduced by [E.D. Demaine, F.V. Fomin, M.Hajiaghayi, and D.M. Thilikos. Subexponential parameterized algorithms on graphs of bounded genus and H-minor-free graphs, J. ACM, 52 (2005), pp.866--893] as a tool to obtain sub-exponential time parameterized algorithms on H-minor-free graphs. In [E.D. Demaine and M.Hajiaghayi, Bidimensionality: new connections between FPT algorithms and PTASs, in Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), SIAM, 2005, pp.590--601] this theory was extended in order to obtain polynomial time approximatio…
We consider one-sided error property testing of $\mathcal{F}$-minor freeness in bounded-degree graphs for any finite family of graphs $\mathcal{F}$ that contains a minor of $K_{2,k}$, the $k$-circus graph, or the $(k\times 2)$-grid for any $k\in\mathbb{N}$. This includes, for instance, testing whether a graph is outerplanar or a cactus graph. The query complexity of our algorithm in terms of the number of vertices in the graph, $n$, is $\tilde{O}(n^{2/3} / ε^5)$. Czumaj et~al.\ showed that cycle-freeness and $C_k$-minor freeness can be tested with query complexity $\tilde{O}(\sqrt{n})$ by usin…
We give an $\tilde{O}(n^{7/5} \log (nC))$-time algorithm to compute a minimum-cost maximum cardinality matching (optimal matching) in $K_h$-minor free graphs with $h=O(1)$ and integer edge weights having magnitude at most $C$. This improves upon the $\tilde{O}(n^{10/7}\log{C})$ algorithm of Cohen et al. [SODA 2017] and the $O(n^{3/2}\log (nC))$ algorithm of Gabow and Tarjan [SIAM J. Comput. 1989]. For a graph with $m$ edges and $n$ vertices, the well-known Hungarian Algorithm computes a shortest augmenting path in each phase in $O(m)$ time, yielding an optimal matching in $O(mn)$ time. The Hop…
An $n$-vertex graph is equitably $k$-colorable if there is a proper coloring of its vertices such that each color is used either $\left\lfloor n/k \right\rfloor$ or $\left\lceil n/k \right\rceil$ times. While classic Vertex Coloring is fixed parameter tractable under well established parameters such as pathwidth and feedback vertex set, Equitable Coloring is $\mathsf{W}[1]$-$\mathsf{hard}$. We present an extensive study of structural parameterizations of Equitable Coloring, tackling both tractability and kernelization questions. We begin by showing that the problem is fixed parameter tractable…
A cactus is a connected graph that does not contain $K_4 - e$ as a minor. Given a graph $G = (V, E)$ and integer $k \ge 0$, Cactus Vertex Deletion (also known as Diamond Hitting Set) is the problem of deciding whether $G$ has a vertex set of size at most $k$ whose removal leaves a forest of cacti. The current best deterministic parameterized algorithm for this problem was due to Bonnet et al. [WG 2016], which runs in time $26^kn^{O(1)}$, where $n$ is the number of vertices of $G$. In this paper, we design a deterministic algorithm for Cactus Vertex Deletion, which runs in time $17.64^kn^{O(1)}…
We design the first subexponential-time (parameterized) algorithms for several cut and cycle-hitting problems on $H$-minor free graphs. In particular, we obtain the following results (where $k$ is the solution-size parameter). 1. $2^{O(\sqrt{k}\log k)} \cdot n^{O(1)}$ time algorithms for Edge Bipartization and Odd Cycle Transversal; 2. a $2^{O(\sqrt{k}\log^4 k)} \cdot n^{O(1)}$ time algorithm for Edge Multiway Cut and a $2^{O(r \sqrt{k} \log k)} \cdot n^{O(1)}$ time algorithm for Vertex Multiway Cut, where $r$ is the number of terminals to be separated; 3. a $2^{O((r+\sqrt{k})\log^4 (rk))} \cd…
For a fixed finite family of graphs $\mathcal{F}$, the $\mathcal{F}$-Minor-Free Deletion problem takes as input a graph $G$ and an integer $\ell$ and asks whether there exists a set $X \subseteq V(G)$ of size at most $\ell$ such that $G-X$ is $\mathcal{F}$-minor-free. For $\mathcal{F}=\{K_2\}$ and $\mathcal{F}=\{K_3\}$ this encodes Vertex Cover and Feedback Vertex Set respectively. When parameterized by the feedback vertex number of $G$ these two problems are known to admit a polynomial kernelization. Such a polynomial kernelization also exists for any $\mathcal{F}$ containing a planar graph b…
We investigate the computational complexity of finding temporally disjoint paths or walks in temporal graphs. There, the edge set changes over discrete time steps and a temporal path (resp. walk) uses edges that appear at monotonically increasing time steps. Two paths (or walks) are temporally disjoint if they never use the same vertex at the same time; otherwise, they interfere. This reflects applications in robotics, traffic routing, or finding safe pathways in dynamically changing networks. On the one extreme, we show that on general graphs the problem is computationally hard. The "walk ver…
