Large-Treewidth Graph Decompositions and Applications

Treewidth is a graph parameter that plays a fundamental role in several structural and algorithmic results. We study the problem of decomposing a given graph $G$ into node-disjoint subgraphs, where each subgraph has sufficiently large treewidth. We prove two theorems on the tradeoff between the number of the desired subgraphs $h$, and the desired lower bound $r$ on the treewidth of each subgraph. The theorems assert that, given a graph $G$ with treewidth $k$, a decomposition with parameters $h,r$ is feasible whenever hr^2 \le k/\polylog(k), or h^3r \le k/\polylog(k) holds. We then show a framework for using these theorems to bypass the well-known Grid-Minor Theorem of Robertson and Seymour in some applications. In particular, this leads to substantially improved parameters in some Erdos-Posa-type results, and faster algorithms for a class of fixed-parameter tractable problems.Comment: An extended abstract of the paper is to appear in Proceedings of ACM STOC, 201

Polynomial expansion and sublinear separators

Let $\mathcal{C}$ be a class of graphs that is closed under taking subgraphs. We prove that if for some fixed $0<\delta\le 1$, every $n$-vertex graph of $\mathcal{C}$ has a balanced separator of order $O(n^{1-\delta})$, then any depth-$k$ minor (i.e. minor obtained by contracting disjoint subgraphs of radius at most $k$) of a graph in $\mathcal{C}$ has average degree $O\big((k \text{ polylog }k)^{1/\delta}\big)$. This confirms a conjecture of Dvo\v{r}\'ak and Norin.Comment: 6 pages, no figur

A tight Erd\H{o}s-P\'osa function for wheel minors

Let $W_t$ denote the wheel on $t+1$ vertices. We prove that for every integer $t \geq 3$ there is a constant $c=c(t)$ such that for every integer $k\geq 1$ and every graph $G$, either $G$ has $k$ vertex-disjoint subgraphs each containing $W_t$ as minor, or there is a subset $X$ of at most $c k \log k$ vertices such that $G-X$ has no $W_t$ minor. This is best possible, up to the value of $c$. We conjecture that the result remains true more generally if we replace $W_t$ with any fixed planar graph $H$.Comment: 15 pages, 1 figur

On Routing Disjoint Paths in Bounded Treewidth Graphs

We study the problem of routing on disjoint paths in bounded treewidth graphs with both edge and node capacities. The input consists of a capacitated graph $G$ and a collection of $k$ source-destination pairs $\mathcal{M} = \{(s_1, t_1), \dots, (s_k, t_k)\}$. The goal is to maximize the number of pairs that can be routed subject to the capacities in the graph. A routing of a subset $\mathcal{M}'$ of the pairs is a collection $\mathcal{P}$ of paths such that, for each pair $(s_i, t_i) \in \mathcal{M}'$, there is a path in $\mathcal{P}$ connecting $s_i$ to $t_i$. In the Maximum Edge Disjoint Paths (MaxEDP) problem, the graph $G$ has capacities $\mathrm{cap}(e)$ on the edges and a routing $\mathcal{P}$ is feasible if each edge $e$ is in at most $\mathrm{cap}(e)$ of the paths of $\mathcal{P}$. The Maximum Node Disjoint Paths (MaxNDP) problem is the node-capacitated counterpart of MaxEDP. In this paper we obtain an $O(r^3)$ approximation for MaxEDP on graphs of treewidth at most $r$ and a matching approximation for MaxNDP on graphs of pathwidth at most $r$. Our results build on and significantly improve the work by Chekuri et al. [ICALP 2013] who obtained an $O(r \cdot 3^r)$ approximation for MaxEDP

Constant Congestion Routing of Symmetric Demands in Planar Directed Graphs

We study the problem of routing symmetric demand pairs in planar digraphs. The input consists of a directed planar graph G = (V, E) and a collection of k source-destination pairs M = {s_1t_1, ..., s_kt_k}. The goal is to maximize the number of pairs that are routed along disjoint paths. A pair s_it_i is routed in the symmetric setting if there is a directed path connecting s_i to t_i and a directed path connecting t_i to s_i. In this paper we obtain a randomized poly-logarithmic approximation with constant congestion for this problem in planar digraphs. The main technical contribution is to show that a planar digraph with directed treewidth h contains a constant congestion crossbar of size Omega(h/polylog(h))

Packing Directed Cycles Quarter- and Half-Integrally

The celebrated Erd\H{o}s-P\'osa theorem states that every undirected graph that does not admit a family of $k$ vertex-disjoint cycles contains a feedback vertex set (a set of vertices hitting all cycles in the graph) of size $O(k \log k)$. After being known for long as Younger's conjecture, a similar statement for directed graphs has been proven in 1996 by Reed, Robertson, Seymour, and Thomas. However, in their proof, the dependency of the size of the feedback vertex set on the size of vertex-disjoint cycle packing is not elementary. We show that if we compare the size of a minimum feedback vertex set in a directed graph with the quarter-integral cycle packing number, we obtain a polynomial bound. More precisely, we show that if in a directed graph $G$ there is no family of $k$ cycles such that every vertex of $G$ is in at most four of the cycles, then there exists a feedback vertex set in $G$ of size $O(k^4)$. Furthermore, a variant of our proof shows that if in a directed graph $G$ there is no family of $k$ cycles such that every vertex of $G$ is in at most two of the cycles, then there exists a feedback vertex set in $G$ of size $O(k^6)$. On the way there we prove a more general result about quarter-integral packing of subgraphs of high directed treewidth: for every pair of positive integers $a$ and $b$, if a directed graph $G$ has directed treewidth $\Omega(a^6 b^8 \log^2(ab))$, then one can find in $G$ a family of $a$ subgraphs, each of directed treewidth at least $b$, such that every vertex of $G$ is in at most four subgraphs.Comment: Accepted to European Symposium on Algorithms (ESA '19

Packing cycles faster than Erdos-Posa

The Cycle Packing problem asks whether a given undirected graph $G=(V,E)$ contains $k$ vertex-disjoint cycles. Since the publication of the classic Erdös--Pósa theorem in 1965, this problem received significant attention in the fields of graph theory and algorithm design. In particular, this problem is one of the first problems studied in the framework of parameterized complexity. The nonuniform fixed-parameter tractability of Cycle Packing follows from the Robertson--Seymour theorem, a fact already observed by Fellows and Langston in the 1980s. In 1994, Bodlaender showed that Cycle Packing can be solved in time $2^{\mathcal{O}(k^2)}\cdot |V|$ using exponential space. In the case a solution exists, Bodlaender's algorithm also outputs a solution (in the same time). It has later become common knowledge that Cycle Packing admits a $2^{\mathcal{O}(k\log^2k)}\cdot |V|$-time (deterministic) algorithm using exponential space, which is a consequence of the Erdös--Pósa theorem. Nowadays, the design of this algorithm is given as an exercise in textbooks on parameterized complexity. Yet, no algorithm that runs in time $2^{o(k\log^2k)}\cdot |V|^{\mathcal{O}(1)}$, beating the bound $2^{\mathcal{O}(k\log^2k)}\cdot |V|^{\mathcal{O}(1)}$, has been found. In light of this, it seems natural to ask whetherthe $2^{\mathcal{O}(k\log^2k)}\cdot |V|^{\mathcal{O}(1)}$ bound is essentially optimal. In this paper, we answer this question negatively by developing a $2^{\mathcal{O}(\frac{k\log^2k}{\log\log k})}\cdot |V|$-time (deterministic) algorithm for Cycle Packing. In the case a solution exists, our algorithm also outputs a solution (in the same time). Moreover, apart from beating the bound $2^{\mathcal{O}(k\log^2k)}\cdot |V|^{\mathcal{O}(1)}$, our algorithm runs in time linear in $|V|$, and its space complexity is polynomial in the input size.publishedVersio