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Julia Chuzhoy

Publications and source records attributed to Julia Chuzhoy.

At least 37 records · Page 2Linked to original sources
arXiv2019

Towards Tight(er) Bounds for the Excluded Grid Theorem

We study the Excluded Grid Theorem, a fundamental structural result in graph theory, that was proved by Robertson and Seymour in their seminal work on graph minors. The theorem states that there is a function $f: \mathbb{Z}^+ \to \mathbb{Z}^+$, such that for every integer $g>0$, every graph of treewidth at least $f(g)$ contains the $(g\times g)$-grid as a minor. For every integer $g>0$, let $f(g)$ be the smallest value for which the theorem holds. Establishing tight bounds on $f(g)$ is an important graph-theoretic question. Robertson and Seymour showed that $f(g)=Ω(g^2\log g)$ must hold. For a long time, the best known upper bounds on $f(g)$ were super-exponential in $g$. The first polynomial upper bound of $f(g)=O(g^{98}\text{poly}\log g)$ was proved by Chekuri and Chuzhoy. It was later improved to $f(g) = O(g^{36}\text{poly} \log g)$, and then to $f(g)=O(g^{19}\text{poly}\log g)$. In this paper we further improve this bound to $f(g)=O(g^{9}\text{poly} \log g)$. We believe that our proof is significantly simpler than the proofs of the previous bounds. Moreover, while there are natural barriers that seem to prevent the previous methods from yielding tight bounds for the theorem, it seems conceivable that the techniques proposed in this paper can lead to even tighter bounds on $f(g)$.

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arXiv2018

Improved Approximation for Node-Disjoint Paths in Grids with Sources on the Boundary

We study the classical Node-Disjoint Paths (NDP) problem: given an undirected $n$-vertex graph G, together with a set {(s_1,t_1),...,(s_k,t_k)} of pairs of its vertices, called source-destination, or demand pairs, find a maximum-cardinality set of mutually node-disjoint paths that connect the demand pairs. The best current approximation for the problem is achieved by a simple greedy $O(\sqrt{n})$-approximation algorithm. A special case of the problem called NDP-Grid, where the underlying graph is a grid, has been studied extensively. The best current approximation algorithm for NDP-Grid achieves an $\tilde{O}(n^{1/4})$-approximation factor. On the negative side, a recent result by the authors shows that NDP is hard to approximate to within factor $2^{Ω(\sqrt{\log n})}$, even if the underlying graph is a sub-graph of a grid, and all source vertices lie on the grid boundary. In a follow-up work, the authors further show that NDP-Grid is hard to approximate to within factor $Ω(2^{\log^{1-ε}n})$ for any constant $ε$ under standard complexity assumptions, and to within factor $n^{Ω(1/(\log\log n)^2)}$ under randomized ETH. In this paper we study NDP-Grid, where all source vertices {s_1,...,s_k} appear on the grid boundary. Our main result is an efficient randomized $2^{O(\sqrt{\log n} \cdot \log\log n)}$-approximation algorithm for this problem. We generalize this result to instances where the source vertices lie within a prescribed distance from the grid boundary. Much of the work on approximation algorithms for NDP relies on the multicommodity flow relaxation of the problem, which is known to have an $Ω(\sqrt n)$ integrality gap, even in grid graphs. Our work departs from this paradigm, and uses a (completely different) linear program only to select the pairs to be routed, while the routing itself is computed by other methods.

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arXiv2017

Almost Polynomial Hardness of Node-Disjoint Paths in Grids

In the classical Node-Disjoint Paths (NDP) problem, we are given an $n$-vertex graph $G=(V,E)$, and a collection $M=\{(s_1,t_1),\ldots,(s_k,t_k)\}$ of pairs of its vertices, called source-destination, or demand pairs. The goal is to route as many of the demand pairs as possible, where to route a pair we need to select a path connecting it, so that all selected paths are disjoint in their vertices. The best current algorithm for NDP achieves an $O(\sqrt{n})$-approximation, while, until recently, the best negative result was a factor $Ω(\log^{1/2-ε}n)$-hardness of approximation, for any constant $ε$, unless $NP \subseteq ZPTIME(n^{poly \log n})$. In a recent work, the authors have shown an improved $2^{Ω(\sqrt{\log n})}$-hardness of approximation for NDP, unless $NP\subseteq DTIME(n^{O(\log n)})$, even if the underlying graph is a subgraph of a grid graph, and all source vertices lie on the boundary of the grid. Unfortunately, this result does not extend to grid graphs. The approximability of the NDP problem on grid graphs has remained a tantalizing open question, with the best current upper bound of $\tilde{O}(n^{1/4})$, and the best current lower bound of APX-hardness. In this paper we come close to resolving the approximability of NDP in general, and NDP in grids in particular. Our main result is that NDP is $2^{Ω(\log^{1-ε} n)}$-hard to approximate for any constant $ε$, assuming that $NP\not\subseteq RTIME(n^{poly\log n})$, and that it is $n^{Ω(1/(\log \log n)^2)}$-hard to approximate, assuming that for some constant $δ>0$, $NP \not \subseteq RTIME(2^{n^δ})$. These results hold even for grid graphs and wall graphs, and extend to the closely related Edge-Disjoint Paths problem, even in wall graphs.

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arXiv2016

New Hardness Results for Routing on Disjoint Paths

In the classical Node-Disjoint Paths (NDP) problem, the input consists of an undirected $n$-vertex graph $G$, and a collection $\mathcal{M}=\{(s_1,t_1),\ldots,(s_k,t_k)\}$ of pairs of its vertices, called source-destination, or demand, pairs. The goal is to route the largest possible number of the demand pairs via node-disjoint paths. The best current approximation for the problem is achieved by a simple greedy algorithm, whose approximation factor is $O(\sqrt n)$, while the best current negative result is an $Ω(\log^{1/2-δ}n)$-hardness of approximation for any constant $δ$, under standard complexity assumptions. Even seemingly simple special cases of the problem are still poorly understood: when the input graph is a grid, the best current algorithm achieves an $\tilde O(n^{1/4})$-approximation, and when it is a general planar graph, the best current approximation ratio of an efficient algorithm is $\tilde O(n^{9/19})$. The best currently known lower bound on the approximability of both these versions of the problem is APX-hardness. In this paper we prove that NDP is $2^{Ω(\sqrt{\log n})}$-hard to approximate, unless all problems in NP have algorithms with running time $n^{O(\log n)}$. Our result holds even when the underlying graph is a planar graph with maximum vertex degree $3$, and all source vertices lie on the boundary of a single face (but the destination vertices may lie anywhere in the graph). We extend this result to the closely related Edge-Disjoint Paths problem, showing the same hardness of approximation ratio even for sub-cubic planar graphs with all sources lying on the boundary of a single face.

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arXiv2016

Polynomial Bounds for the Grid-Minor Theorem

One of the key results in Robertson and Seymour's seminal work on graph minors is the Grid-Minor Theorem (also called the Excluded Grid Theorem). The theorem states that for every grid $H$, every graph whose treewidth is large enough relative to $|V(H)|$ contains $H$ as a minor. This theorem has found many applications in graph theory and algorithms. Let $f(k)$ denote the largest value such that every graph of treewidth $k$ contains a grid minor of size $(f(k)\times f(k))$. The best previous quantitative bound, due to recent work of Kawarabayashi and Kobayashi, and Leaf and Seymour, shows that $f(k)=Ω(\sqrt{\log k/\log \log k})$. In contrast, the best known upper bound implies that $f(k) = O(\sqrt{k/\log k})$. In this paper we obtain the first polynomial relationship between treewidth and grid minor size by showing that $f(k)=Ω(k^δ)$ for some fixed constant $δ> 0$, and describe a randomized algorithm, whose running time is polynomial in $|V(G)|$ and $k$, that with high probability finds a model of such a grid minor in $G$.

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arXiv2016

On Approximating Maximum Independent Set of Rectangles

We study the Maximum Independent Set of Rectangles (MISR) problem: given a set of $n$ axis-parallel rectangles, find a largest-cardinality subset of the rectangles, such that no two of them overlap. MISR is a basic geometric optimization problem with many applications, that has been studied extensively. Until recently, the best approximation algorithm for it achieved an $O(\log \log n)$-approximation factor. In a recent breakthrough, Adamaszek and Wiese provided a quasi-polynomial time approximation scheme: a $(1-ε)$-approximation algorithm with running time $n^{O(\operatorname{poly}(\log n)/ε)}$. Despite this result, obtaining a PTAS or even a polynomial-time constant-factor approximation remains a challenging open problem. In this paper we make progress towards this goal by providing an algorithm for MISR that achieves a $(1 - ε)$-approximation in time $n^{O(\operatorname{poly}(\log\log{n} / ε))}$. We introduce several new technical ideas, that we hope will lead to further progress on this and related problems.

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arXiv2016

Improved Approximation for Node-Disjoint Paths in Planar Graphs

We study the classical Node-Disjoint Paths (NDP) problem: given an $n$-vertex graph $G$ and a collection $M=\{(s_1,t_1),\ldots,(s_k,t_k)\}$ of pairs of vertices of $G$ called demand pairs, find a maximum-cardinality set of node-disjoint paths connecting the demand pairs. NDP is one of the most basic routing problems, that has been studied extensively. Despite this, there are still wide gaps in our understanding of its approximability: the best currently known upper bound of $O(\sqrt n)$ on its approximation ratio is achieved via a simple greedy algorithm, while the best current negative result shows that the problem does not have a better than $Ω(\log^{1/2-δ}n)$-approximation for any constant $δ$, under standard complexity assumptions. Even for planar graphs no better approximation algorithms are known, and to the best of our knowledge, the best negative bound is APX-hardness. Perhaps the biggest obstacle to obtaining better approximation algorithms for NDP is that most currently known approximation algorithms for this type of problems rely on the standard multicommodity flow relaxation, whose integrality gap is $Ω(\sqrt n)$ for NDP, even in planar graphs. In this paper, we break the barrier of $O(\sqrt n)$ on the approximability of the NDP problem in planar graphs and obtain an $\tilde O(n^{9/19})$-approximation. We introduce a new linear programming relaxation of the problem, and a number of new techniques, that we hope will be helpful in designing more powerful algorithms for this and related problems.

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arXiv2016

Improved Bounds for the Excluded Grid Theorem

We study the Excluded Grid Theorem of Robertson and Seymour. This is a fundamental result in graph theory, that states that there is some function $f: Z^+\rightarrow Z^+$, such that for all integers $g>0$, every graph of treewidth at least $f(g)$ contains the $(g\times g)$-grid as a minor. Until recently, the best known upper bounds on $f$ were super-exponential in $g$. A recent work of Chekuri and Chuzhoy provided the first polynomial bound, by showing that treewidth $f(g)=O(g^{98}\operatorname{poly}\log g)$ is sufficient to ensure the existence of the $(g\times g)$-grid minor in any graph. In this paper we improve this bound to $f(g)=O(g^{19}\operatorname{poly}\log g)$. We introduce a number of new techniques, including a conceptually simple and almost entirely self-contained proof of the theorem that achieves a polynomial bound on $f(g)$.

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arXiv2014

Degree-3 Treewidth Sparsifiers

We study treewidth sparsifiers. Informally, given a graph $G$ of treewidth $k$, a treewidth sparsifier $H$ is a minor of $G$, whose treewidth is close to $k$, $|V(H)|$ is small, and the maximum vertex degree in $H$ is bounded. Treewidth sparsifiers of degree $3$ are of particular interest, as routing on node-disjoint paths, and computing minors seems easier in sub-cubic graphs than in general graphs. In this paper we describe an algorithm that, given a graph $G$ of treewidth $k$, computes a topological minor $H$ of $G$ such that (i) the treewidth of $H$ is $Ω(k/\text{polylog}(k))$; (ii) $|V(H)| = O(k^4)$; and (iii) the maximum vertex degree in $H$ is $3$. The running time of the algorithm is polynomial in $|V(G)|$ and $k$. Our result is in contrast to the known fact that unless $NP \subseteq coNP/{\sf poly}$, treewidth does not admit polynomial-size kernels. One of our key technical tools, which is of independent interest, is a construction of a small minor that preserves node-disjoint routability between two pairs of vertex subsets. This is closely related to the open question of computing small good-quality vertex-cut sparsifiers that are also minors of the original graph.

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arXiv2014

Improved Bounds for the Flat Wall Theorem

The Flat Wall Theorem of Robertson and Seymour states that there is some function $f$, such that for all integers $w,t>1$, every graph $G$ containing a wall of size $f(w,t)$, must contain either (i) a $K_t$-minor; or (ii) a small subset $A\subset V(G)$ of vertices, and a flat wall of size $w$ in $G\setminus A$. Kawarabayashi, Thomas and Wollan recently showed a self-contained proof of this theorem with the following two sets of parameters: (1) $f(w,t)=Θ(t^{24}(t^2+w))$ with $|A|=O(t^{24})$, and (2) $f(w,t)=w^{2^{Θ(t^{24})}}$ with $|A|\leq t-5$. The latter result gives the best possible bound on $|A|$. In this paper we improve their bounds to $f(w,t)=Θ(t(t+w))$ with $|A|\leq t-5$. For the special case where the maximum vertex degree in $G$ is bounded by $D$, we show that, if $G$ contains a wall of size $Ω(Dt(t+w))$, then either $G$ contains a $K_t$-minor, or there is a flat wall of size $w$ in $G$. This setting naturally arises in algorithms for the Edge-Disjoint Paths problem, with $D\leq 4$. Like the proof of Kawarabayashi et al., our proof is self-contained, except for using a well-known theorem on routing pairs of disjoint paths. We also provide efficient algorithms that return either a model of the $K_t$-minor, or a vertex set $A$ and a flat wall of size $w$ in $G\setminus A$. We complement our result for the low-degree scenario by proving an almost matching lower bound: namely, for all integers $w,t>1$, there is a graph $G$, containing a wall of size $Ω(wt)$, such that the maximum vertex degree in $G$ is 5, and $G$ contains no flat wall of size $w$, and no $K_t$-minor.

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arXiv2013

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.

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arXiv2012

A Polylogarithimic Approximation Algorithm for Edge-Disjoint Paths with Congestion 2

In the Edge-Disjoint Paths with Congestion problem (EDPwC), we are given an undirected n-vertex graph G, a collection M={(s_1,t_1),...,(s_k,t_k)} of demand pairs and an integer c. The goal is to connect the maximum possible number of the demand pairs by paths, so that the maximum edge congestion - the number of paths sharing any edge - is bounded by c. When the maximum allowed congestion is c=1, this is the classical Edge-Disjoint Paths problem (EDP). The best current approximation algorithm for EDP achieves an $O(\sqrt n)$-approximation, by rounding the standard multi-commodity flow relaxation of the problem. This matches the $Ω(\sqrt n)$ lower bound on the integrality gap of this relaxation. We show an $O(poly log k)$-approximation algorithm for EDPwC with congestion c=2, by rounding the same multi-commodity flow relaxation. This gives the best possible congestion for a sub-polynomial approximation of EDPwC via this relaxation. Our results are also close to optimal in terms of the number of pairs routed, since EDPwC is known to be hard to approximate to within a factor of $\tildeΩ((\log n)^{1/(c+1)})$ for any constant congestion c. Prior to our work, the best approximation factor for EDPwC with congestion 2 was $\tilde O(n^{3/7})$, and the best algorithm achieving a polylogarithmic approximation required congestion 14.

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arXiv2012

On Vertex Sparsifiers with Steiner Nodes

Given an undirected graph $G=(V,E)$ with edge capacities $c_e\geq 1$ for $e\in E$ and a subset $T$ of $k$ vertices called terminals, we say that a graph $H$ is a quality-$q$ cut sparsifier for $G$ iff $T\subseteq V(H)$, and for any partition $(A,B)$ of $T$, the values of the minimum cuts separating $A$ and $B$ in graphs $G$ and $H$ are within a factor $q$ from each other. We say that $H$ is a quality-$q$ flow sparsifier for $G$ iff $T\subseteq V(H)$, and for any set $D$ of demands over the terminals, the values of the minimum edge congestion incurred by fractionally routing the demands in $D$ in graphs $G$ and $H$ are within a factor $q$ from each other. So far vertex sparsifiers have been studied in a restricted setting where the sparsifier $H$ is not allowed to contain any non-terminal vertices, that is $V(H)=T$. For this setting, efficient algorithms are known for constructing quality-$O(\log k/\log\log k)$ cut and flow vertex sparsifiers, as well as a lower bound of $\tildeΩ(\sqrt{\log k})$ on the quality of any flow or cut sparsifier. We study flow and cut sparsifiers in the more general setting where Steiner vertices are allowed, that is, we no longer require that $V(H)=T$. We show algorithms to construct constant-quality cut sparsifiers of size $O(C^3)$ in time $\poly(n)\cdot 2^C$, and constant-quality flow sparsifiers of size $C^{O(\log\log C)}$ in time $n^{O(\log C)}\cdot 2^C$, where $C$ is the total capacity of the edges incident on the terminals.

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arXiv2011

Approximation Algorithms and Hardness of the k-Route Cut Problem

We study the k-route cut problem: given an undirected edge-weighted graph G=(V,E), a collection {(s_1,t_1),(s_2,t_2),...,(s_r,t_r)} of source-sink pairs, and an integer connectivity requirement k, the goal is to find a minimum-weight subset E' of edges to remove, such that the connectivity of every pair (s_i, t_i) falls below k. Specifically, in the edge-connectivity version, EC-kRC, the requirement is that there are at most (k-1) edge-disjoint paths connecting s_i to t_i in G \ E', while in the vertex-connectivity version, NC-kRC, the same requirement is for vertex-disjoint paths. Prior to our work, poly-logarithmic approximation algorithms have been known for the special case where k >= 3, but no non-trivial approximation algorithms were known for any value k>3, except in the single-source setting. We show an O(k log^{3/2}r)-approximation algorithm for EC-kRC with uniform edge weights, and several polylogarithmic bi-criteria approximation algorithms for EC-kRC and NC-kRC, where the connectivity requirement k is violated by a constant factor. We complement these upper bounds by proving that NC-kRC is hard to approximate to within a factor of k^{eps} for some fixed eps>0. We then turn to study a simpler version of NC-kRC, where only one source-sink pair is present. We give a simple bi-criteria approximation algorithm for this case, and show evidence that even this restricted version of the problem may be hard to approximate. For example, we prove that the single source-sink pair version of NC-kRC has no constant-factor approximation, assuming Feige's Random k-AND assumption.

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arXiv2011

Routing in Undirected Graphs with Constant Congestion

Given an undirected graph G=(V,E), a collection (s_1,t_1),...,(s_k,t_k) of k source-sink pairs, and an integer c, the goal in the Edge Disjoint Paths with Congestion problem is to connect maximum possible number of the source-sink pairs by paths, so that the maximum load on any edge (called edge congestion) does not exceed c. We show an efficient randomized algorithm to route $Ω(OPT/\poly\log k)$ source-sink pairs with congestion at most 14, where OPT is the maximum number of pairs that can be simultaneously routed on edge-disjoint paths. The best previous algorithm that routed $Ω(OPT/\poly\log n)$ pairs required congestion $\poly(\log \log n)$, and for the setting where the maximum allowed congestion is bounded by a constant c, the best previous algorithms could only guarantee the routing of $OPT/n^{O(1/c)}$ pairs.

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arXiv2010

An Algorithm for the Graph Crossing Number Problem

We study the Minimum Crossing Number problem: given an $n$-vertex graph $G$, the goal is to find a drawing of $G$ in the plane with minimum number of edge crossings. This is one of the central problems in topological graph theory, that has been studied extensively over the past three decades. The first non-trivial efficient algorithm for the problem, due to Leighton and Rao, achieved an $O(n\log^4n)$-approximation for bounded degree graphs. This algorithm has since been improved by poly-logarithmic factors, with the best current approximation ratio standing on $O(n \poly(d) \log^{3/2}n)$ for graphs with maximum degree $d$. In contrast, only APX-hardness is known on the negative side. In this paper we present an efficient randomized algorithm to find a drawing of any $n$-vertex graph $G$ in the plane with $O(OPT^{10}\cdot \poly(d \log n))$ crossings, where $OPT$ is the number of crossings in the optimal solution, and $d$ is the maximum vertex degree in $G$. This result implies an $\tilde{O}(n^{9/10} \poly(d))$-approximation for Minimum Crossing Number, thus breaking the long-standing $\tilde{O}(n)$-approximation barrier for bounded-degree graphs.

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arXiv2010

On Graph Crossing Number and Edge Planarization

Given an n-vertex graph G, a drawing of G in the plane is a mapping of its vertices into points of the plane, and its edges into continuous curves, connecting the images of their endpoints. A crossing in such a drawing is a point where two such curves intersect. In the Minimum Crossing Number problem, the goal is to find a drawing of G with minimum number of crossings. The value of the optimal solution, denoted by OPT, is called the graph's crossing number. This is a very basic problem in topological graph theory, that has received a significant amount of attention, but is still poorly understood algorithmically. The best currently known efficient algorithm produces drawings with $O(\log^2 n)(n + OPT)$ crossings on bounded-degree graphs, while only a constant factor hardness of approximation is known. A closely related problem is Minimum Edge Planarization, in which the goal is to remove a minimum-cardinality subset of edges from G, such that the remaining graph is planar. Our main technical result establishes the following connection between the two problems: if we are given a solution of cost k to the Minimum Edge Planarization problem on graph G, then we can efficiently find a drawing of G with at most $\poly(d)\cdot k\cdot (k+OPT)$ crossings, where $d$ is the maximum degree in G. This result implies an $O(n\cdot \poly(d)\cdot \log^{3/2}n)$-approximation for Minimum Crossing Number, as well as improved algorithms for special cases of the problem, such as, for example, k-apex and bounded-genus graphs.

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arXiv2009

On Allocating Goods to Maximize Fairness

Given a set of $m$ agents and a set of $n$ items, where agent $A$ has utility $u_{A,i}$ for item $i$, our goal is to allocate items to agents to maximize fairness. Specifically, the utility of an agent is the sum of its utilities for items it receives, and we seek to maximize the minimum utility of any agent. While this problem has received much attention recently, its approximability has not been well-understood thus far: the best known approximation algorithm achieves an $\tilde{O}(\sqrt{m})$-approximation, and in contrast, the best known hardness of approximation stands at 2. Our main result is an approximation algorithm that achieves an $\tilde{O}(n^{\eps})$ approximation for any $\eps=Ω(\log\log n/\log n)$ in time $n^{O(1/\eps)}$. In particular, we obtain poly-logarithmic approximation in quasi-polynomial time, and for any constant $\eps > 0$, we obtain $O(n^{\eps})$ approximation in polynomial time. An interesting aspect of our algorithm is that we use as a building block a linear program whose integrality gap is $Ω(\sqrt m)$. We bypass this obstacle by iteratively using the solutions produced by the LP to construct new instances with significantly smaller integrality gaps, eventually obtaining the desired approximation. We also investigate the special case of the problem, where every item has a non-zero utility for at most two agents. We show that even in this restricted setting the problem is hard to approximate upto any factor better tha 2, and show a factor $(2+\eps)$-approximation algorithm running in time $poly(n,1/\eps)$ for any $\eps>0$. This special case can be cast as a graph edge orientation problem, and our algorithm can be viewed as a generalization of Eulerian orientations to weighted graphs.

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