Photo of Chao Xu

Chao Xu | 许超

Ph.D. student in
Algorithms and Theory group
Department of Computer Science
University of Illinois at Urbana-Champaign
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Conference Publications

Hypergraph $k$-Cut in Randomized Polynomial Time
(with Karthekeyan Chandrasekaran and Xilin Yu)
Accepted. SODA 2018.

In the hypergraph $k$-cut problem, the input is a hypergraph, and the goal is to find a smallest subset of hyperedges whose removal ensures that the remaining hypergraph has at least $k$ connected components. This problem is known to be at least as hard as the densest $k$-subgraph problem when k is part of the input (Chekuri-Li, 2015). We present a randomized polynomial time algorithm to solve the hypergraph $k$-cut problem for constant $k$.

Our algorithm solves the more general hedge $k$-cut problem when the subgraph induced by every hedge has a constant number of connected components. In the hedge $k$-cut problem, the input is a hedgegraph specified by a vertex set and a disjoint set of hedges, where each hedge is a subset of edges defined over the vertices. The goal is to find a smallest subset of hedges whose removal ensures that the number of connected components in the remaining underlying (multi-)graph is at least $k$.

Our algorithm is based on random contractions akin to Karger's min cut algorithm. Our main technical contribution is a distribution over the hedges (hyperedges) so that random contraction of hedges (hyperedges) chosen from the distribution succeeds in returning an optimum solution with large probability.

Global and fixed-terminal cuts in digraphs
(with Kristóf Bérczi, Karthekeyan Chandrasekaran, Tamás Király and Euiwoong Lee)
APPROX 2017.

The computational complexity of multicut-like problems may vary significantly depending on whether the terminals are fixed or not. In this work we present a comprehensive study of this phenomenon in two types of cut problems in directed graphs: double cut and bicut.
  1. The fixed-terminal edge-weighted double cut is known to be solvable efficiently. We show a tight approximability factor of $2$ for the fixed-terminal node-weighted double cut. We show that the global node-weighted double cut cannot be approximated to a factor smaller than $\frac{3}{2}$ under the Unique Games Conjecture (UGC).
  2. The fixed-terminal edge-weighted bicut is known to have a tight approximability factor of $2$. We show that the global edge-weighted bicut is approximable to a factor strictly better than $2$, and that the global node-weighted bicut cannot be approximated to a factor smaller than $\frac{3}{2}$ under UGC.
  3. In relation to these investigations, we also prove two results on undirected graphs which are of independent interest. First, we show NP-completeness and a tight inapproximability bound of $\frac{4}{3}$ for the node-weighted $3$-cut problem. Second, we show that for constant $k$, there exists an efficient algorithm to solve the minimum $\{s,t\}$-separating $k$-cut problem.
Our techniques for the algorithms are combinatorial, based on LPs and based on enumeration of approximate min-cuts. Our hardness results are based on combinatorial reductions and integrality gap instances.
This image was used to prove some bold claim...

A Faster Pseudopolynomial Time Algorithm for Subset Sum
(with Konstantinos Koiliaris)
SODA 2017.
Note: Not practical. Discussion/implementation.

Given a multiset $S$ of $n$ positive integers and a target integer $t$, the subset sum problem is to decide if there is a subset of $S$ that sums up to $t$. We present a new divide-and-conquer algorithm that computes all the realizable subset sums up to an integer $u$ in $\tilde{O}\left(\min\{n\sqrt{u},u^{4/3},\sigma\}\right)$, where $\sigma$ is the sum of all elements in $S$ and $\tilde{O}$ hides polylogarithmic factors. This result improves upon the standard dynamic programming algorithm that runs in $O(nu)$ time. To the best of our knowledge, the new algorithm is the fastest general algorithm for this problem. We also present a modified algorithm for cyclic groups, which computes all the realizable subset sums within the group in $\tilde{O}\left(\min\{n\sqrt{m},m^{5/4}\}\right)$ time, where m is the order of the group.
Covering $\mathbb{Z}^*_{11}$ with segments of length $3$.

Computing minimum cuts in hypergraphs
(with Chandra Chekuri)
SODA 2017.
Note: The SODA camera ready version has a bug in the sparsification section that is fixed in the arXiv update.

We study algorithmic and structural aspects of connectivity in hypergraphs. Given a hypergraph $H=(V,E)$ with $n=|V|$, $m=|E|$ and $p=\sum_{e\in E}|e|$ the best known algorithm to compute a global minimum cut in $H$ runs in time $O(np)$ for the uncapacitated case and in $O(np+n^2\log n)$ time for the capacitated case. We show the following new results.
  1. Given an uncapacitated hypergraph $H$ and an integer $k$ we describe an algorithm that runs in $O(p)$ time to find a subhypergraph $H'$ with sum of degrees $O(kn)$ that preserves all edge-connectivities up to $k$ (a $k$-sparsifier). This generalizes the corresponding result of Nagamochi and Ibaraki from graphs to hypergraphs. Using this sparsification we obtain an $O(p+\lambda n^2)$ time algorithm for computing a global minimum cut of $H$ where $\lambda$ is the minimum cut value.
  2. We generalize Matula's argument for graphs to hypergraphs and obtain a $(2+\epsilon)$-approximation to the global minimum cut in a capacitated hypergraph in $O(\frac{1}{\epsilon}(p+n \log n)\log n)$ time.
  3. We show that a hypercactus representation of all the global minimum cuts of a capacitated hypergraph can be computed in $O(np+n^2\log n)$ time and $O(p)$ space.
We utilize vertex ordering based ideas to obtain our results. Unlike graphs we observe that there are several different orderings for hypergraphs which yield different insights.
Example of hyperedge insertion operation on vertex $v$.

On Element-Connectivity Preserving Graph Simplification
(with Chandra Chekuri and Thapanapong Rukkanchanunt)
ESA 2015.

The notion of element-connectivity has found several important applications in network design and routing problems. We focus on a reduction step that preserves the element-connectivity, which when applied repeatedly allows one to reduce the original graph to a simpler one. This pre-processing step is a crucial ingredient in several applications. In this paper we revisit this reduction step and provide a new proof via the use of setpairs. Our main contribution is algorithmic results for several basic problems on element-connectivity including the problem of achieving the aforementioned graph simplification. We utilize the underlying submodularity properties of element-connectivity to derive faster algorithms.
The black vertices are the terminals. The left image shows $4$ element-disjoint $st$-paths. The right image shows removing $4$ elements disconnects $s$ and $t$. $\kappa(s,t) = 4$.

Detecting Weakly Simple Polygons
(with Jeff Erickson and Hsien-Chih Chang)
SODA 2015.

A closed curve in the plane is weakly simple if it is the limit (in the Fréchet metric) of a sequence of simple closed curves. We describe an algorithm to determine whether a closed walk of length n in a simple plane graph is weakly simple in $O(n \log n)$ time, improving an earlier $O(n^3)$-time algorithm of Cortese et al.. As an immediate corollary, we obtain the first efficient algorithm to determine whether an arbitrary n-vertex polygon is weakly simple; our algorithm runs in $O(n^2 \log n)$ time. We also describe algorithms that detect weak simplicity in $O(n \log n)$ time for two interesting classes of polygons. Finally, we discuss subtle errors in several previously published definitions of weak simplicity.

Dedicated with thanks to our colleague Ferran Hurtado (1951–2014).

A polygon $(a, b, c, a, b, c, a, x, y, z, x, y, z, x)$ that is not weakly simple, even though its rotation number is $1$ and every pair of vertices splits the polygon into two paths that do not cross.
Journal Publications

Reconstructing edge-disjoint paths faster
Operations Research Letters, 44 (2) (2016), pp. 174-176.

For a simple undirected graph with $n$ vertices and $m$ edges, we consider a data structure that given a query of a pair of vertices $u$, $v$ and an integer $k\geq 1$, it returns $k$ edge-disjoint $uv$-paths. The data structure takes $\tilde{O}(n^{3.375})$ time to build, using $O(mn^{1.5}\log n)$ space, and each query takes $O(kn)$ time, which is optimal and beats the previous query time of $O(kn\alpha(n))$.

Champion spiders in the game of Graph Nim
(with Neil J. Calkin and Janine E. Janoski and Allison Nelson and Sydney Ryan)
Congr. Numer., 218:5-19, 2013.

In the game of Graph Nim, players take turns removing one or more edges incident to a chosen vertex in a graph. The player that removes the last edge in the graph wins. A spider graph is a champion if it has a Sprague-Grundy number equal to the number of edges in the graph. We investigate the the Sprague-Grundy numbers of various spider graphs when the number of paths or length of paths increase.
A spider graph.

The shortest kinship description problem
(with Qian Zhang)

We consider a problem in descriptive kinship systems, namely finding the shortest sequence of terms that describes the kinship between a person and and their relatives. The problem reduces to finding the minimum weight path in a labeled graph where the label of the path comes from a regular language. The running time of the algorithm is $O(n^3+s)$, where $n$ and $s$ is the input size and the output size of the algorithm, respectively.

To the memories of Jiaqi Zhao(1994–2016).

A note on approximate strengths of edges in a hypergraph
(with Chandra Chekuri)

Let $H=(V,E)$ be an edge-weighted hypergraph of rank $r$. Kogan and Krauthgamer extended Benczúr and Karger's random sampling scheme for cut sparsification from graphs to hypergraphs. The sampling requires an algorithm for computing the approximate strengths of edges. In this note we extend the algorithm for graphs to hypergraphs and describe a near-linear time algorithm to compute approximate strengths of edges; we build on a sparsification result for hypergraphs from our recent work. Combined with prior results we obtain faster algorithms for finding $(1+\epsilon)$-approximate mincuts when the rank of the hypergraph is small.

Marking Streets to Improve Parking Density
(with Steven Skiena)

Street parking spots for automobiles are a scarce commodity in most urban environments. The heterogeneity of car sizes makes it inefficient to rigidly define fixed-sized spots. Instead, unmarked streets in cities like New York leave placement decisions to individual drivers, who have no direct incentive to maximize street utilization. In this paper, we explore the effectiveness of two different behavioral interventions designed to encourage better parking, namely (1) educational campaigns to encourage parkers to "kiss the bumper" and reduce the distance between themselves and their neighbors, or (2) painting appropriately-spaced markings on the street and urging drivers to "hit the line". Through analysis and simulation, we establish that the greatest densities are achieved when lines are painted to create spots roughly twice the length of average-sized cars. Kiss-the-bumper campaigns are in principle more effective than hit-the-line for equal degrees of compliance, although we believe that the visual cues of painted lines induce better parking behavior.
Representative street landscapes for random/Rényi (top), kiss-the-bumper (center), and hit-the-line (bottom) for a street of length $l = 20$ with the optimal line spacing of $k = 2$, for $α$ values of 0, 0.5, and 0.5 respectively.