I'm a Software Engineer at Voleon. (See open positions) My research interests are algorithms, combinatorial optimization and computational geometry. I'm also interested in more applied problems with nice theoretical components. Here is my CV(last updated Oct 2020) and old research statement.

I was a software engineer at Grab in Seattle, and was a research scientist at Yahoo! Research in New York.

I obtained my PhD in Computer Science from University of Illinois at Urbana-Champaign in May 2018. My advisors were Karthik Chandrasekaran and Chandra Chekuri. I finished my undergrad in mathematics at Stony Brook University in 2013, and spent Spring 2012 in Budapest Semesters in Mathematics.

I maintain a github account and a **blog**. I frequent cstheory.

You can contact me through email `the.chao.xu@gmail.com`.

Selected Publications

*See all publications.*

Conference Publications

Marking Streets to Improve Parking Density

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Urban Complex Systems, .
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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.

Multicritera Cuts and Size-Constrained k-cuts in Hypergraphs

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RANDOM, .
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We address counting and optimization variants of multicriteria global min-cut and size-constrained min-k-cut in hypergraphs.

- For an r-rank n-vertex hypergraph endowed with t hyperedge-cost functions, we show that the number of multiobjective min-cuts is O(r2^{tr}n^{3t−1}). In particular, this shows that the number of parametric min-cuts in constant rank hypergraphs for a constant number of criteria is strongly polynomial, thus resolving an open question by Aissi, Mahjoub, McCormick, and Queyranne (Math Programming, 2015). In addition, we give randomized algorithms to enumerate all multiobjective min-cuts and all pareto-optimal cuts in strongly polynomial-time.
- We also address node-budgeted multiobjective min-cuts: For an n-vertex hypergraph endowed with t vertex-weight functions, we show that the number of node-budgeted multiobjective min-cuts is O(r2^{r}nt+2), where r is the rank of the hypergraph, and the number of node-budgeted b-multiobjective min-cuts for a fixed budget-vector b is O(n^2).
- We show that min-k-cut in hypergraphs subject to constant lower bounds on part sizes is solvable in polynomial-time for constant k, thus resolving an open problem posed by Queyranne. Our technique also shows that the number of optimal solutions is polynomial. All of our results build on the random contraction approach of Karger (SODA, 1993). Our techniques illustrate the versatility of the random contraction approach to address counting and algorithmic problems concerning multiobjective min-cuts and size-constrained k-cuts in hypergraphs.

LP Relaxation and Tree Packing for Minimum k-cuts

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SOSA, .
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Karger used spanning tree packings to derive a near linear-time randomized algorithm for the global minimum cut problem as well as a bound on the number of approximate minimum cuts. This is a different approach from his well-known random contraction algorithm. Thorup developed a fast deterministic algorithm for the minimum k-cut problem via greedy *recursive* tree packings.
In this paper we revisit properties of an LP relaxation for k-cut proposed by Naor and Rabani, and analyzed by Chekuri, Guha and Naor. We show that the dual of the LP yields a tree packing, that when combined with an upper bound on the integrality gap for the LP, easily and transparently extends Karger's analysis for mincut to the k-cut problem. In addition to the simplicity of the algorithm and its analysis, this allows us to improve the running time of Thorup's algorithm by a factor of n. We also improve the bound on the number of \alpha-approximate k-cuts. Second, we give a simple proof that the integrality gap of the LP is 2(1-1/n). Third, we show that an optimum solution to the LP relaxation, for all values of k, is fully determined by the principal sequence of partitions of the input graph. This allows us to relate the LP relaxation to the Lagrangean relaxation approach of Barahona and Ravi and Sinha; it also shows that the idealized recursive tree packing considered by Thorup gives an optimum dual solution to the LP. This work arose from an effort to understand and simplify the results of Thorup.

Hypergraph k-Cut in Randomized Polynomial Time

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SODA, .
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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

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APPROX, .
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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.

- 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).
- 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.
- 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.

A Faster Pseudopolynomial Time Algorithm for Subset Sum

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SODA, .
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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.

Computing minimum cuts in hypergraphs

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SODA, .
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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.

- 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.
- We generalize Matula's argument for graphs to hypergraphs and obtain a (2+\e)-approximation to the global minimum cut in a capacitated hypergraph in O(\frac{1}{\e}(p+n \log n)\log n) time.
- 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.

On Element-Connectivity Preserving Graph Simplification

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ESA, .
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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.

Detecting Weakly Simple Polygons

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SODA, .
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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.

Journal Publications

Minimum violation vertex maps and their applications to cut problems

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SIAM Journal on Discrete Mathematics .
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The minimum violation problem asks for a vertex map from a digraph to a pattern digraph that minimizes violation, the total weight of the edges not mapped to an edge. We are interested in surjective mappings. We characterize all patterns where a minimum violation map that fixes some vertices can be computed in polynomial time. We also make progress in the case where we do not fix any vertex in the mapping, including when the digraph is disconnected, when the graph is in the variety of finite paths. Moreover, we obtain a dichotomy result for trees. We apply the result to some cut problems, including k-cut with size lower bounds and length bounded k-cuts.

High multiplicity asymmetric traveling salesman problem with feedback vertex set and its application to storage/retrieval system

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European Journal of Operational Research .
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We describe an algorithm for the high multiplicity asymmetric traveling salesman problem with feedback vertex set of size k (HMTSPFVS-kFVS) where each vertex can be visited a certain number of times and each cycle in a solution contains at least one vertex from the feedback vertex set. We show how it can be used to improve algorithms in automated storage and retrieval systems. An automated storage and retrieval system includes storage blocks and storage and retrieval machines that either move to retrieve unit loads from their current locations in the system to a depot or take unit loads from a depot and store them to specific locations in the system. Given n storage and retrieval requests in a system with k depots and one storage and retrieval machine, we show that our algorithm for HMTSPFVS-kFVS can solve the problem of minimizing total traveling time of the storage and retrieval machine in O(n^k+n^3) time when all depots are specialized (each depot fulfills one type of requests) and in O(n^{2k}+n^3) time when depots are regular (each depot fulfills both types of requests). The best previous algorithm only solves the special case of the problem with 2 regular depots in O(n^6) time. The applicability of our algorithm for several generalizations and special cases of the problem is also discussed. Furthermore, to evaluate the performance of our solution method, we perform extensive numerical experiments.

LP Relaxation and Tree Packing for Minimum k-Cut

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SIAM Journal on Discrete Mathematics .
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Karger used spanning tree packings [D. R. Karger, J. ACM, 47 (2000), pp. 46--76] to derive a near linear-time randomized algorithm for the global minimum cut problem as well as a bound on the number of approximate minimum cuts. This is a different approach from his well-known random contraction algorithm [D. R. Karger, Random Sampling in Graph Optimization Problems, Ph.D. thesis, Stanford University, Stanford, CA, 1995, D. R. Karger and C. Stein, J. ACM, 43 (1996), pp. 601--640]. Thorup developed a fast deterministic algorithm for the minimum k-cut problem via greedy recursive tree packings [M. Thorup, Minimum k-way cuts via deterministic greedy tree packing, in Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing, ACM, 2008, pp. 159--166]. In this paper we revisit properties of an LP relaxation for cͅut proposed by Naor and Rabani [Tree packing and approximating k-cuts, in Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms, Vol. 103, SIAM, Philadelphia, 2001, pp. 26--27], and analyzed in [C. Chekuri, S. Guha, and J. Naor, SIAM J. Discrete Math., 20 (2006), pp. 261--271]. We show that the dual of the LP yields a tree packing that, when combined with an upper bound on the integrality gap for the LP, easily and transparently extends Karger's analysis for mincut to the k-cut problem. In addition to the simplicity of the algorithm and its analysis, this allows us to improve the running time of Thorup's algorithm by a factor of n. We also improve the bound on the number of \alpha-approximate k-cuts. Second, we give a simple proof that the integrality gap of the LP is 2(1-1/n). Third, we show that an optimum solution to the LP relaxation, for all values of k, is fully determined by the principal sequence of partitions of the input graph. This allows us to relate the LP relaxation to the Lagrangean relaxation approach of Barahona [Oper. Res. Lett., 26 (2000), pp. 99--105] and Ravi and Sinha [European J. Oper. Res., 186 (2008), pp. 77--90]; it also shows that the idealized recursive tree packing considered by Thorup gives an optimum dual solution to the LP.

Hypergraph k-Cut in Randomized Polynomial Time

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Mathematical Programming .
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For a fixed integer k\geq 2, the hypergraph k-cut problem asks for a smallest subset of hyperedges whose removal leads to at least k connected components in the remaining hypergraph. While graph k-cut is solvable efficiently (Goldschmidt and Hochbaum in Math. Oper. Res. 19(1):24–37, 1994), the complexity of hypergraph k-cut has been open. In this work, we present a randomized polynomial time algorithm to solve the hypergraph k-cut problem. Our algorithmic technique extends to solve the more general hedge k-cut problem when the subgraph induced by every hedge has a constant number of connected components. Our algorithm is based on random contractions akin to Karger’s min cut algorithm. Our main technical contribution is a non-uniform 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. In addition, we present an alternative contraction based randomized polynomial time approximation scheme for hedge k-cut in arbitrary hedgegraphs (i.e., hedgegraphs whose hedges could have a large number of connected components). Our algorithm and analysis also lead to bounds on the number of optimal solutions to the respective problems.

Faster Pseudopolynomial Time Algorithms for Subset Sum

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ACM Transactions on Algorithms .
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Given a (multi) set S of n positive integers and a target integer u, the subset sum problem is to decide if there is a subset of S that sums up to u. We present a series of new algorithms that compute and return

Beating the 2-approximation factor for global bicut

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Mathematical Programming .
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In the fixed-terminal bicut problem, the input is a directed graph with two specified nodes s and t and the goal is to find a smallest subset of edges whose removal ensures that s cannot reach t and t cannot reach s. In the global bicut problem, the input is a directed graph and the goal is to find a smallest subset of edges whose removal ensures that there exist two nodes s and t such that s cannot reach t and t cannot reach s. Fixed-terminal bicut and global bicut are natural extensions of \{s,t\}-min cut and global min-cut respectively, from undirected graphs to directed graphs. Fixed-terminal bicut is NP-hard, admits a simple 2-approximation, and does not admit a (2-\e)-approximation for any constant \e>0 assuming the unique games conjecture. In this work, we show that global bicut admits a (2-1/448)-approximation, thus improving on the approximability of the global variant in comparison to the fixed-terminal variant.

Minimum cuts and sparsification in hypergraphs

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SIAM Journal on Computing .
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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 fastest known algorithm to compute a global minimum cut in H runs in O(np) time for the uncapacitated case, and in O(np + n^2 \log n) time for the capacitated case. We show the following new results.

- Given an uncapacitated hypergraph H and an integer k we describe an algorithm that runs in O(p) time to find a (trimmed) subhypergraph H' with sum of degrees O(kn) that preserves all edge-connectivities up to k (a k-sparse certificate). 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.
- 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 matching the asymptotic time to find a single minimum cut. - We obtain a (2+\e)-approximation to the global minimum cut of a capacitated hypergraph in O(\frac{1}{\e} (p \log n + n \log^2 n)) time, and for uncapacitated hypergraphs in O(p/\e) time. We achieve this by generalizing Matula's algorithm for graphs to hypergraphs.
- We describe an algorithm to compute approximate strengths of all the edges of a hypergraph in O(p \log^2 n \log p) time. This gives a near linear time algorithm for finding a (1+\e)-cut sparsifier based on the work of Kogan and Krauthgamer. As a byproduct we obtain faster algorithms for various cut and flow problems in hypergraphs of small rank.

Our results build upon properties of vertex orderings that were inspired by the maximum adjacency ordering for graphs due to Nagamochi and Ibaraki. Unlike graphs we observe that there are several orderings for hypergraphs and these yield different insights.

The shortest kinship description problem

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Information Processing Letters .
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We consider a problem in descriptive kinship systems, namely finding the shortest sequence of terms that describes the kinship between a person and his/her 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 are the input size and the output size of the algorithm, respectively.

Reconstructing edge-disjoint paths faster

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Operations Research Letters .
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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

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Congressus Numerantium .
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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.

Manuscripts

*Some manuscripts are available upon request.*

A near-linear time algorithm for computing the optimal landing times of a fixed sequence of planes

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, Submitted.
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An algorithm for the metric multiple depots capacitated vehicle routing problem with restocking and capacity two

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2019, Submitted.
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The capacitated vehicle routing problem (CVRP) is one of the most well known NP-hard combinatorial optimization problems. Single depot CVRP with a general metric is NP-hard even for fixed capacity 3, while polynomial time solvable for fixed capacity 2. We consider the variant of CVRP where restocking is available. We show that if there is a constant number of depots, then the problem can be solved in polynomial time when capacity is 2.

Thesis

Cuts and Connectivity in Graphs and Hypergraphs

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In this thesis, we consider cut and connectivity problems on graphs, digraphs, hypergraphs and hedgegraphs. The main results are the following:

- We introduce a faster algorithm for finding the reduced graph in element-connectivity computations. We also show its application to node separation.
- We present several results on hypergraph cuts, including (a) a near linear time algorithm for finding a (2 + ε)-approximate min-cut, (b) an algorithm to find a representation of all min-cuts in the same time as finding a single min-cut, (c) a sparse subgraph that preserves connectivity for hypergraphs and (d) a near linear-time hypergraph cut sparsifier.
- We design the first randomized polynomial time algorithm for the hypergraph k-cut problem whose complexity has been open for over 20 years. The algorithm generalizes to hedgegraphs with constant span.
- We address the complexity gap between global vs. fixed-terminal cuts problems in digraphs by presenting a 2-\frac{1}{448} approximation algorithm for the global bicut problem.