# Approximation Algorithms and Hardness of the k-Route Cut Problem

@article{Chuzhoy2011ApproximationAA, title={Approximation Algorithms and Hardness of the k-Route Cut Problem}, author={Julia Chuzhoy and Yury Makarychev and Aravindan Vijayaraghavan and Yuan Zhou}, journal={ArXiv}, year={2011}, volume={abs/1112.3611} }

We study the <i>k</i>-route cut problem: given an undirected edge-weighted graph <i>G</i> = (<i>V</i>, <i>E</i>), a collection {(<i>s</i><sub>1</sub>, <i>t</i><sub>1</sub>), (<i>s</i><sub>2</sub>, <i>t</i><sub>2</sub>), …, (<i>s<sub>r</sub></i>, <i>t<sub>r</sub></i>)} of source-sink pairs, and an integer connectivity requirement <i>k</i>, the goal is to find a minimum-weight subset <i>E</i>′ of edges to remove, such that the connectivity of every pair (<i>s<sub>i</sub></i>, <i>t<sub>i</sub></i…

## 31 Citations

### Improved Region-Growing and Combinatorial Algorithms for k-Route Cut Problems (Extended Abstract)

- Computer ScienceSODA
- 2015

The main technical innovation is the definition of a new, powerful \emph{region growing} lemma that allows us to perform region-growing in a recursive fashion even though the LP solution yields a {\em different metric} for each source-sink pair.

### Vertex Sparsifiers: New Results from Old Techniques

- Computer ScienceSIAM J. Comput.
- 2014

Efficient algorithms for constructing a flow-sparsifier H that maintains congestion up to a factor ofO(log k/log log k) where k = |K| and a convex combination of trees over the terminals K that maintains congested graphs is given.

### A (k + 1)-Approximation Robust Network Flow Algorithm and a Tighter Heuristic Method Using Iterative Multiroute Flow

- Computer ScienceWALCOM
- 2014

A polynomial-time heuristic algorithm for both cases of a max-flow problem against k edge failures, called the iterative multiroute flow, which shows that the average improvement made by the heuristic method can be up to 10% better than the multirout flow algorithm.

### Approximation Algorithms for Hypergraph Small-Set Expansion and Small-Set Vertex Expansion

- Computer Science, MathematicsTheory Comput.
- 2016

The Hypergraph Small Set Expansion problem, which asks to compute the cut having the least expansion while having at most $\delta$ fraction of the vertices on the smaller side of the cut, is studied and two algorithms are presented that give an $\tilde O(\delta^{-1} \sqrt{\log n})$ approximation and an algorithm that finds a set with vertex expansion.

### The Densest $k$-Subhypergraph Problem. SIAM on Discrete Mathematics

- Mathematics
- 2018

. The densest k -subgraph (D k S) problem and its corresponding minimization problem smallest p -edge subgraph (S p ES) have come to play a central role in approximation algorithms. This is due both…

### A Note on Degree vs Gap of Min-Rep Label Cover and Improved Inapproximability for Connectivity Problems

- Computer Science, MathematicsInf. Process. Lett.
- 2019

### Almost-polynomial ratio ETH-hardness of approximating densest k-subgraph

- Mathematics, Computer ScienceSTOC
- 2017

It is shown, assuming the exponential time hypothesis (ETH), that there is no polynomial-time algorithm that approximates Densest k-Subgraph to within n1/(loglogn)c factor of the optimum, where c > 0 is a universal constant independent of n.

### Hardness and approximation for network flow interdiction

- Computer ScienceNetworks
- 2017

The first such approximation algorithm, which has approximation ratio at most 2(n−1) for any graph with n vertices, is presented, and it is shown that any nϵ‐approximation algorithm for Network Flow Interdiction, or one of several variants, would imply a 2n4ϵ-approximating algorithm for Densest k‐Subgraph, which is an improvement over past work in terms of the polynomial factor.

### Improved approximation algorithm for the Dense-3-Subhypergraph Problem

- Mathematics, Computer ScienceArXiv
- 2017

A new algorithm is given that approximates Dense-$3$-Subhypergraph within a ratio of $\tilde O(n/k)$, which improves the ratio of $O(n^2/k^2)$ of Chlamt{\'{a}}c et al.

### Downgrading to Minimize Connectivity

- Mathematics, GeologyArXiv
- 2019

We study the problem of interdicting a directed graph by deleting nodes with the goal of minimizing the local edge connectivity of the remaining graph from a given source to a sink. We show hardness…

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