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## Filters

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Multiway cuts in node weighted graphs

2004
*
Journal of Algorithms
*

A (2 -2/

doi:10.1016/s0196-6774(03)00111-1
fatcat:jxfpih7nezhebbrj5q7p2bcxzq
*k*)*approximation*algorithm is presented for*the*node multiway*cut*problem, thus matching*the*result*of*Dahlhaus et al. (SIAM J. ... Comput. 23 (4) (1994) 864-894) for*the*edge version*of*this problem. This is done by showing that*the*associated LP-relaxation always has a half-integral optimal solution. ... For*k*= 3 or more terminals, as in*the*case*of**the*edge multiway*cut*problem,*the*node multiway*cut*problem is NP-*hard**and**MAX*SNP-*hard*. ...##
###
Minimal multicut and maximal integer multiflow: A survey

2005
*
European Journal of Operational Research
*

Most

doi:10.1016/j.ejor.2003.10.037
fatcat:pxc4bhuvefe6di7abxmy6l57x4
*of**the*results are very recent*and*some are new. ... We present a survey about*the*maximum integral multiflow*and*minimum multicut problems*and*their subproblems, such as*the*multiterminal*cut**and**the*unsplittable flow problems. ... In*the*case*of*a single commodity (*K*¼ 1)*the*vertices*of**the*primal*and**dual*polyhedrons are integral*and**the**max*flow-min*cut*theorem is a direct consequence*of*this integrality. ...##
###
Flows with unit path capacities and related packing and covering problems

2009
*
Journal of combinatorial optimization
*

*The*resulting (fractional) flow problem is NP-

*hard*;

*its*integral version is strongly NP-

*hard*already

*on*very simple classes

*of*graphs. ... We derive several results

*on*

*the*complexity

*and*

*approximability*

*of*

*the*new problem. ...

*It*is not difficult to observe that

*the*

*dual*separation problem

*of*

*the*

*max*-1FP can be solved by computing

*the*

*k*shortest s-t-paths with respect to

*the*

*dual*arc lengths y a , where

*k*is

*the*number

*of*paths ...

##
###
A Quantitative Geometric Approach to Neural Network Smoothness
[article]

2022
*
arXiv
*
pre-print

By adopting this framework, we can immediately obtain several theoretical results, including

arXiv:2203.01212v1
fatcat:3dxlxwunjfbeba2tskdvwtueca
*the*computational*hardness**of*Lipschitz constant estimation*and**its**approximability*. ... We believe that this unified quantitative geometric perspective can bring new insights*and*theoretical tools to*the*investigation*of*neural-network smoothness*and*robustness. ... Alon*and*Naor (2004) showed that both*the**cut*-norm*and**the*mixed-norm*of*a matrix A is NP-*hard*, or specifically MAXSNP-*hard*, via a reduction to*the**Max*-*Cut*problem. ...##
###
Canonical dual approach to solving the maximum cut problem

2012
*
Journal of Global Optimization
*

This paper presents a canonical

doi:10.1007/s10898-012-9881-8
fatcat:uzuvgc24zvdojnrog7wec56xp4
*dual*approach for finding either an optimal or*approximate*solution to*the*maximum*cut*problem (*MAX**CUT*). ... We show that, by introducing a linear perturbation term to*the*objective function,*the*maximum*cut*problem is perturbed to have a*dual*problem which is a concave maximization problem over a convex feasible ... Acknowledgments*The*authors are grateful to*the*anonymous referees for their valuable comments. References ...##
###
Fault Tolerant Max-Cut
[article]

2021
*
arXiv
*
pre-print

*The*non-linear nature

*of*

*the*fault tolerant objective makes

*the*design

*and*analysis

*of*algorithms harder when compared to

*the*classic

*Max*

*Cut*. ... For any constant number

*of*failures

*k*we present an

*approximation*

*of*(0.878-ϵ) against an adaptive adversary

*and*

*of*α_GW≈ 0.8786 against an oblivious adversary (here α_GW is

*the*

*approximation*achieved ... Thus, either

*one*finds a different algorithm for

*k*-AF T

*cut*that does not rely

*on*Simultaneous

*Max*-

*Cut*

*and*achieves an

*approximation*

*of*α GW , or

*one*can extend

*the*

*hardness*result

*of*[11] to

*k*-AF T ...

##
###
Classification on the Computational Complexity of Spin Models
[article]

2019
*
arXiv
*
pre-print

In this note, we provide a unifying framework to investigate

arXiv:1911.04122v1
fatcat:b3sefrnk5ff2lbp7e2j2qpr2pm
*the*computational complexity*of*classical spin models*and*give*the*full classification*on*spin models in terms*of*system dimensions, randomness ... We conclude by a brief discussion*on**the*picture when quantum computation*and*quantum complexity theory are included. ... Acknowledgment: We thank Hong Yao*and*Zi-Xiang Li for useful discussions. ...##
###
The Minimum Vulnerability Problem
[chapter]

2012
*
Lecture Notes in Computer Science
*

We revisit

doi:10.1007/978-3-642-35261-4_41
fatcat:sy5uck5irbc4xdduwnazvb6geu
*the*problem*of*finding*k*paths with a minimum number*of*shared edges between two vertices*of*a graph. An edge is called shared if*it*is used in more than*one**of**the**k*paths. ... While*the*problem is NP-*hard*,*and*even*hard*to*approximate*to within an O(log n) factor, we show that*the*problem is polynomially solvable when*k*is a constant. ... Our results are mainly based*on*a clever use*of**max*-flow min-*cut*duality. ...##
###
The Minimum Vulnerability Problem

2014
*
Algorithmica
*

We revisit

doi:10.1007/s00453-014-9927-z
fatcat:q64tdoecxfbhzhx7xdf7ymtbli
*the*problem*of*finding*k*paths with a minimum number*of*shared edges between two vertices*of*a graph. An edge is called shared if*it*is used in more than*one**of**the**k*paths. ... While*the*problem is NP-*hard*,*and*even*hard*to*approximate*to within an O(log n) factor, we show that*the*problem is polynomially solvable when*k*is a constant. ... Our results are mainly based*on*a clever use*of**max*-flow min-*cut*duality. ...##
###
Primal-dual approximation algorithms for integral flow and multicut in trees

1997
*
Algorithmica
*

*It*is shown that both

*the*maximum integral multicommodity flow

*and*

*the*minimum multicut problem are NP-

*hard*

*and*

*MAX*SNP-

*hard*

*on*trees, although

*the*maximum integral flow can be computed in polynomial time ... We present an efficient algorithm that computes a multicut

*and*integral flow such that

*the*weight

*of*

*the*multicut is at most twice

*the*value

*of*

*the*flow. ... We wish to thank Clyde Momna

*and*Alex Schaffer for providing references for

*the*tree-representable set systems. ...

##
###
Approximability of the Minimum Bisection Problem: An Algorithmic Challenge
[chapter]

2002
*
Lecture Notes in Computer Science
*

We survey some recent results

doi:10.1007/3-540-45687-2_4
fatcat:k7art7ukvrgshkquf65he4juha
*on**the*complexity*of*computing*approximate*solutions for instances*of**the*Minimum Bisection problem*and*formulate some very intriguing*and*still open questions about*the**approximability*... status*of*that problem. ... Fernandez de la Vega, Ravi Kannan,*and*Claire Kenyon for many stimulating discussions. ...##
###
An O(log k) Approximate Min-Cut Max-Flow Theorem and Approximation Algorithm

1998
*
SIAM journal on computing (Print)
*

*It*is shown that

*the*minimum

*cut*ratio is within a factor

*of*O(log

*k*)

*of*

*the*maximum concurrent flow for

*k*-commodity flow instances with arbitrary capacities

*and*demands. ... An algorithm for finding a

*cut*with ratio within a factor

*of*O(log

*k*)

*of*

*the*maximum concurrent flow,

*and*thus

*of*

*the*optimal min

*cut*ratio, is presented. ... They show a 2 − 2

*k*

*approximation*algorithm.

*Of*related interest is

*the*beautiful

*approximate*

*max*-

*cut*algorithm given by Goemans

*and*Williamson [13] . ...

##
###
The label cut problem with respect to path length and label frequency

2016
*
Theoretical Computer Science
*

Given a graph with labels defined

doi:10.1016/j.tcs.2016.08.006
fatcat:2lvdwbzounhpplvtcfkt4zcf5e
*on*edges*and*a source-sink pair (s, t),*the*Label s-t*Cut*problem asks for a minimum number*of*labels such that*the*removal*of*edges with these labels disconnects s*and*... Furthermore, we give (i) an O * (c*k*) time parameterized algorithm for Label s-t*Cut*with l*max*bounded from above, where parameter*k*is*the*number*of*labels in a solution,*and*c is a constant with l ... after Theorem 2.1),*and*improve*the*presentation*of**the*paper. ...##
###
A Bundle Approach for SDPs with Exact Subgraph Constraints
[article]

2019
*
arXiv
*
pre-print

Computational experiments

arXiv:1902.05345v1
fatcat:ra4ckvzbdvaklpulpc5hzxhfdm
*on**the**Max*-*Cut*, stable set*and*coloring problem show*the*efficiency*of*this approach. ... We suggest a partial Lagrangian*dual*,*and*exploit*the*fact that*its*evaluation decomposes into two independent subproblems. ... So*it*is possible to*approximate*z*k*mc , z*k*ss*and*z*k*c by forcing only some subgraphs*of*order*k*to be exact. ...##
###
The prize-collecting generalized steiner tree problem via a new approach of primal-dual schema

2006
*
Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm - SODA '06
*

We also consider

doi:10.1145/1109557.1109626
fatcat:fly6chxikjdw5c4gl3as75f76q
*the**k*-forest problem, a generalization*of**k*-MST*and**k*-Steiner tree,*and*we show that in spite*of*these problems for which there are constant factor*approximation*algorithms,*the**k*-forest ... We note that*k*-forest*and*prize-collecting version*of*Generalized Steiner Tree are closely related to each other, since*the*latter is*the*Lagrangian relaxation*of**the*former. ... However, such a simple dynamic-programming approach does not work for*the*PCGST problem*on*trees,*and*we reduce*the*problem to an instance*of**max*flow/min*cut*to solve*it*polynomially (see Appendix D). ...
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