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Multiway cuts in node weighted graphs

Naveen Garg, Vijay V. Vazirani, Mihalis Yannakakis
2004 Journal of Algorithms  
A (2 -2/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.  ... 
doi:10.1016/s0196-6774(03)00111-1 fatcat:jxfpih7nezhebbrj5q7p2bcxzq

Minimal multicut and maximal integer multiflow: A survey

Marie-Christine Costa, Lucas Létocart, Frédéric Roupin
2005 European Journal of Operational Research  
Most 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.  ... 
doi:10.1016/j.ejor.2003.10.037 fatcat:pxc4bhuvefe6di7abxmy6l57x4

Flows with unit path capacities and related packing and covering problems

Maren Martens, Martin Skutella
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  ... 
doi:10.1007/s10878-009-9225-x fatcat:on5fprvm6batpbvh54qdpsj7jq

A Quantitative Geometric Approach to Neural Network Smoothness [article]

Zi Wang, Gautam Prakriya, Somesh Jha
2022 arXiv   pre-print
By adopting this framework, we can immediately obtain several theoretical results, including 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.  ... 
arXiv:2203.01212v1 fatcat:3dxlxwunjfbeba2tskdvwtueca

Canonical dual approach to solving the maximum cut problem

Zhenbo Wang, Shu-Cherng Fang, David Y. Gao, Wenxun Xing
2012 Journal of Global Optimization  
This paper presents a canonical 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  ... 
doi:10.1007/s10898-012-9881-8 fatcat:uzuvgc24zvdojnrog7wec56xp4

Fault Tolerant Max-Cut [article]

Keren Censor-Hillel and Noa Marelly and Roy Schwartz and Tigran Tonoyan
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  ... 
arXiv:2105.01138v1 fatcat:nd6yd2t4jrflzieve2logxyoeq

Classification on the Computational Complexity of Spin Models [article]

Shi-Xin Zhang
2019 arXiv   pre-print
In this note, we provide a unifying framework to investigate 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.  ... 
arXiv:1911.04122v1 fatcat:b3sefrnk5ff2lbp7e2j2qpr2pm

The Minimum Vulnerability Problem [chapter]

Sepehr Assadi, Ehsan Emamjomeh-Zadeh, Ashkan Norouzi-Fard, Sadra Yazdanbod, Hamid Zarrabi-Zadeh
2012 Lecture Notes in Computer Science  
We revisit 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.  ... 
doi:10.1007/978-3-642-35261-4_41 fatcat:sy5uck5irbc4xdduwnazvb6geu

The Minimum Vulnerability Problem

Sepehr Assadi, Ehsan Emamjomeh-Zadeh, Ashkan Norouzi-Fard, Sadra Yazdanbod, Hamid Zarrabi-Zadeh
2014 Algorithmica  
We revisit 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.  ... 
doi:10.1007/s00453-014-9927-z fatcat:q64tdoecxfbhzhx7xdf7ymtbli

Primal-dual approximation algorithms for integral flow and multicut in trees

N. Garg, V. V. Vazirani, M. Yannakakis
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.  ... 
doi:10.1007/bf02523685 fatcat:vthvisrmcrh5nkcvwnh7kqkhay

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

Marek Karpinski
2002 Lecture Notes in Computer Science  
We survey some recent results 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.  ... 
doi:10.1007/3-540-45687-2_4 fatcat:k7art7ukvrgshkquf65he4juha

An O(log k) Approximate Min-Cut Max-Flow Theorem and Approximation Algorithm

Yonatan Aumann, Yuval Rabani
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] .  ... 
doi:10.1137/s0097539794285983 fatcat:i6bjolss7bf4lf4tng53dmckma

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

Peng Zhang, Bin Fu
2016 Theoretical Computer Science  
Given a graph with labels defined 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.  ... 
doi:10.1016/j.tcs.2016.08.006 fatcat:2lvdwbzounhpplvtcfkt4zcf5e

A Bundle Approach for SDPs with Exact Subgraph Constraints [article]

Elisabeth Gaar, Franz Rendl
2019 arXiv   pre-print
Computational experiments 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.  ... 
arXiv:1902.05345v1 fatcat:ra4ckvzbdvaklpulpc5hzxhfdm

The prize-collecting generalized steiner tree problem via a new approach of primal-dual schema

Mohammad Taghi Hajiaghayi, Kamal Jain
2006 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm - SODA '06  
We also consider 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).  ... 
doi:10.1145/1109557.1109626 fatcat:fly6chxikjdw5c4gl3as75f76q
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