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A Mechanized Proof of the Max-Flow Min-Cut Theorem for Countable Networks

Andreas Lochbihler, Liron Cohen, Cezary Kaliszyk
2021
., 2010] proved the max-flow min-cut theorem for countable networks, namely that in every countable network with finite edge capacities, there exists a flow and a cut such that the flow saturates all outgoing  ...  We also provide a simpler proof for networks where the total outgoing capacity of all vertices other than the source is finite. This proof is based on the max-flow min-cut theorem for finite networks.  ...  Acknowledgements We thank Ron Aharoni and Eli Berger for helping to clarify the weaknesses in the original proofs. S. Reza Sefidgar and the anonymous reviewers helped to improve the presentation.  ... 
doi:10.4230/lipics.itp.2021.25 fatcat:eonhznahkzcg3nrl6pwn7yr4ay

Quantum Max-flow/Min-cut

Shawn X. Cui, Michael H. Freedman, Or Sattath, Richard Stong, Greg Minton
2016 Journal of Mathematical Physics  
The classical max-flow min-cut theorem describes transport through certain idealized classical networks. We consider the quantum analog for tensor networks.  ...  ., when the capacity on each edge is some power of a fixed integer, the quantum max-flow is proved to equal the quantum min-cut.  ...  In Section 2, we give a review of the classical max-flow min-cut theorem and Menger's theorem. In Section 3 and 4, we provide two versions of quantum analogues of the max-flow min-cut theorem.  ... 
doi:10.1063/1.4954231 fatcat:hitmk5nvfbhn3nsg3wdgvrupai

Capacities of repeater-assisted quantum communications [article]

Stefano Pirandola
2017 arXiv   pre-print
For the converse part, we adopt a teleportation-inspired simulation of a quantum network which leads to upper bounds in terms of the relative entropy of entanglement.  ...  Depending on the type of routing (single or multiple), optimal strategies corresponds to finding the widest path or the maximum flow in the quantum network.  ...  Now, from the classical max-flow min-cut theorem, we know that the maximum value of the flow in the network |R| max is equal to the capacity of the minimum cut [59, 66] , i.e., we may write |R| max =  ... 
arXiv:1601.00966v4 fatcat:ra3vshgxonaihkb2zh7riq6kea

Relational ⋆-Liftings for Differential Privacy

Gilles Barthe, Thomas Espitau, Justin Hsu, Tetsuya Sato, Pierre-Yves Strub
2018 Logical Methods in Computer Science  
By a maximum flow/minimum cut argument, we show that -liftings are equivalent to Sato's lifting over discrete distributions.  ...  Our work unifies all known notions of approximate lifting, yielding cleaner properties, more general constructions, and more precise composition theorems for both styles of lifting, enabling richer proofs  ...  Acknowledgments We thank the anonymous reviewers for their helpful suggestions.  ... 
doi:10.23638/lmcs-15(4:18)2019 fatcat:br7pgyikijce5obdwca6emnymq

On Ergodicity, Infinite Flow, and Consensus in Random Models

Behrouz Touri, Angelia Nedic
2011 IEEE Transactions on Automatic Control  
The central result is the infinite flow theorem establishing the equivalence between the infinite flow and the ergodicity for a class of independent random models, where the matrices in the model have  ...  We consider the ergodicity and consensus problem for a discrete-time linear dynamic model driven by random stochastic matrices, which is equivalent to studying these concepts for the product of such matrices  ...  INTRODUCTION There is an evidence of growing number of applications in decentralized control of networked agents, as well as social and other networks where the consensus is used as a mechanism for decentralized  ... 
doi:10.1109/tac.2010.2091174 fatcat:m2hcxkjq3vamzm34xzxtlvkz3e

Bit Threads and Holographic Entanglement

Michael Freedman, Matthew Headrick
2016 Communications in Mathematical Physics  
In a mathematical appendix, we review the max flow-min cut theorem on networks and on Riemannian manifolds, and prove in the network case that the set of max flows varies Lipshitz continuously in the network  ...  Using the max flow-min cut principle, a theorem from network theory, we rewrite the RT formula in a way that does not make reference to the minimal surface.  ...  A Max-flow min-cut on manifolds We put the max flow/min cut principle into a larger context.  ... 
doi:10.1007/s00220-016-2796-3 fatcat:r3qrdsifffeadf6dgeu77z53la

Optimal nonparametric testing of Missing Completely At Random, and its connections to compatibility [article]

Thomas B Berrett, Richard J Samworth
2022 arXiv   pre-print
Given a set of incomplete observations, we study the nonparametric problem of testing whether data are Missing Completely At Random (MCAR).  ...  We define an incompatibility index as a natural measure of ease of detectability, establish its key properties, and show how it can be computed exactly in some cases and bounded in others.  ...  Vlach (1986) gives a halfspace representation for P 0, * S using the max-flow min-cut theorem for a single-commodity flow through a simpler network; since there is no general max-flow min-cut theorem  ... 
arXiv:2205.08627v1 fatcat:vruavyyv75fnboiondnvgu7o6a

Bounds for multi-end communication over quantum networks

Stefano Pirandola
2019 Quantum Science and Technology  
Within a quantum network, other problems and features may appear when we move from the basic unicast setting of single-sender/single-receiver to more complex multi-end scenarios, where multiple senders  ...  These bounds cannot be surpassed by the most general adaptive protocols of quantum network communication.  ...  This work has been supported by the EPSRC via the 'UK Quantum Communications HUB' (EP/M013472/1) and by the European Union via the project 'Continuous Variable Quantum Communications' (CiViQ, no 820466  ... 
doi:10.1088/2058-9565/ab3f66 fatcat:hjb7ia35tbgprbdrdvcb57mbf4

Dynamic Traffic Assignment for Electric Vehicles [article]

Lukas Graf, Tobias Harks, Prashant Palkar
2022 arXiv   pre-print
properties of the network loading as a result of injecting flow into walks.  ...  There are three key modeling-ingredients for obtaining this existence result: * We introduce a walk-based definition of dynamic traffic flows which allows for cyclic routing behavior as a result of recharging  ...  There is a constant M ≥ 0 such that for any h ∈ C all edge flows of the network loading corresponding to h are supported on [0, M ]. Proof.  ... 
arXiv:2207.04454v1 fatcat:edjoey3glnca5gavqrzlijcqsa

The Price of Privacy in Untrusted Recommendation Engines [article]

Siddhartha Banerjee, Nidhi Hegde, Laurent Massoulié
2014 arXiv   pre-print
The techniques we develop for bounding mutual information may be of broader interest.  ...  However, the information-scarce regime, where each user rates only a vanishing fraction of items, is found to require a fundamentally different approach both for lower bounds and algorithms.  ...  To this end, we use the max flow-min cut theorem.  ... 
arXiv:1207.3269v2 fatcat:soq3oknygvcvlm2rvexi5prln4

Stability analysis of quota allocation access protocols in ring networks with spatial reuse

L. Georgiadis, W. Szpankowski, L. Tassiulas
1997 IEEE Transactions on Information Theory  
These conditions are derived by extending a technique developed by two of us for some networks of queues satisfying a monotonicity property.  ...  For the case of two nodes, we provide closed-form expressions for the essential stability region.  ...  Theorem 3 Let us define T G max and T G min as in (22) and (25) respectively.  ... 
doi:10.1109/18.568702 fatcat:3d7hhoy2w5gxjmhnzqwrpc3i44

The notion of a rational convex program, and an algorithm for the arrow-debreu Nash bargaining game

Vijay V. Vazirani
2012 Journal of the ACM  
We show that the convex program for ADNB is a logarithmic RCP, but unlike other known members of this class, it is non-total. Our main result is a combinatorial, polynomial time algorithm for ADNB.  ...  Finally, we present a number of interesting questions that the new notion of RCP raises.  ...  Proof : Since the Invariant holds and f is a max-flow in N (p), the cut (s, G ∪ B ∪ t) is saturated by f , and hence by x · f in N (xp).  ... 
doi:10.1145/2160158.2160160 fatcat:ymvyez7cqnd3zhh3oy22mooxim

The Notion of a Rational Convex Program, and an Algorithm for the Arrow-Debreu Nash Bargaining Game [chapter]

Vijay V. Vazirani
2012 Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms  
We show that the convex program for ADNB is a logarithmic RCP, but unlike other known members of this class, it is non-total. Our main result is a combinatorial, polynomial time algorithm for ADNB.  ...  Finally, we present a number of interesting questions that the new notion of RCP raises.  ...  Proof : Since the Invariant holds and f is a max-flow in N (p), the cut (s, G ∪ B ∪ t) is saturated by f , and hence by x · f in N (xp).  ... 
doi:10.1137/1.9781611973099.78 dblp:conf/soda/Vazirani12 fatcat:yioctv4rgbcpllc4dsflagfrrm

Axiomatic Construction of Hierarchical Clustering in Asymmetric Networks [article]

Gunnar Carlsson, Facundo Mémoli, Alejandro Ribeiro, Santiago Segarra
2014 arXiv   pre-print
The paper also develops algorithms for the computation of hierarchical clusters using matrix powers on a min-max dioid algebra and studies the stability of the methods proposed.  ...  The goal is to study methods for the determination of hierarchical clusters, i.e., a family of nested partitions indexed by a connectivity parameter, induced by the given dissimilarity structures.  ...  The cut at δ U 2 corresponds to a migrational flow of 10 . 45% .  ... 
arXiv:1301.7724v2 fatcat:mfyoakowqvhs7g2kiru4kwjygy

Recurrence of Markov chain traces [article]

Itai Benjamini, Jonathan Hermon
2019 arXiv   pre-print
For d=2 we present a relevant example due to Gady Kozma, while the general statement for transient graphs is obtained by proving that for every transient irreducible Markov chain on a countable state space  ...  We exploit recent results in potential theory of non-reversible Markov chains in order to extend their result to the non-reversible setup.  ...  Acknowledgements The examples from the proofs of Theorems 3 and 4 are due to Gady Kozma. We are tremendously grateful to him for allowing us to present them.  ... 
arXiv:1711.03479v4 fatcat:ong6of5xajbmxb7bokxmaawr2m
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