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

2021

., 2010] proved

doi:10.4230/lipics.itp.2021.25
fatcat:eonhznahkzcg3nrl6pwn7yr4ay
*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. ...##
###
Quantum Max-flow/Min-cut

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*. ...

##
###
Capacities of repeater-assisted quantum communications
[article]

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*= ...

##
###
Relational ⋆-Liftings for Differential Privacy

2018
*
Logical Methods in Computer Science
*

By

doi:10.23638/lmcs-15(4:18)2019
fatcat:br7pgyikijce5obdwca6emnymq
*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. ...##
###
On Ergodicity, Infinite Flow, and Consensus in Random Models

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 ...

##
###
Bit Threads and Holographic Entanglement

2016
*
Communications in Mathematical Physics
*

In

doi:10.1007/s00220-016-2796-3
fatcat:r3qrdsifffeadf6dgeu77z53la
*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. ...##
###
Optimal nonparametric testing of Missing Completely At Random, and its connections to compatibility
[article]

2022
*
arXiv
*
pre-print

Given

arXiv:2205.08627v1
fatcat:vruavyyv75fnboiondnvgu7o6a
*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*...##
###
Bounds for multi-end communication over quantum networks

2019
*
Quantum Science and Technology
*

Within

doi:10.1088/2058-9565/ab3f66
fatcat:hjb7ia35tbgprbdrdvcb57mbf4
*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 ...##
###
Dynamic Traffic Assignment for Electric Vehicles
[article]

2022
*
arXiv
*
pre-print

properties

arXiv:2207.04454v1
fatcat:edjoey3glnca5gavqrzlijcqsa
*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*. ...##
###
The Price of Privacy in Untrusted Recommendation Engines
[article]

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*. ...

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

1997
*
IEEE Transactions on Information Theory
*

These conditions are derived by extending

doi:10.1109/18.568702
fatcat:3d7hhoy2w5gxjmhnzqwrpc3i44
*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. ...##
###
The notion of a rational convex program, and an algorithm for the arrow-debreu Nash bargaining game

2012
*
Journal of the ACM
*

We show that

doi:10.1145/2160158.2160160
fatcat:ymvyez7cqnd3zhh3oy22mooxim
*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). ...##
###
The Notion of a Rational Convex Program, and an Algorithm for the Arrow-Debreu Nash Bargaining Game
[chapter]

2012
*
Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms
*

We show that

doi:10.1137/1.9781611973099.78
dblp:conf/soda/Vazirani12
fatcat:yioctv4rgbcpllc4dsflagfrrm
*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). ...##
###
Axiomatic Construction of Hierarchical Clustering in Asymmetric Networks
[article]

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% . ...

##
###
Recurrence of Markov chain traces
[article]

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. ...

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