Integer Maximum Flow (Maximum Flow)

Description

Maximum flow problems involve finding a feasible flow through a flow network that is maximum. In this variant, the capacities must be integers.

Related Problems

Generalizations: Non-Integer Maximum Flow

Subproblem: Unweighted Maximum Flow

Related: st-Maximum Flow, Non-integer Maximum Flow, Minimum-Cost Flow, All-Pairs Maximum Flow, Maximum Local Edge Connectivity

Parameters

V: number of vertices

E: number of edges

U: maximum edge capacity

Table of Algorithms

Name	Year	Time	Space	Approximation Factor	Model	Reference
Ford & Fulkerson	1955	$O(E^{2}U)$	$O(E)$	Exact	Deterministic	Time & Space
Dinitz	1970	$O(V^{2}E)$	$O(E)$	Exact	Deterministic	Time & Space
Edmonds & Karp	1972	$O(E^{2}LogU)$	$O(E)$	Exact	Deterministic	Time & Space
Karzanov	1974	$O(V^3)$	$O(V^2)$	Exact	Deterministic	Time & Space
Galil & Naamad	1980	$O(VELo{g}^{2}V)$	$O(E)$	Exact	Deterministic	Time & Space
Dantzig	1951	$O(V^{2}EU)$	$O(VE)$?	Exact	Deterministic
Dinitz (with dynamic trees)	1973	$O(VELogU)$	$O(E)$	Exact	Deterministic	Time
Cherkassky	1977	$O(V^2{E}^{0.5})$	$O(E)$	Exact	Deterministic	Time & Space
Sleator & Tarjan	1983	$O(VELogV)$	$O(E)$	Exact	Deterministic	Time
Goldberg & Tarjan	1986	$O(VELog(V^2/E))$	$O(E)$	Exact	Deterministic	Time
Ahuja & Orlin	1987	$O(VE+V^{2}LogU)$	$O(ELogU)$	Exact	Deterministic	Time
Goldberg & Rao	1997	$O(E^{1.5}Log(V^2/E)LogU)$	$O(V+E)$	Exact	Deterministic	Time
Goldberg & Rao	1997	$O(V^{0.66}ELog(V^2/E)LogU)$	$O(V+E)$	Exact	Deterministic	Time
Ahuja et al.	1987	$O(VELog(V(LogU)$^{0.5} / E))		Exact	Deterministic	Time
MKM Algorithm	1978	$O(V^3)$	$O(E)$	Exact	Deterministic	Time & Space
Galil	1978	$O(V^{(}5/3)$E^({2}/{3}))	$O(E)$	Exact	Deterministic	Time & Space
Shiloach	1981	$O(V^3\ast log(V)$/p)	$O(E)$	Exact	Parallel	Time
Gabow	1985	$O(VE\ast logU)$	$O(E)$	Exact	Deterministic	Time
Lee, Sidford	2014	$O(E\ast sqrt(V)$*log^{2}(U)*polylog(E, V, log(U))	$O(E)$	Exact	Deterministic	Time
Madry	2016	$O(E^{(}10/7)$U^({1}/{7})polylog(V, E, log U))	$O(E)$	Exact	Deterministic	Time
Kathuria, Liu, Sidford	2020	$O(E^{(}1+o(1)$)/sqrt(eps))	$O(E)$ or $O(V^2)$ ?	1+eps	Deterministic	Time
Kathuria, Liu, Sidford	2020	$O(E^{(}4/3+o(1)$)U^({1}/{3}))	$O(E)$ or $O(V^2)$ ?	Exact	Deterministic	Time
Brand et al	2021	$O((E+V^{1.5})$log(U)polylog(V, E, log U))	$O(E)$	Exact	Randomized	Time
Gao, Liu, Peng	2021	$O(E^{(}3/2-1/328)$*log(U)*polylog(E))	$O(E)$	Exact	Deterministic	Time
Chen et al	2022	$O(E^{(}1+o(1)$)*log(U))	$O(E)$	Exact	Deterministic	Time
Goldberg & Rao (Parallel)	1997	$O(V^{1.66}log(V)$ log(U))	$O(V^2)$	Exact	Parallel	Time & Space
Goldberg & Rao (Parallel)	1997	$O(E^{0.5}Vlog(V)$ log(U))	$O(V^2)$	Exact	Parallel	Time & Space

Time Complexity Graph

Space Complexity Graph

Space-Time Tradeoff Improvements

References/Citation

https://arxiv.org/abs/2203.00671,