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A New Approach to the Maximum-Flow ProblemAndrew V. Goldberg, Robert E. Tarjan
Presented by Andrew Guillory
Outline
Background Definitions Push-Relabel Algorithm Correctness / Termination Proofs Sequential Implementation Dynamic Tree Implementation
Maximum Flow Problem
Classic problem in operations research Many problems reduce to max flow
Maximum cardinality bipartite matching Maximum number of edge disjoint paths Minimum cut (Max-Flow Min-Cut Theorem)
Machine learning applications Structured Prediction, Dual Extragradient and Bregman
Projections (Taskar, Lacoste-Julien, Jordan JMLR 2006) Local Search for Balanced Submodular Clusterings
(Narasimhan, Bilmes, IJCAI 2007)
Relation to Optimization
Special case of submodular function minimization
Special case of linear programming Integer edge capacities permit integer
maximum flows (constructive proof)
History of Algorithms
Augmenting Paths based algorithmsFord-Fulkerson (1962) O(mU)Edmonds-Karp (1969) O(nm3)… O(n3) O(nmlog(n)) O(nmlog(U))
Push-Relabel based algorithmsGoldberg (1985) O(n3)Goldberg and Tarjan (1986) O(nmlog(n2/m))Ahuja and Orlin O(nm + n2log(U))
Outline
Background Definitions Push-Relabel Algorithm Correctness / Termination Proofs Sequential Implementation Dynamic Tree Implementation
Definitions
Graph G = (V, E) |V| = n |E| = m
G is a flow network if it hassource s and sink tcapacity c(v,w) for each edge (v,w) in Ec(v,w) = 0 for (v,w) not in E
Definitions (continued)
A flow f on G is a real value function on vertex pairs f(v,w) <= c(v,w) for all (v,w) f(v,w) = -f(w,v)∑uf(u,v) = 0 for all v in V - {s,t}
Value of a flow |f| is ∑vf(v,t) Maximum flow is a flow of maximum value
Definitions (continued again)
A preflow f on G is a real value function on vertex pairs f(v,w) <= c(v,w) for all (v,w) f(v,w) = -f(w,v)∑uf(u,v) >= 0 for all v in V - {s}
Flow excess e(v) = ∑uf(u,v) Intuition: flow into a vertex can exceed flow out
Outline
Background Definitions Push-Relabel Algorithm Correctness / Termination Proofs Sequential Implementation Dynamic Tree Implementation
Intuition
Starting with a preflow, push excess flow closer towards sink
If excess flow cannot reach sink, push it backwards to source
Eventually, preflow becomes a flow and in fact the maximum flow
Residual Graph
Residual capacity rf(v, w) of a vertex pair is c(v, w) – f(v, w)
If v has positive excess and (v,w) has residual capacity, can push
δ = min(e(v), rf(v, w)) flow from v to w
Edge (v,w) is saturated if rf(v, w) = 0
Residual graph Gf = (V, Ef) where Ef is the set of residual edges (v,w) with rf(v, w) > 0
Labeling
A valid labeling is a function d from vertices to nonnegative integersd(s) = nd(t) = 0d(v) <= d(w) + 1 for every residual edge
If d(v) < n, d(v) is a lower bound on distance to sink
If d(v) >= n, d(v) - n is a lower bound on distance to source
Push Operation
Push(v,w)Precondition: v is active (e(v) > 0) and
rf(v, w) > 0 and d(v) = d(w) + 1
Action: Push δ = min(e(v), rf(v, w)) from v to w
f(v,w) = f(v,w) + δ; f(w,v) = f(w,v) – δ;e(v) = e(v) - δ; e(w) = e(w) + δ;
Relabel Operation
Relabel(v)
Precondition: v is active (e(v) > 0) and
rf(v, w) > 0 implies d(v) <= d(w)
Action: d(v) = min{d(w) + 1 | (v,w) in Ef}
Generic Push-Relabel Algorithm
Starting from an initial preflow
<<loop>>
While there is an active vertex
Chose an active vertex v
Apply Push(v,w) for some w or Relabel(v)
Outline
Background Definitions Push-Relabel Algorithm Correctness / Termination Proofs Sequential Implementation Dynamic Tree Implementation
Correctness
Lemma 2.1 If f is a preflow, d is a valid labeling, and v is active, either push or relabel is applicable to v
Lemma 3.1 The algorithm maintains a valid labeling d
Theorem 3.2 A flow is maximum iff there is no path from s to t in Gf (Ford and Fulkerson [7])
Correctness (continued)
Lemma 3.3 If f is a preflow and d is a valid labeling for f, there is no path from s to t in Gf
Proof by contradictionPath s, v0, v1, …, vl, t implies that
d(s) <= d(v0) + 1 <= d(v1) + 2 <= …
<= d(t) + l < nWhich contradicts d(s) = n
Correctness (continued)
Theorem 3.4 If the algorithm terminates with a valid labeling, the preflow is a maximum flow If the algorithm terminates, all vertices have
zero excess (preflow is a flow)By Lemma 3.3 the sink is not reachable from
the sourceBy Theorem 3.2 the flow is maximum
Termination
Lemma 3.5 If f is a preflow and v is an active vertex then the source is reachable from v in Gf Let S be the set of vertices reachable in Gf Suppose s is not in S For every u,w, with w in S and u not in S, f(u,w) <= 0 ∑w in S e(w) = ∑u in V, w in S f(u,w)
= ∑u not in S, w in S f(u,w) + ∑u in S, w in S f(u,w) = ∑u not in S, w in S f(u,w) <= 0
e(w) = 0 for all w in S Lemma 3.6 A vertex’s label never decreases
Termination (continued)
Lemma 3.7 At any time the label of any vertex is at most 2n – 1Only active vertex labels are changedActive vertices can reach sPath v, v0, v1, …, vl, s implies that
d(v) <= d(v0) + 1 <= d(v1) + 2 <= …
<= d(s) + l <= n + n - 1
Termination (continued)
Lemma 3.8 There are at most 2n2 labeling operationsOnly the labels corresponding to V-{s,t} may
be relabeledEach of these n – 2 labels can only increaseAt most (2n – 1) (n – 2) relabelings
Termination (continued)
Lemma 3.9 The number of saturating pushes is at most 2nmFor any pair (v,w) d(w) must increase by 2 between
saturating pushes from v to wSimilarly d(v) must increase by 2 between pushes
from w to vd(v) + d(w) >= 1 on the first saturating pushd(v) + d(w) <= 4n - 3 on the lastAt most 2n - 1 saturating pushes per edge
Termination (continued)
Lemma 3.10 The number of nonsaturating pushes is at most 4n2m Φ = ∑v d(v) where v is active
Each nonsaturating push causes Φ to decrease by at least 1 The total increase in Φ from saturating pushes is
(2n – 1) 2nm The total increase in Φ from relabeling is
(2n – 1)(n – 2) Φ is 0 initially and 0 at termination
Termination
Theorem 3.11 The algorithm terminates in O(n2m)
Total time =
# nonsaturating pushes
+ #saturating pushes
+ #relabeling operations
4n2m + 2nm + 2n2 = O(n2m)
Outline
Background Definitions Push-Relabel Algorithm Correctness / Termination Proofs Sequential Implementation Dynamic Tree Implementation
Implementation
At each step select an active vertex and apply either Push or Relabel
Problem: Determining which operation to perform and in the case of Push finding a residual edge
Solution: For each vertex maintain a list of edges which touch that vertex and a current edge
Push/Relabel Operation
Push/Relabel(v)
Precondition: v is active
Action:
If Push(v,w) is applicable to current edge (v,w) then Push(v,w)
Else if (v,w) is not the last edge advance current edge
Else reset the current edge and Relabel(v)
Push/Relabel Operation
Lemma 4.1 The push/relabel operation does a relabeling only when relabeling is applicable
Theorem 4.2 The push/relabel implementation runs in O(nm) time plus O(1) time per nonsaturating push operation
O(n3) bound
We can select vertices in arbitrary order Certain vertex selection strategies give
O(n3) boundsMaximum distance method (proved here)First-in, first-out method (proved in paper)Wave method
Maximum distance method
Theorem The maximum distance method performs at most 4n3 nonsaturating pushes Consider D = maxx d(x) where x is active D only increases because of relabeling D increases at most 2n2 times D starts at 0 and ends nonnegative D changes at most 4n2 times There is at most one nonsaturating push per node per
value of D
Maximum distance method
Theorem The maximum distance method runs in time O(n3) using the push/relabel implementationPrevious theorem and Theorem 4.2
First-In First-Out Method
Discharge()Precondition: Queue is not emptyAction: Push/Relabel the vertex v at the front of the queue
until e(v) = 0 or d(v) increasesIf w becomes active during the Push/Relabel add
w to the back of the QueueIf v is still active add v to the back of the Queue
First-In First-Out Method
Lemma 4.3 The number of passes over the queue is at most 4n2
Proof very similar to the proof of O(n3) bound for maximum distance method
Corollary 4.4 The number of non saturating pushes is at most 4n3
One per vertex per pass
First-In First-Out Method
Theorem 4.5 The first-in, first-out method runs in O(n3) timeCorollary 4.4 and Theorem 4.2
Outline
Background Definitions Push-Relabel Algorithm Correctness / Termination Proofs Sequential Implementation Dynamic Tree Implementation
Dynamic Tree Implementation
Intuition: Maintain trees such that connections between child nodes and parent nodes correspond to edges in the residual graph which permit push operations
Send flow up branches of trees Queue contains trees with active roots
Send Operation
Send(v)
Precondition: v is active
Action:
While v is not the root of its tree and e(v) > 0
Send flow up the tree from v
Cut the tree along the bottleneck edge(s)
Tree-Push/Relabel Operation
Tree-Push/Relabel(v)Precondition: v is a root of a tree and activeAction:1) If Push is applicable to current edge (v,w):
1a) If we can combine v and w’s trees without making the tree > size k, make w v’s parent and Send(v)1b) Else Push(v,w) and Send(w)
2) Else2a) If (v,w) isn’t the last edge advance the edge2b) Else cut v’s children out of the tree, relabel v, and reset the current edge
Dynamic Tree Implementation
Lemma 5.1 The dynamic tree algorithm runs in O(nm log k) time plus O(log k) time per addition of a vertex to the queue Trees are kept at most size k by 1a) Tree operations take time O(log k) Each Tree-Push/Relabel operation takes O(1) tree
operations plus O(1) tree operations per cut Relabeling takes time O(nm) There are O(nm) cuts Tree-Push/Relabel is performed O(nm) times plus
once per addition to the Queue
Dynamic Tree Implementation
Lemma 5.2 The number of times a vertex is added to the queue is O(nm + n3/k) A vertex is added only after d(v) changes or e(v)
increases from zero d(v) changes at most n2 times e(v) increased only in 1a) or 1b) Number of vertices added to queue in 1a) or 1b) is the
number of cuts performed (2nm) plus one per occurrence of each subcase
Dynamic Tree Implementation
Lemma 5.2 The number of times a vertex is added to the queue is O(nm + n3/k) (Continued)Subcase 1a) occurs at most 2nm times (the
number of links)Subcase 1b) occurs at most 2nm times when
it causes a cut and at most 2nm times when the push from v to w is saturating
Dynamic Tree Implementation
Lemma 5.2 The number of times a vertex is added to the queue is O(nm + n3/k) (Continued) Subcase 1b) is nonsaturating if it doesn’t cause a cut
or a saturating push from v to w In a nonsaturating occurrence of 1b), either v or w’s
tree is large (size greater than k/2) There are at most 2n/k large trees in the queue at the
beginning of a pass If the large tree has changed since the beginning of
this pass, charge the operation to the cut / link that changed it (at most one per link, 2 per cuts, 6nm)
Else charge the operation to that tree (at most 2n/k per pass, 2n2 passes, 4n3/k)
Dynamic Tree Implementation
Theorem 5.3 The dynamic tree algorithm runs in O(nm log(n2/m)) time if k is chosen to be n2/m