Min Cost Flow: Polynomial Algorithms

Overview• Recap:

• Min Cost Flow, Residual Network• Potential and Reduced Cost

• Polynomial Algorithms• Approach• Capacity Scaling

• Successive Shortest Path Algorithm Recap

• Incorporating Scaling• Cost Scaling

• Preflow/Push Algorithm Recap

• Incorporating Scaling• Double Scaling Algorithm - Idea

Min Cost Flow - Recap• G=(V,E) is a directed graph

• Capacity u(v,w) for every

• Balances: For every we will have a number b(v)

• Cost c(v,w) for every

• can assume non-negative

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Min Cost Flow - Recap fdsfds

Compute feasible flow with min cost ij ijx c

Residual Network - Recap

• We replace each arc by two arcs and .

• The arc has cost and residual capacity .

• The arc has cost and residual capacity .

• The residual network consists only of arcs with positive residual capacity.

• A For a feasible flow x, G(x) is the residual network that corresponds to the flow x.

Reduced Cost - Recap

• Let be a potential function of the nodes in V.• Reduced cost: of an edge (u,v) :

Reduced Cost - Recap

• Theorem (Reduced Cost Optimality):A feasible solution x* is an optimal solution of the minimum cost flow problem Exists Node potential function that satisfies the following reduced cost optimality conditions:

• Idea: No negative cycles in G(x*)

Min Cost Flow: Polynomial Algorithms

Approach

• We have seen several algorithm for the MCF problem, but none polynomial – in logU, logC.

• Idea: Scaling!• Capacity/Flow values• Costs• both

• Next Week: Algorithms with running time independent of logU, logC• Strongly Polynomial• Will solve problems with irrational data

Capacity Scaling

Successive Shortest Path - Recap

• Pseudo-Flow – Flow which doesn’t necessarily satisfy balance constraints

• Define imbalance function e for node i:

• Define E and D as the sets of excess and Deficit nodes.

• Observation:

Successive Shortest Path - Recap

• Lemma: Suppose pseudoflow x has potential , satisfying reduced cost optimality.Letting d be the vector of distances from some node s in G(x) with as the length of (i,j), then:• Potential also satisfies optimality conditions• for edges in shortest paths from s.

• Idea: triangle inequality

Successive Shortest Path - Recap

• Algorithm:• Maintain pseudoflow x with pot. , satisfying

reduced cost optimality.• Iteratively, choose node s with positive

excess – e(s) > 0, until there none.• compute shortest path distances d from s

in G(x) with as the length of (i,j).• Update . also satisfies optimality

conditions• Push as much flow as possible along path

from s to some node t with e(t) < 0. (retains optimality conditions w.r.t. ).

Successive Shortest Path - Recap

• Algorithm Complexity:• Assuming integrality, at most nU iterations.• In each iteration, compute shortest paths,

• Using Dijkstra, bounded by O(m+nlogn) per iteration

Capacity Scaling - Scheme

• Successive Shortest Path Algorithm may push very little in each iteration.

• Fix idea: use scaling• Modify algorithm to push units of flow

• Ignore edges with residual capacity < • until there is no node with excess or no

node with deficit • Decrease by factor 2, and repeat

• Until < 1.

Definitions

• Define - nodes with access at least - nodes with deficit at least

• The -residual network G(x, ) is the subgraph of G(x) with edges with residual capacity at least .

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Definitions

• Define - nodes with access at least - nodes with deficit at least

• The -residual network G(x, ) is the subgraph of G(x) with edges with residual capacity at least .

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G(x, 3)

Main Observation in Algorithm

• Observation: Augmentation of units must start at a node in S(), along a path in G(x, ), and end at a node in T().

• In the -phase, we find shortest paths in G(x, ), and augment over them.

• Thus, edges in G(x, ), will satisfy the reduced optimality conditions.

• We will consider edges with less residual capacity later.

Initializing phases

• In later -phases, “new” edges with residual capacity rij < 2 may be introduced to G(x, ).

• We ignored them until now, so possibly

• Solution: saturate such edges.

i j

Capacity Scaling Algorithm

Capacity Scaling Algorithm

Initial values. 0 pseudoflow and potential (optimal!)Large enough

Capacity Scaling Algorithm

In beginning of -phase, fix optimality condition on “new” edges with resid. Cap. rij < 2 by saturation

Capacity Scaling Algorithm

augment path in G(x,) from node inS() to node in T()

Capacity Scaling Algorithm - Correctness

• Inductively, The algorithm maintains a reduced cost optimal flow x w.r.t. in G(x, ).• Initially this is clear.• At the beginning of the -phase, “new” arcs

s.t. rij < 2 are introduced, may not satisfy optimality.Saturation of edges such that < 0 suffices, since the reversal satisfies .

• Augmenting flow in shortest path procedure in G(x, ) retains optimality

Capacity Scaling Algorithm - Correctness

• When =1, G(x, ) = G(x).• The algorithm ends with , or .

• By integrality, we have a feasible flow.

Capacity Scaling Algorithm - Assumption

• We assume path from k to l in G(x, ) exists.• And we assume we can compute shortest

distances from k to rest of nodes.

• Quick fix: initially, add dummy node D with artificial edges (1,D) and (D,1) with infinite capacity and very large cost.

Capacity Scaling Algorithm – Complexity

• The algorithm has O(log U) phases.• We analyze each phase separately.

Capacity Scaling Algorithm – Phase Complexity

• Let’s assess the sum of excesses at the beginning.

• Observe that when -phase begins, either, or .

Thus the sum of excesses (= sum of deficits) is less than

DE

Capacity Scaling Algorithm – Phase Complexity – Cont.

• Saturation of edges in the beginning of the phase saturate edges with rij < 2. This may add at most to the sum of excesses

Capacity Scaling Algorithm – Phase Complexity – Cont.

• Thus the sum of excesses is less than • Therefore, at most augmentations can

be performed in a phase.

• In total: per phase.

Capacity Scaling Algorithm –Complexity

• per phase.

Cost Scaling

Approximate Optimality

• A pseudoflow (flow) x is said to be -optimal for some >0 if for some pot. , for every edge (i,j) in G(x).

Approximate Optimality Properties

• Lemma: For a min. cost flow problem with integer costs, any feasible flow is C-optimal. In addition, if , then any -optimal feasible flow is optimal.

• Proof:• Part 1 is easy: set .• Part 2: for any cycle W in G(x),

.applying integrality, it follows . The lemma follows.

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Algorithm Strategy

• The previous lemma suggests the following strategy:1. Begin with feasible flow x and , which is C-

optimal2. Iteratively improve from and -optimal flow

(x, ) to an /2-optimal flow (x’, ), until < 1/n.

• We discuss two methods to implement the underlying improvement procedure.

• The first is a variation of the Preflow Push Algorithm.

Preflow Push Recap

Distance Labels

• Distance Labels Satisfy:• d(t) = 0, d(s) = n• d(v) ≤ d(w) + 1 if r(v,w) > 0

• d(v) is at most the distance from v to t in the residual network.

• s must be disconnected from t …

Terms

• Nodes with positive excess are called active.

• Admissible arc in the residual graph:

w

v

d(v) = d(w) + 1

The preflow push algorithm

While there is an active node {

pick an active node v and push/relabel(v)

}

Push/relabel(v) {

If there is an admissible arc (v,w) then {

push = min {e(v) , r(v,w)} flow from v to w

} else {

d(v) := min{d(w) + 1 | r(v,w) > 0} (relabel)

}

Running Time

• The # of relabelings is (2n-1)(n-2) < 2n2

• The # of saturating pushes is at most 2nm

• The # of nonsaturating pushes is at most 4n2m – using potential Φ = Σv active d(v)

Back to Min Cost Flow…

Applying Preflow Push’s technique

• Our motivation was to find a method to improve an -optimal flow (x, ) to an /2-optimal flow (x’, ).

• We use Push Preflow’s technique:

• Labels: (i)• Admissible edge in residual network:

if • Relabel: Increase (i) by

j

i𝑐 i , j𝜋

𝑐 i , j❑ +𝜋 ( j )− 𝜋 (i )<0

Initialization

• We first transform the input -optimal flow (x, ) to an /2-optimal pseduoflow (x’, ).

• This is easy• Saturate edges with negative reduced

cost.• Clear flow from edges with positive

reduced cost.

v w-10

Obtain a - optimal pseudoflow

Push/relabel until no active nodes exist

Correctness

• Lemma 1: Let x be pseudo-flow, and x’ a feasible flow.

Then, for every node v with excess in x, there exists a path P in G(x) ending at a node w with deficit, and its reversal is a path in G(x’).

• Proof: Look at the difference x’-x, and observe the underlying graph (edges with negative difference are reversed).

Lemma 1: Proof Cont.

• Proof: Look at the difference x’-x, and observe the underlying graph (edges with negative difference are reversed).

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Lemma 1: Proof Cont.

• Proof: Look at the difference x’-x, and observe the underlying graph (edges with negative difference are reversed).

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• There must be a node with deficit reachable, otherwise x’ isn’t feasible

Correctness (cont)

Corollary: There is an outgoing residual arc incident with every active vertex

Corollary: So we can push/relabel as long as there is an active vertex

Correctness – Cont.

• Lemma 2: The algorithm maintains an /2-optimal pseudo-flow (x’, ),

• Proof: By induction on the number of operations.

• Initially, optimal preflow.

Correctness – Cont.

• Push on edge where may introduce (j,i) to G(x), but that’s OK: .

Correctness – Cont.

• Relabel operation at node i occurs when there’s no admissible arc - for every edge emanating from i in the residual network.

• Relabel decreases by , so it satisfies -optimality condition.

• Relabel increases , so it clearly still satisfies -optimality condition.

Correctness (cont)

• Lemma 3: When (and if) the algorithm stops the preflow is an /2-optimal feasible flow (x’, ).

Proof:

It is a feasible flow since there is no active vertex.

It is /2-optimal since ’ satisfies /2-optimality conditions.

Complexity

• Lemma : a node is relabeled at most 3n times.

Lemma 2 – Cont.

• Proof: Let x’ be the -opt. input flow, f and x’ an intermediate -opt. pseudo-flow. Let , ’ be the respective potentials. Take the path P from v to w as described in Lemma 1.

-opt. gives:

-opt. gives:

Lemma 2 – Cont.

−3 ε2∙ 𝑙𝑒𝑛 (𝑃 )≤ (𝜋 (𝑤 )−𝜋 ′ (𝑤 ) )− (𝜋 (𝑣 )−𝜋 ′ (𝑣 ) )

Nodes never become deficit nodes, so is untouched

Lemma 2 – Cont.

3 ε2∙𝑙𝑒𝑛 (𝑃 )≥ 𝜋 (𝑣 )−𝜋 ′ (𝑣 )

Complexity Analysis (Cont.)

• Lemma: The # of saturating pushes is at most O(nm).

• Proof: same as in Preflow Push.

Complexity Analysis – non Saturating Pushes

• Def: The admissible network is the graph of admissible edges.

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Complexity Analysis – non Saturating Pushes

• Def: The admissible network is the graph of admissible edges.

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Complexity Analysis – non Saturating Pushes

• Def: The admissible network is the graph of admissible edges.

• Lemma: The admissible network is acyclic throughout the algorithm.

• Proof: induction.

Complexity Analysis – non Saturating Pushes – Cont.

• Lemma: The # of nonsaturating pushes is O(n2m).

• Proof: Let g(i) be the # of nodes reachable from i in admissible network

• Let Φ = Σi active g(i)

Complexity Analysis – non Saturating Pushes – Cont.

• Φ = Σi active g(i)

• By acyclicity, decreases (by at least one) by every nonsaturating push

i j

Complexity Analysis – non Saturating Pushes – Cont.

• Φ = Σi active g(i)

• Initially g(i) = 1.

• Increases by at most n by a saturating push : total increase O(n2m)

• Increases by each relabeling by at most n (no incoming edges become admissible): total increase < O(n3)

• O(n2m) non saturating pushes

Cost Scaling Summary

• Total complexity O(n2mlog(nC)).• Can be improved using ideas used to

improve preflow push

Double Scaling (If Time Permits)

Double Scaling Idea

• Use capacity scaling to implement improve-approximation

• New ‘tricks’ used:• Transform network to bipartite

network on nodes .• Augment on admissible paths

(generalization of shortest paths and push on edges)

• Find admissible paths greedily (DFS), and relabel potentials in retreats.

Network Transformation

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b(i)

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b(j)Cij, uij

xij

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b(i)

(i,j)Cij,

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b(j)-uij0,

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• Bipartite network on nodes . For Convenience, shall refer to V as N1, and to E as N2.• All edges from N1 to N2.

• No capacities

Improve approximation Initialization

• Recall: receives (x, ) which is -optimal.• If we reset x’:=0, then G(x’) = G.• No capacities – G is a subgraph of G(x).

• -optimality means for all edges in G(x), and in particular in G.

• Adding to for every j in N2, obtains an optimal pseudoflow.

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Capacity Scaling - Scheme

• While S() • Pick a node k S().• search for an admissible path using DFS

• On retreat from node i, relabel: increase (i) by

• Upon finding a deficit node, augment units of flow in path.

• Decrease by factor 2, and repeat• Until < 1.

Double Scaling - Correctness

• Assuming algorithm ends, immediate, since we augment along admissible path.

• (Residual path from excess to node to deficit node always exists – see cost scaling algorithm)

• We relabel when indeed no outgoing admissible edges.

Double Scaling - Complexity

• O(log U) phases.• In each phase, each augmentation clears a

node from S() and doesn’t introduce a new one.

so O(m) augmentations per phase.

Double Scaling Complexity – Cont.

• In each augmentation, • We find a path of length at most 2n

(admissible network is acyclic and bipartite)• Need to account retreats.

Double Scaling Complexity – Cont.

• In each retreat, we relabel.• Using above lemma, potential cannot grow

more than O(l), where l is a length of path to a deficit node.

• Since graph is bipartite, l = O(n).• So in all augmentations, O(n (m+n)) = O(mn)

retreats.

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Double Scaling Complexity – Cont.

• To sum up:• Implemented improve-approximation using

capacity scaling• O(log U) phases.

• O(m) augmentations per phase.• O(n) nodes in path

• O(mn) node retreats in total.

• Total complexity: O(log(nC) log(U) mn)

Thank You