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Min Cost Flow: Polynomial Algorithms
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-negativev1
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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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1G(x)
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 AlgorithmInitial 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
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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.
a
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d
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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 algorithmWhile 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
i
b(i)
j
b(j)Cij, uij
xij
i
b(i)
(i,j)Cij,
xij
j
b(j)-uij 0,
rij
• 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.
N1N2
+
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.N1
N2
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
