Minimum-Cost Flow Network
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1 Minimum-Cost Flow Network Smoothed flow network: G = (V,E) balance values: b : V Z costs: c : E R capacities: u : E N 2 1/2 3/1-1 1/3 3/2 3/1 1 3/1 1/2-2 cost/capacity 2/ 17
2 Smoothed 2 1 1/2 3/ /3 1 3/2 3/ /1 1/ flow: f : E R capacity constraints: e E : f(e) u(e) Kirchhoff s law: v V : b(v) = out(v) in(v) 3/17
3 Smoothed 2 1 1/2 3/ /3 1 3/2 3/ /1 1/ flow: f : E R capacity constraints: e E : f(e) u(e) Kirchhoff s law: v V : b(v) = out(v) in(v) Goal: min flow f e E f(e) c(e) 3/17
4 Smoothed 2 1/2 3/ /3 3/2 3/ /1 1/2-2 flow: f : E R capacity constraints: e E : f(e) u(e) Kirchhoff s law: v V : b(v) = out(v) in(v) Goal: min flow f e E f(e) c(e) 3/17
5 Short History Smoothed Pseudo-Polynomial Algorithms: Out-of-Kilter algorithm [Minty 6, Fulkerson 61] Cycle Canceling algorithm Successive Shortest Path algorithm 4/17
6 Short History Smoothed Pseudo-Polynomial Algorithms: Out-of-Kilter algorithm [Minty 6, Fulkerson 61] Cycle Canceling algorithm Successive Shortest Path algorithm Polynomial Time Algorithms: Capacity Scaling algorithm [Edmonds and Karp 72] Cost Scaling algorithm 4/17
7 Short History Smoothed Pseudo-Polynomial Algorithms: Out-of-Kilter algorithm [Minty 6, Fulkerson 61] Cycle Canceling algorithm Successive Shortest Path algorithm Polynomial Time Algorithms: Capacity Scaling algorithm [Edmonds and Karp 72] Cost Scaling algorithm Strongly Polynomial Algorithms: Tardos algorithm [Tardos 85] Minimum-Mean Cycle Canceling algorithm Network Simplex algorithm Enhanced Capacity Scaling algorithm [Orlin 93] 4/17
8 Theory vs. Practice Smoothed Theory Practice Fastest algorithm: Enhanced Capacity Scaling Fastest algorithm: Network Simplex 5/17
9 Theory vs. Practice Smoothed Theory Practice Fastest algorithm: Enhanced Capacity Scaling Fastest algorithm: Network Simplex Successive Shortest Path: exponential in worst case Minimum-Mean Cycle Canceling: strongly polynomial Successive Shortest Path much faster than Minimum-Mean Cycle Canceling 5/17
10 Smoothed Reason for Gap between Theory and Practice Adversary Worst-case complexity is too pessimistic! There are artificial worst-case inputs. These inputs, however, do not occur in practice. I will trick your algorithm! 6/17
11 Smoothed Reason for Gap between Theory and Practice Adversary Worst-case complexity is too pessimistic! There are artificial worst-case inputs. These inputs, however, do not occur in practice. This phenomenon occurs also for many other problems and algorithms. I will trick your algorithm! 6/17
12 Smoothed Reason for Gap between Theory and Practice Adversary Worst-case complexity is too pessimistic! There are artificial worst-case inputs. These inputs, however, do not occur in practice. This phenomenon occurs also for many other problems and algorithms. I will trick your algorithm! Goal Find a more realistic performance measure that is not just based on the worst case. 6/17
13 Smoothed Smoothed Observation: In worst-case analysis, the adversary is too powerful. Idea: Let s weaken him! Input model: Adversarial choice of flow network Adversarial real arc capacities u e and node balance values b v Adversarial densities f e : [,1] [,φ] Arc costs c e independently drawn according to f e 7/17
14 Smoothed Smoothed Observation: In worst-case analysis, the adversary is too powerful. Idea: Let s weaken him! Input model: Adversarial choice of flow network Adversarial real arc capacities u e and node balance values b v Adversarial densities f e : [,1] [,φ] Arc costs c e independently drawn according to f e Randomness models, e.g., measurement errors, numerical imprecision, rounding,... 7/17
15 Smoothed Smoothed Worst-case : max ce T Smoothed : max fe E[T] 8/17
16 Smoothed Smoothed Worst-case : max ce T Smoothed : max fe E[T] f(x) φ = 1: Average-case analysis 1 1 x 8/17
17 Smoothed Smoothed Worst-case : max ce T Smoothed : max fe E[T] f(x) φ 1/φ 1 x 8/17
18 Smoothed Smoothed Worst-case : max ce T Smoothed : max fe E[T] f(x) φ : Worst-case analysis φ x 1 x 8/17
19 Smoothed Smoothed Worst-case : max ce T Smoothed : max fe E[T] φ : Worst-case analysis f(x) φ x 1 x 8/17
20 Initial Transformation Smoothed Successive Shortest Path algorithm 2 1/2 3/1-1 1/3 3/2 3/1 1 3/1 1/2-2 9/17
21 Initial Transformation Smoothed Successive Shortest Path algorithm /2 1/2 3/1 /1 3 1/3 3/2 3/1-3 /1 3/1 1/2 /2 9/17
22 Augmenting Steps Smoothed Successive Shortest Path algorithm 3 /2 2 /1 1/2 3/1 2 21/3 3/2 3/1 2 3/1 1/2 /1 2 /2-3 path length: 3, augmenting flow value: 2 1/17
23 Augmenting Steps Smoothed Successive Shortest Path algorithm /2-1/2 3/1 /1 1 1/1-1/2 3/2 3/1-1 /1 3/1-1/2 /2 update residual network 1/17
24 Augmenting Steps Smoothed Successive Shortest Path algorithm 1 /2 1 /1-1/2 1/1 1-1/2 3/2 3/1 1 3/1 3/1 1-1/2 /1 1 /2-1 path length: 5, augmenting flow value: 1 1/17
25 Augmenting Steps Smoothed Successive Shortest Path algorithm /2-1/2-3/1 /1 1/2-1/1 3/2 3/1 /1-3/1-1/2 /2 update residual network 1/17
26 Resulting Flow Smoothed Successive Shortest Path algorithm 3 /2 2 1 /1 1/2 3/ /3 3/2 3/ /1 1/2 /1 1 2 /2-3 flow cost: 11, flow value: 3 11/17
27 Resulting Flow Smoothed Successive Shortest Path algorithm 2 1/2 3/ /3 3/2 3/ /1 1/2-2 flow cost: 11, flow value: 3 11/17
28 Results Smoothed Theorem (Upper Bound) In expectation, the SSP algorithm requires O(mnφ) iterations and has a running time of O(mnφ(m+nlogn)). 12/17
29 Results Smoothed Theorem (Upper Bound) In expectation, the SSP algorithm requires O(mnφ) iterations and has a running time of O(mnφ(m+nlogn)). Theorem (Lower Bound) There are smoothed instances on which the SSP algorithm requires Ω(m min{n,φ} φ) iterations in expectation. upper bound tight for φ = Ω(n) 12/17
30 Useful Properties Observations Discretization Flow Reconstruction Lemma The distances from the source to any node increase monotonically. cost initial solution: empty flow value 13/17
31 Useful Properties Observations Discretization Flow Reconstruction Lemma The distances from the source to any node increase monotonically. cost slope = path length value length value after 1 iteration value 13/17
32 Useful Properties Observations Discretization Flow Reconstruction Lemma The distances from the source to any node increase monotonically. cost after 2 iterations value 13/17
33 Useful Properties Observations Discretization Flow Reconstruction Lemma The distances from the source to any node increase monotonically. cost after 3 iterations value 13/17
34 Useful Properties Observations Discretization Flow Reconstruction Lemma The distances from the source to any node increase monotonically. cost after 4 iterations value 13/17
35 Useful Properties Observations Discretization Flow Reconstruction Lemma The distances from the source to any node increase monotonically. cost value after 5 iterations #iterations = #distinct slopes 13/17
36 Useful Properties Observations Discretization Flow Reconstruction Lemma The distances from the source to any node increase monotonically. cost value after 5 iterations #iterations = #distinct slopes Lemma Every intermediate flow is optimal for its flow value. 13/17
37 Counting the Number of Slopes Observations Discretization Flow Reconstruction slope = augmenting path length (,n] n 14/17
38 Counting the Number of Slopes Observations Discretization Flow Reconstruction slope = augmenting path length (,n] = k I l, I l = n k l=1 n 14/17
39 Counting the Number of Slopes Observations Discretization Flow Reconstruction slope = augmenting path length (,n] = k I l, I l = n k l=1 n = #slopes = k l=1 #slopes I l 14/17
40 Counting the Number of Slopes Observations Discretization Flow Reconstruction slope = augmenting path length (,n] = k I l, I l = n k l=1 n = E[#slopes]= k E[#slopes I l ] l=1 14/17
41 Counting the Number of Slopes Observations Discretization Flow Reconstruction slope = augmenting path length (,n] = k I l, I l = n k l=1 n = E[#slopes]= k E[#slopes I l ] l=1 k Pr[ slope I l ] l=1 14/17
42 Counting the Number of Slopes Observations Discretization Flow Reconstruction slope = augmenting path length (,n] = k I l, I l = n k l=1 n = E[#slopes]= k E[#slopes I l ] l=1 k Pr[ slope I l ] l=1 Main Lemma d : ε : Pr[ slope (d,d +ε]] = O(mφε) 14/17
43 Counting the Number of Slopes Observations Discretization Flow Reconstruction slope = augmenting path length (,n] = k I l, I l = n k l=1 n = E[#slopes]= k E[#slopes I l ] Main Lemma l=1 k Pr[ slope I l ] l=1 = O(mnφ) d : ε : Pr[ slope (d,d +ε]] = O(mφε) 14/17
44 Flow Reconstruction Observations Discretization Flow Reconstruction Main Lemma d : ε : Pr[ slope (d,d +ε]] = O(mφε) cost value 15/17
45 Flow Reconstruction Observations Discretization Flow Reconstruction Main Lemma d : ε : Pr[ slope (d,d +ε]] = O(mφε) d - slope threshold cost d > d > d d d value 15/17
46 Flow Reconstruction Observations Discretization Flow Reconstruction Main Lemma d : ε : Pr[ slope (d,d +ε]] = O(mφε) d - slope threshold F - flow at breakpoint cost d > d d > d F d value 15/17
47 Flow Reconstruction Observations Discretization Flow Reconstruction Main Lemma d : ε : Pr[ slope (d,d +ε]] = O(mφε) d - slope threshold cost F - flow at breakpoint d > d P - next augmenting path e - empty arc of P in G f d c(p) > d d F value 15/17
48 Flow Reconstruction Observations Discretization Flow Reconstruction Main Lemma d : ε : Pr[ slope (d,d +ε]] = O(mφε) d - slope threshold cost F - flow at breakpoint d > d P - next augmenting path e - empty arc of P in G f d c(p) > d d F value slope (d,d +ε] c(p) (d,d +ε] 15/17
49 Flow Reconstruction Observations Discretization Flow Reconstruction Main Lemma d : ε : Pr[ slope (d,d +ε]] = O(mφε) d - slope threshold cost F - flow at breakpoint d > d P - next augmenting path e - empty arc of P in G f d c(p) > d d F value slope (d,d +ε] c(p) (d,d +ε] Goal: Reconstruct F and P without knowing c e 15/17
50 Principle of Deferred Decisions Observations Discretization Flow Reconstruction Main Lemma d : ε : Pr[ slope (d,d +ε]] = O(mφε) Phase 1: Reveal all c e for e e. Assume this suffices to uniquely identify F and P. cost d > d c(p) > d d F d value 16/17
51 Principle of Deferred Decisions Observations Discretization Flow Reconstruction Main Lemma d : ε : Pr[ slope (d,d +ε]] = O(mφε) Phase 1: Reveal all c e for e e. Assume this suffices to uniquely identify F and P. cost d Phase 2: > d c(p) > d Pr[c(P) (d,d +ε]] d F = Pr[c(e) (z,z +ε]] φε, d value where z is fixed if c e for e e is fixed. 16/17
52 Flow Reconstruction Observations Discretization Flow Reconstruction Case 1: e forward arc Set c (e) = 1 and for all e e set c (e ) = c(e ). Run SSP with modified costs c. 17/17
53 Flow Reconstruction Observations Discretization Flow Reconstruction Case 1: e forward arc Set c (e) = 1 and for all e e set c (e ) = c(e ). Run SSP with modified costs c. cost d > d c(p) > d d F d value 17/17
54 Flow Reconstruction Observations Discretization Flow Reconstruction Case 1: e forward arc Set c (e) = 1 and for all e e set c (e ) = c(e ). Run SSP with modified costs c. cost c d c > d d > d F d value 17/17
55 Flow Reconstruction Observations Discretization Flow Reconstruction Case 1: e forward arc Set c (e) = 1 and for all e e set c (e ) = c(e ). Run SSP with modified costs c. cost c d c > d d > d F d value F is the same for c and c 17/17
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