On real and on bad Linear Programs Günter M. Ziegler.
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1 On real and on bad Linear Programs Günter M. Ziegler
2 On real and on bad LPs... but when they had taken their seats Her Majesty turned to the president and resumed.... was he nevertheless as bad as he was painted? Or more to the point, and she took up her soup spoon,
3 On real and on bad LPs... but when they had taken their seats Her Majesty turned to the president and resumed.... was he nevertheless as bad as he was painted? Or more to the point, and she took up her soup spoon, was he as good? Alan Bennett: The Uncommon Reader, 2007
4 On real and on bad LPs Manfred Padberg, Linear Optimization and Extensions, 1995
5 On real LPs On bad 3D LPs On RandomEdge on KM-cubes
6 On real LPs... based on joint work with Stefan Fischer (1998) and with Dietmar Weber (1999)
7 On real LPs
8 On real LPs afiro.lp: 32 variables 8 equations
9 On real LPs afiro.lp: 32 variables 8 equations dimension: 24
10 On real LPs afiro.lp: 32 variables 8 equations dimension: explicit inequality constraints 32 non-negativities
11 On real LPs afiro.lp: 32 variables 8 equations dimension: explicit inequality constraints 32 non-negativities 29 facets
12 On real LPs \Problem name: afiro.lp Minimize COST: X X X X X39 Subject To R09: - X01 + X02 + X03 = 0 R10: X01 + X04 = 0 X05: X01 <= 80 X21: - X X14 <= 0 R12: - X06 - X07 - X08 - X09 + X14 + X15 = 0 R13: X X X X09 + X16 = 0 X17: X06 - X10 <= 80 X18: X07 - X11 <= 0 X19: X08 - X12 <= 0 X20: X09 - X13 <= 0 R19: - X22 + X23 + X24 + X25 = 0 R20: X22 + X26 = 0 X27: X22 <= 500 X44: - X X36 <= 0 R22: X X X X31 + X38 = 0 R23: X28 + X29 + X30 + X31 - X36 + X37 + X39 = 44 X40: X28 - X32 <= 500 X41: X29 - X33 <= 0 X42: X30 - X34 <= 0 X43: X31 - X35 <= 0 X45: X X X X13 - X X X X X35 <= 0 X46: - X X22 <= 0 X47: - X X X X X31 <= 0 X48: X01 - X24 <= 0 X49: X X X X09 - X37 <= 0 X50: X04 + X26 <= 310 X51: X16 + X38 <= 300 End
13 On real LPs: afiro.lp afiro.lp: dimension: facets
14 On real LPs: afiro.lp afiro.lp: dimension: facets 1654 vertices 78 degenerate vertices minimal vertex degree = 24 maximal vertex degree = 39 average vertex degree = 24.71
15 On real LPs: afiro.lp afiro.lp: dimension: facets 1654 vertices 78 degenerate vertices minimal vertex degree = 24 maximal vertex degree = 39 average vertex degree = edges horizontal edges (more than half of the edges horizontal!)
16 On real LPs: afiro.lp afiro.lp: dimension: facets 1654 vertices 78 degenerate vertices minimal vertex degree = 24 maximal vertex degree = 39 average vertex degree = edges horizontal edges (more than half of the edges horizontal!) maximal vertex: unique, degree 39 minimal vertex: 4 of them, 2-face
17 On real LPs: afiro.lp afiro.lp: dimension: facets 1654 vertices 78 degenerate vertices minimal vertex degree = 24 maximal vertex degree = 39 average vertex degree = edges horizontal edges (more than half of the edges horizontal!) maximal vertex: unique, degree 39 minimal vertex: 4 of them, 2-face graph distance minimal vertices to maximal vertex: 2 graph diameter: 5 (Hirsch conjecture sharp!)
18 On real LPs: Pictures??
19 On real LPs: afiro.lp 200 afiro
20 On real LPs: afiro.lp random1 random
21 On real LPs: boeing1.lp boeing
22 On real LPs: boeing2.lp boeing
23 On real LPs: fit1d.lp fit1d
24 On real LPs: kb2.lp 0 kb
25 On real LPs: sc105.lp 4000 sc
26 On bad 3D LPs... joint work with Volker Kaibel, Rafael Mechtel and Micha Sharir (SIAM J. Comp. 2005)
27 On bad 3D LPs
28 On bad 3D LPs Geometry vs. Combinatorial Model [Steinitz 1922]: 3-polytope P planar, 3-connected graph G (no loops, no parallel edges)
29 On bad 3D LPs Geometry vs. Combinatorial Model [Mihalisin & Klee 2000]: v max v max v min ϕ 3-polytopal graph G v min 3-polytope P generic linear function ϕ : R 3 R acyclic orientation with unique sink in every face (AOF) three disjoint monotone paths from v max to v min
30 On bad 3D LPs Geometry vs. Combinatorial Model [Mihalisin & Klee 2000]: v max v max v min ϕ 3-polytopal graph G v min 3-polytope P generic linear function ϕ : R 3 R acyclic orientation with unique sink in every face (AOF) three disjoint monotone paths from v max to v min
31 On bad 3D LPs: RandomEdge RandomEdge takes a step to an improving neighbor chosen uniformly at random: v max v start v min
32 On bad 3D LPs: RandomEdge RandomEdge takes a step to an improving neighbor chosen uniformly at random: v max v start v min Theorem: The worst-case running time of RandomEdge on a 3D polytope with n facets is bounded by n RandomEdge(n) n.
33 On bad 3D LPs: RandomEdge For n 12, we enumerated 3-connected cubic graphs with n faces using plantri by [Brinkmann & McKay]... and all the abstract objective functions on each of these to see what worst-case examples look like: v max v start v min
34 On bad 3D LPs: RandomEdge the backbone : v 0 = v min v max v k 1 v k 2 v k 3 v 2 v 1
35 On bad 3D LPs: RandomEdge the backbone : v 0 = v min v max v k 1 v k 2 v k 3 v 2 v 1 a configuration: v i,start v i,min
36 On bad 3D LPs: RandomEdge the backbone : v 0 = v min v max v k 1 v k 2 v k 3 v 2 v 1 a configuration: v i,start flow costs per configuration: 8 facets per configuration: = RandomEdge(n)/n v i,min
37 On bad 3D LPs: RandomEdge a worse configuration: 1897 flow costs per configuration: 128 facets per configuration: = RandomEdge(n)/n
38 On bad 3D LPs: RandomEdge Recursion formula for the expected number of pivot steps E(v) starting from v : E(v) = δ + (v) u:(v,u) δ + (v) E(u), v max v start v min
39 On bad 3D LPs: RandomEdge Recursion formula for the expected number of pivot steps E(v) starting from v : E(v) = δ + (v) u:(v,u) δ + (v) E(u), v start 0 v max v min
40 On bad 3D LPs: RandomEdge Recursion formula for the expected number of pivot steps E(v) starting from v : E(v) = δ + (v) u:(v,u) δ + (v) E(u), v start 1 0 v max v min
41 On bad 3D LPs: RandomEdge Recursion formula for the expected number of pivot steps E(v) starting from v : E(v) = δ + (v) u:(v,u) δ + (v) E(u), 3 v start v max v min
42 On bad 3D LPs: RandomEdge Recursion formula for the expected number of pivot steps E(v) starting from v : E(v) = δ + (v) u:(v,u) δ + (v) E(u), 5 3 v start v max v min
43 On bad 3D LPs: RandomEdge Recursion formula for the expected number of pivot steps E(v) starting from v : E(v) = δ + (v) u:(v,u) δ + (v) E(u), v start v max v min
44 On bad 3D LPs: RandomEdge Recursion formula for the expected number of pivot steps E(v) starting from v : E(v) = δ + (v) u:(v,u) δ + (v) E(u), v start v max v min 13 4
45 On bad 3D LPs: RandomEdge Recursion formula for the expected number of pivot steps E(v) starting from v : E(v) = δ + (v) u:(v,u) δ + (v) E(u), v start v max v min 13 4
46 On bad 3D LPs: RandomEdge Theorem: The worst-case running time of RandomEdge on a 3D polytope with n facets is bounded by n RandomEdge(n) n. Proof of Upper bound: Complicated, LP-optimized induction based on the numbers of 1- and 2-vertices ahead by case analysis of possible local structure.
47 On RandomEdge on KM-cubes... joint work with Bernd Gärtner (Combinatorica 1998)
48 RandomEdge on KM-cubes [Klee & Minty 1972]: min x n : 0 x 1 1, εx i 1 x i 1 εx i 1 for 2 i n.
49 RandomEdge on KM-cubes
50 RandomEdge on KM-cubes [Klee & Minty 1972] Geometry: deformed product [Amenta & Z. 1998] [Sanyal & Z. 2010]
51 RandomEdge on KM-cubes [Klee & Minty 1972] Geometry: deformed product [Amenta & Z. 1998] [Sanyal & Z. 2010]... not a projective cube, cf. [Gritzmann & Klee 1993]
52 RandomEdge on KM-cubes [Klee & Minty 1972] Combinatorial model:
53 RandomEdge on KM-cubes [Klee & Minty 1972] Combinatorial model:
54 RandomEdge on KM-cubes [Klee & Minty 1972] Combinatorial model:
55 RandomEdge on KM-cubes [Klee & Minty 1972] Combinatorial model:
56 RandomEdge on KM-cubes [Klee & Minty 1972] Combinatorial model:
57 RandomEdge on KM-cubes [Klee & Minty 1972] Combinatorial model:
58 RandomEdge on KM-cubes [Klee & Minty 1972] Combinatorial model:
59 RandomEdge on KM-cubes Recursion for running time of RandomEdge: E(v) = δ + (v) u:(v,u) δ + (v) E(u),
60 RandomEdge on KM-cubes Recursion for running time of RandomEdge: E(v) = δ + (v) u:(v,u) δ + (v) E(u), Conjectures/claims to deal with: E(KM n ) = O(n log 2 n) [Kelly 1981] E(KM n ) = O(n) [Papadimitriou & Steiglitz 1982] Worst starting vertex is [Kelly 1981]
61 RandomEdge on KM-cubes Theorem: [Gärtner, Henk & Z. 1998] n 2 4 ln n E(KM n).32 n 2
62 RandomEdge on KM-cubes Theorem: [Gärtner, Henk & Z. 1998] n 2 4 ln n E(KM n).32 n 2 Theorem: [Balogh & Pemantle 2007] E(KM n ) = Θ(n 2 )
63 RandomEdge on KM-cubes Theorem: [Gärtner, Henk & Z. 1998] For n = 18, the worst starting vertex is not , / *** ROUNDBF turned on to increase accuracy #
64 RandomEdge on KM-cubes Theorem: [Gärtner, Henk & Z. 1998] For n = 18, the worst starting vertex is not , but / #
65 RandomEdge on KM-cubes Conclusions: RandomEdge is charming and simple, but awful to analyze.
66 RandomEdge on KM-cubes Conclusions: RandomEdge is charming and simple, but awful to analyze. We don t understand really bad linear programs, e.g. in dimension 4.
67 RandomEdge on KM-cubes Conclusions: RandomEdge is charming and simple, but awful to analyze. We don t understand really bad linear programs, e.g. in dimension 4. Experiments help.
68 On real and on bad LPs
69 On real and on bad LPs Manfred Padberg, Linear Optimization and Extensions, 1995
70
71 Discrete & Computational Geometry V O L U M E Volume 45 Number 1 January 2011 DCG Discrete & Computational Geomet ry An International Journal of Mathematics and Computer Science 4 5 N U M B E R 1 J A N U A R Y Springer 454 ISSN (1) (2011) Special Issue: Xxxxx Xxxxx Xxxxxx Xxxxxx Xxxxxxx Founding Editors Jacob E. Goodman Richard Pollack Edited by Herbert Edelsbrunner János Pach Günter M. Ziegler
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