On “real” and on “bad”Linear Programs

Günter M. Zieglerziegler@math.fu-berlin.de

On “real” and on “bad” LPs

. . . but when they had taken their seats Her Majestyturned 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

On “real” and on “bad” LPs

. . . but when they had taken their seats Her Majestyturned 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

On “real” and on “bad” LPs

Manfred Padberg, “Linear Optimization and Extensions”, 1995

On “real” LPs

On “bad” 3D LPs

On RandomEdge on KM-cubes

On “real” LPs. . . based on joint work

with Stefan Fischer (1998)and with Dietmar Weber (1999)

On “real” LPs

On “real” LPs

afiro.lp:32 variables8 equations

dimension: 2419 explicit inequality constraints32 non-negativities29 facets

On “real” LPs

afiro.lp:32 variables8 equationsdimension: 24

19 explicit inequality constraints32 non-negativities29 facets

On “real” LPs

afiro.lp:32 variables8 equationsdimension: 2419 explicit inequality constraints32 non-negativities

29 facets

On “real” LPs

afiro.lp:32 variables8 equationsdimension: 2419 explicit inequality constraints32 non-negativities29 facets

On “real” LPs\Problem name: afiro.lp

MinimizeCOST: - 0.4 X02 - 0.32 X14 - 0.6 X23 - 0.48 X36 + 10 X39

Subject ToR09: - X01 + X02 + X03 = 0R10: - 1.06 X01 + X04 = 0X05: X01 <= 80X21: - X02 + 1.4 X14 <= 0R12: - X06 - X07 - X08 - X09 + X14 + X15 = 0R13: - 1.06 X06 - 1.06 X07 - 0.96 X08 - 0.86 X09 + X16 = 0X17: X06 - X10 <= 80X18: X07 - X11 <= 0X19: X08 - X12 <= 0X20: X09 - X13 <= 0R19: - X22 + X23 + X24 + X25 = 0R20: - 0.43 X22 + X26 = 0X27: X22 <= 500X44: - X23 + 1.4 X36 <= 0R22: - 0.43 X28 - 0.43 X29 - 0.39 X30 - 0.37 X31 + X38 = 0R23: X28 + X29 + X30 + X31 - X36 + X37 + X39 = 44X40: X28 - X32 <= 500X41: X29 - X33 <= 0X42: X30 - X34 <= 0X43: X31 - X35 <= 0X45: 2.364 X10 + 2.386 X11 + 2.408 X12 + 2.429 X13 - X25 + 2.191 X32

+ 2.219 X33 + 2.249 X34 + 2.279 X35 <= 0X46: - X03 + 0.109 X22 <= 0X47: - X15 + 0.109 X28 + 0.108 X29 + 0.108 X30 + 0.107 X31 <= 0X48: 0.301 X01 - X24 <= 0X49: 0.301 X06 + 0.313 X07 + 0.313 X08 + 0.326 X09 - X37 <= 0X50: X04 + X26 <= 310X51: X16 + X38 <= 300

End

On “real” LPs: afiro.lp

afiro.lp:dimension: 2429 facets

1654 vertices78 degenerate verticesminimal vertex degree = 24maximal vertex degree = 39average vertex degree = 24.7120433 edges11718 horizontal edges (more than half of the edges horizontal!)maximal vertex: unique, degree 39minimal vertex: 4 of them, 2-facegraph distance minimal vertices to maximal vertex: 2graph diameter: 5 (Hirsch conjecture sharp!)

On “real” LPs: afiro.lp

afiro.lp:dimension: 2429 facets

1654 vertices78 degenerate verticesminimal vertex degree = 24maximal vertex degree = 39average vertex degree = 24.71

20433 edges11718 horizontal edges (more than half of the edges horizontal!)maximal vertex: unique, degree 39minimal vertex: 4 of them, 2-facegraph distance minimal vertices to maximal vertex: 2graph diameter: 5 (Hirsch conjecture sharp!)

On “real” LPs: afiro.lp

afiro.lp:dimension: 2429 facets

1654 vertices78 degenerate verticesminimal vertex degree = 24maximal vertex degree = 39average vertex degree = 24.7120433 edges11718 horizontal edges (more than half of the edges horizontal!)

maximal vertex: unique, degree 39minimal vertex: 4 of them, 2-facegraph distance minimal vertices to maximal vertex: 2graph diameter: 5 (Hirsch conjecture sharp!)

On “real” LPs: afiro.lp

afiro.lp:dimension: 2429 facets

1654 vertices78 degenerate verticesminimal vertex degree = 24maximal vertex degree = 39average vertex degree = 24.7120433 edges11718 horizontal edges (more than half of the edges horizontal!)maximal vertex: unique, degree 39minimal vertex: 4 of them, 2-face

graph distance minimal vertices to maximal vertex: 2graph diameter: 5 (Hirsch conjecture sharp!)

On “real” LPs: afiro.lp

afiro.lp:dimension: 2429 facets

1654 vertices78 degenerate verticesminimal vertex degree = 24maximal vertex degree = 39average vertex degree = 24.7120433 edges11718 horizontal edges (more than half of the edges horizontal!)maximal vertex: unique, degree 39minimal vertex: 4 of them, 2-facegraph distance minimal vertices to maximal vertex: 2graph diameter: 5 (Hirsch conjecture sharp!)

On “real” LPs: Pictures??

On “real” LPs: afiro.lp

-1200

-1000

-800

-600

-400

-200

0

200

-500 0 500 1000 1500 2000 2500 3000 3500

afiro

On “real” LPs: afiro.lp

-0.16

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

random1random2

On “real” LPs: boeing1.lp

0

10000

20000

30000

40000

50000

60000

70000

-300 -200 -100 0 100 200 300

boeing1

On “real” LPs: boeing2.lp

-82000

-80000

-78000

-76000

-74000

-72000

-70000

-68000

-300 -250 -200 -150 -100

boeing2

On “real” LPs: fit1d.lp

-20000

-10000

0

10000

20000

30000

40000

50000

60000

-10000 0 10000 20000 30000 40000 50000 60000 70000 80000

fit1d

On “real” LPs: kb2.lp

-14000

-12000

-10000

-8000

-6000

-4000

-2000

0

-1800 -1600 -1400 -1200 -1000 -800 -600 -400 -200 0

kb2

On “real” LPs: sc105.lp

-5000

-4000

-3000

-2000

-1000

0

1000

2000

3000

4000

-50 -40 -30 -20 -10 0

sc105

On “bad” 3D LPs. . . joint work with

Volker Kaibel, Rafael Mechteland Micha Sharir

(SIAM J. Comp. 2005)

On “bad” 3D LPs

On “bad” 3D LPs

Geometry vs. Combinatorial Model [Steinitz 1922]:

3-polytope P planar, 3-connected graph G(no loops, no parallel edges)

On “bad” 3D LPs

Geometry vs. Combinatorial Model [Mihalisin & Klee 2000]:

vmin

vmax

ϕ

3-polytope Pgeneric linear functionϕ : R3 → R

vminvmax

• 3-polytopal graph G

• acyclic orientation withunique sink in every face (AOF)

• three disjoint monotone pathsfrom vmax to vmin

On “bad” 3D LPs

Geometry vs. Combinatorial Model [Mihalisin & Klee 2000]:

vmin

vmax

ϕ

3-polytope Pgeneric linear functionϕ : R3 → R

vminvmax

• 3-polytopal graph G

• acyclic orientation withunique sink in every face (AOF)

• three disjoint monotone pathsfrom vmax to vmin

On “bad” 3D LPs: RandomEdgeRandomEdge takes a step to an improving neighbor chosenuniformly at random:

vminvstart

vmax

Theorem:The worst-case running time of RandomEdge on a 3D polytopewith n facets is bounded by

1.3473 n ≤ RandomEdge(n) ≤ 1.4943 n.

On “bad” 3D LPs: RandomEdgeRandomEdge takes a step to an improving neighbor chosenuniformly at random:

vminvstart

vmax

Theorem:The worst-case running time of RandomEdge on a 3D polytopewith n facets is bounded by

1.3473 n ≤ RandomEdge(n) ≤ 1.4943 n.

On “bad” 3D LPs: RandomEdge

For n ≤ 12, we enumerated 3-connected cubic graphs with nfaces using plantri by [Brinkmann & McKay]

. . . and all the “abstract objective functions” on each ofthese . . .

. . . to see what worst-case examples look like:

vminvstartvmax

On “bad” 3D LPs: RandomEdgethe “backbone”:

v0 = vmin

v1v2vk−1 vk−2 vk−3vmax

a configuration:

vi ,min

vi ,start

4

4

622

63

3

58

flow costs per configuration:438

facets per configuration: 3 + 1

=⇒ RandomEdge(n)/n ≥ 438 · 4

≈ 1.3437

On “bad” 3D LPs: RandomEdgethe “backbone”:

v0 = vmin

v1v2vk−1 vk−2 vk−3vmax

a configuration:

vi ,min

vi ,start

4

4

622

63

3

58

flow costs per configuration:438

facets per configuration: 3 + 1

=⇒ RandomEdge(n)/n ≥ 438 · 4

≈ 1.3437

On “bad” 3D LPs: RandomEdgethe “backbone”:

v0 = vmin

v1v2vk−1 vk−2 vk−3vmax

a configuration:

vi ,min

vi ,start

4

4

622

63

3

58

flow costs per configuration:438

facets per configuration: 3 + 1

=⇒ RandomEdge(n)/n ≥ 438 · 4

≈ 1.3437

On “bad” 3D LPs: RandomEdge

a worse configuration:

flow costs per configuration:1897128

facets per configuration: 10 + 1

=⇒ RandomEdge(n)/n ≥ 18971408

≈ 1.3473.

On “bad” 3D LPs: RandomEdge

Recursion formula for the expected number of pivot steps E(v)“starting from v ”:

E(v) = 1 +1

| δ+(v)|∑

u:(v ,u)∈δ+(v)

E(u),

vmin

vstartvmax

0132

52

72

134

358

On “bad” 3D LPs: RandomEdge

Recursion formula for the expected number of pivot steps E(v)“starting from v ”:

E(v) = 1 +1

| δ+(v)|∑

u:(v ,u)∈δ+(v)

E(u),

vmin

vstartvmax

0

132

52

72

134

358

On “bad” 3D LPs: RandomEdge

Recursion formula for the expected number of pivot steps E(v)“starting from v ”:

E(v) = 1 +1

| δ+(v)|∑

u:(v ,u)∈δ+(v)

E(u),

vmin

vstartvmax

01

32

52

72

134

358

On “bad” 3D LPs: RandomEdge

Recursion formula for the expected number of pivot steps E(v)“starting from v ”:

E(v) = 1 +1

| δ+(v)|∑

u:(v ,u)∈δ+(v)

E(u),

vmin

vstartvmax

0132

52

72

134

358

On “bad” 3D LPs: RandomEdge

Recursion formula for the expected number of pivot steps E(v)“starting from v ”:

E(v) = 1 +1

| δ+(v)|∑

u:(v ,u)∈δ+(v)

E(u),

vmin

vstartvmax

0132

52

72

134

358

On “bad” 3D LPs: RandomEdge

Recursion formula for the expected number of pivot steps E(v)“starting from v ”:

E(v) = 1 +1

| δ+(v)|∑

u:(v ,u)∈δ+(v)

E(u),

vmin

vstartvmax

0132

52

72

134

358

On “bad” 3D LPs: RandomEdge

Recursion formula for the expected number of pivot steps E(v)“starting from v ”:

E(v) = 1 +1

| δ+(v)|∑

u:(v ,u)∈δ+(v)

E(u),

vmin

vstartvmax

0132

52

72

134

358

On “bad” 3D LPs: RandomEdge

Recursion formula for the expected number of pivot steps E(v)“starting from v ”:

E(v) = 1 +1

| δ+(v)|∑

u:(v ,u)∈δ+(v)

E(u),

vmin

vstartvmax

0132

52

72

134

358

On “bad” 3D LPs: RandomEdge

Theorem:The worst-case running time of RandomEdge on a 3D polytopewith n facets is bounded by

1.3473 n ≤ RandomEdge(n) ≤ 1.4943 n.

Proof of Upper bound:Complicated, LP-optimized inductionbased on the numbers of 1- and 2-vertices “ahead”by case analysis of possible local structure.

On RandomEdge on KM-cubes. . . joint work with Bernd Gärtner

(Combinatorica 1998)

RandomEdge on KM-cubes

[Klee & Minty 1972]:

min xn : 0 ≤ x1 ≤ 1,εxi−1 ≤ xi ≤ 1− εxi−1 for 2 ≤ i ≤ n.

RandomEdge on KM-cubes

RandomEdge on KM-cubes

[Klee & Minty 1972] – Geometry:

“deformed product” [Amenta & Z. 1998] [Sanyal & Z. 2010]

. . . not a projective cube, cf. [Gritzmann & Klee 1993]

RandomEdge on KM-cubes

[Klee & Minty 1972] – Geometry:

“deformed product” [Amenta & Z. 1998] [Sanyal & Z. 2010]

. . . not a projective cube, cf. [Gritzmann & Klee 1993]

RandomEdge on KM-cubes[Klee & Minty 1972] – Combinatorial model:

RandomEdge on KM-cubes[Klee & Minty 1972] – Combinatorial model:

11010100010101010111001↓

11010100001010101000110↓

11010100001010010111001↓

10101011110101101000110↓

10101011001010010111001↓

10101010110101101000110↓

. . .

RandomEdge on KM-cubes[Klee & Minty 1972] – Combinatorial model:

11010100010101010111001↓

11010100001010101000110↓

11010100001010010111001↓

10101011110101101000110↓

10101011001010010111001↓

10101010110101101000110↓

. . .

RandomEdge on KM-cubes[Klee & Minty 1972] – Combinatorial model:

11010100010101010111001↓

11010100001010101000110↓

11010100001010010111001↓

10101011110101101000110↓

10101011001010010111001↓

10101010110101101000110↓

. . .

RandomEdge on KM-cubes[Klee & Minty 1972] – Combinatorial model:

11010100010101010111001↓

11010100001010101000110↓

11010100001010010111001↓

10101011110101101000110↓

10101011001010010111001↓

10101010110101101000110↓

. . .

RandomEdge on KM-cubes[Klee & Minty 1972] – Combinatorial model:

11010100010101010111001↓

11010100001010101000110↓

11010100001010010111001↓

10101011110101101000110↓

10101011001010010111001↓

10101010110101101000110↓

. . .

RandomEdge on KM-cubes[Klee & Minty 1972] – Combinatorial model:

11010100010101010111001↓

11010100001010101000110↓

11010100001010010111001↓

10101011110101101000110↓

10101011001010010111001↓

10101010110101101000110↓

. . .

RandomEdge on KM-cubes

Recursion for running time of RandomEdge:

E(v) = 1 +1

| δ+(v)|∑

u:(v ,u)∈δ+(v)

E(u),

Conjectures/claims to deal with:

E(KMn) = O(n log2 n) [Kelly 1981]

E(KMn) = O(n) [Papadimitriou & Steiglitz 1982]

Worst starting vertex is “111111111” [Kelly 1981]

RandomEdge on KM-cubes

Recursion for running time of RandomEdge:

E(v) = 1 +1

| δ+(v)|∑

u:(v ,u)∈δ+(v)

E(u),

Conjectures/claims to deal with:

E(KMn) = O(n log2 n) [Kelly 1981]

E(KMn) = O(n) [Papadimitriou & Steiglitz 1982]

Worst starting vertex is “111111111” [Kelly 1981]

RandomEdge on KM-cubes

Theorem: [Gärtner, Henk & Z. 1998]

n2

4 ln n≤ E(KMn) ≤ .32 n2

Theorem: [Balogh & Pemantle 2007]

E(KMn) = Θ(n2)

RandomEdge on KM-cubes

Theorem: [Gärtner, Henk & Z. 1998]

n2

4 ln n≤ E(KMn) ≤ .32 n2

Theorem: [Balogh & Pemantle 2007]

E(KMn) = Θ(n2)

RandomEdge on KM-cubesTheorem: [Gärtner, Henk & Z. 1998]For n = 18, the “worst” starting vertex is not “111111111”,13616631777775844683363020123127094102813560533249201592684030464015036597749164177937892497054636200148618292482860171750895424612282188029893901861623777485911206277962056237212747724106971466780610966750343428884227862863263992240626944189474092321690578465850140349559978346803788901676121822279514510337366767357850445503507313736074535300137075905658614012808891118656316408374481670049167271660724973549966381898781024671174519123371033460330957586555415033425077727004460772400592700649563057465490900379364876075017619331015784049036558215285038489260535168237679310834438980211261828416142728106103764871709282502595038844295608788896317984140105479983862039598375987130214282003033020521051527717363707514678161308836221258641232146849278090766755354459331000620203946310574593680591647128707656257843986034927310159488243185598011054353166122873565317236418489392762400450466591101169432128401829430964496067737309072179851815110185854606923338019104432978507203552547721528956985400094903678817126979242634733

/24962914469583191566367899923960035966404195848061867674084257325351443541635707108155961226253248787890779064193993572350407540882488313462480442009188322965308175793834908350711775470723063600215807657623374362126466957217083277804322543021220684978006202697645006112925727562688499951019008352273202998698433082923670410460471087620616165741606476723494106039072694978995397627362457886096950309963504449900266130281466812904611199098539805421932601086188714264657278379296255216638772840224103572864219332581506845376087212926568647114428532255047582757200833981470518008818805688306634147913338380858098047897556191777014597833213397490582583305285642961484103228645506879443810132259225138635152335311879233430440856986173430296778743350033604723187899699712826261542556246067548824288988199191266967980956809764206176935319256985166746436097191235722924204644128666213286229783071506832436219870537040937793596789158704489369975002974080695684303196676875492248579163093862959138359780655177475508923698668685694367

*** ROUNDBF turned on to increase accuracy

54.5474439468

2#101010101010101010

RandomEdge on KM-cubesTheorem: [Gärtner, Henk & Z. 1998]For n = 18, the “worst” starting vertex is not “111111111”, but

13616684425655993868809930342829681416513745972117542059691282558789265359381405592027738235067168526490944745020032821831458348414708370359959721284479519471305069796673823323749468285051705771273197431466120774988286761021877147886424485186429677466830230656505140112567299734199737747316914047149456525668905990843395292570627973505504153803354783689366237724788621187438316245307594694578844829661290767878882426158862400116464580283005287051721523550761838202480537960286313206215370104486533179185155951187425478644148272361015209552862194206039184472942296180543643639536783031340491471882644253089850167358829961270644689650011819021006889213587618695068079416895590094361423303966234997939240089858719693593594374858225574919421676770475805291256813831033322745894364021986396433890467502668449390681018004891572703264811384653350462845288862763661943911510871076870345393271182732960163244935717489369649608477918928915351385767044907310095700518582096630823172413069321398119400382068757071877644544929382637646

/24962914469583191566367899923960035966404195848061867674084257325351443541635707108155961226253248787890779064193993572350407540882488313462480442009188322965308175793834908350711775470723063600215807657623374362126466957217083277804322543021220684978006202697645006112925727562688499951019008352273202998698433082923670410460471087620616165741606476723494106039072694978995397627362457886096950309963504449900266130281466812904611199098539805421932601086188714264657278379296255216638772840224103572864219332581506845376087212926568647114428532255047582757200833981470518008818805688306634147913338380858098047897556191777014597833213397490582583305285642961484103228645506879443810132259225138635152335311879233430440856986173430296778743350033604723187899699712826261542556246067548824288988199191266967980956809764206176935319256985166746436097191235722924204644128666213286229783071506832436219870537040937793596789158704489369975002974080695684303196676875492248579163093862959138359780655177475508923698668685694367

54.5476548512

2#101010101010101011

RandomEdge on KM-cubes

Conclusions:

RandomEdge is charming and simple, but awful to analyze.

We don’t understand really bad linear programs, e.g. indimension 4.

Experiments help.

RandomEdge on KM-cubes

Conclusions:

RandomEdge is charming and simple, but awful to analyze.

We don’t understand really bad linear programs, e.g. indimension 4.

Experiments help.

RandomEdge on KM-cubes

Conclusions:

RandomEdge is charming and simple, but awful to analyze.

We don’t understand really bad linear programs, e.g. indimension 4.

Experiments help.

On “real” and on “bad” LPs

Manfred Padberg, “Linear Optimization and Extensions”, 1995

On “real” and on “bad” LPs

Manfred Padberg, “Linear Optimization and Extensions”, 1995

Special Issue:Xxxxx Xxxxx Xxxxxx XxxxxxXxxxxxx

454 • ISSN 0179-537645(1) 001-000 (2011)

Springer

An International Journal of Mathematics and Computer Science

Founding EditorsJacob E. GoodmanRichard Pollack

Edited byHerbert Edelsbrunner

János PachGünter M. Ziegler

Volume 45 • Number 1 January 2011

DCGDiscrete & Computational Geomet ry

Discrete &

Computational Geom

etry

VOLUME

45

—

NUMBER

1

—

JANUARY

2011