concise decision diagrams for branching and storing … · 2017. 9. 5. · from trees to diagrams....

CONCISE DECISION DIAGRAMS FOR

BRANCHING AND STORING

NEAR-OPTIMAL SOLUTIONS

Thiago Serra John Hooker

Carnegie Mellon University

MOPTA 2016

IF FINDING ONE IS EASY, WHAT ABOUT ALL?

WE CARE ABOUT THAT EVEN IN FICTION

Motivation

4

Poké Ball Belt Bag

PresenterPresentation Noteshttps://www.youtube.com/watch?v=3xTyohQhZmI

ONE OPTIMAL SOLUTION MAY NOT BE ENOUGH

(Real) Motivation

6

Alternative courses of action

7

Alternative courses of action

How much is left on the table?

8

Truthful bidding in combinatorial auctions

PresenterPresentation Noteshttp://www.picgifs.com/disney-gifs/pinocchio/http://blog.winspireme.com/5-reasons-not-to-hire-a-benefit-auctioneer

9

Truthful bidding in combinatorial auctions

Prices defined by solving problem many times

PresenterPresentation Noteshttp://www.picgifs.com/disney-gifs/pinocchio/http://blog.winspireme.com/5-reasons-not-to-hire-a-benefit-auctioneer

10

Solvers are already doing that

PresenterPresentation Noteshttp://yetanothermathprogrammingconsultant.blogspot.com/2016/01/finding-all-optimal-lp-solutions-using.html

11

Solvers are already doing that

Where is my Poké Belt?

PresenterPresentation Noteshttp://yetanothermathprogrammingconsultant.blogspot.com/2016/01/finding-all-optimal-lp-solutions-using.html

USING AND RELAXING DECISION DIAGRAMS

Our approach

13

From trees to diagrams Too many solutions,

not much information

X1

X2

X3

Variable for branching:

0-arc

1-arc

Branching tree

14

From trees to diagrams Too many solutions,

not much information

X1

X2

X3

Variable for branching:

0-arc

1-arc

r

t Branching tree Decision diagram

15

From trees to diagrams Anything other than a path gets smaller

Branching tree Decision diagram

16


Branching tree Decision diagram

17


Solutions ≡ Leaves → Solutions ≡ Paths

18



19



Optimal solutions ≡ Shortest (Longest) paths

20

Sound relaxations Exact for Δ–optimal solutions

Decision diagram

3

2

3

Weights of 1-arcs:

21


Min problem: optimal value z* to z* + Δ

Decision diagram (z* = 3 Δ = 3)

3

2

3

Weights of 1-arcs:

22



Decision diagram Sound relaxation (z* = 3 Δ = 3)

3

2

3

Weights of 1-arcs:

23




3

2

3

Weights of 1-arcs:

24

Sound relaxations Decision diagram ≡ Paths within a budget


3

2

3

25

Sound relaxations Decision diagram ≡ Paths within a budget

Some arcs forbidden after worse choices


3

2

3

26

How it works Reduction

Merges nodes with same children

27



28



29



Converges to minimum size diagram

30

How it works Sound-reduction

Redirects incoming arcs from node u to v

v u

31



v u v u

32



(i) Node v has a superset of completions

v u v u

33



(ii) Min cost r-u path

+ Min cost extra v-t path

> z* + Δ

(i) Node v has a superset of completions

v u v u

34


4

1

1

2

2

z* = 1 Δ = 6

v1 u1

v2 q u2

v3 u3

r

t

35


4

1

1

2

2

v1 u1

v2 q u2

v3 u3

z* = 1 Δ = 6

Min r-u2 path: 6

+ Min extra q-t path: 2

> 7

r

t

36


4

1

1

2

2

v1 u1

v2 q

v3 u3

z* = 1 Δ = 6

Min r-u2 path: 6

+ Min extra q-t path: 2

> 7

r

t

37


4

1

1

2

2

v1 u1

v2 q u2

v3 u3

z* = 1 Δ = 6

Min r-u2 path: 6

+ Min extra v2-t path: 2

> 7

r

t

38


4

1

1

2

2

v1 u1

v2 q

v3 u3

z* = 1 Δ = 6

Min r-u2 path: 6


> 7

r

t

39


4

1

1

2

2

v1 u1

v2 q u2

v3 u3

z* = 1 Δ = 6

Min r-u1 path: 5


> 7

r

t

40


4

1

1

2

2

v1

v2 q

v3 u3

z* = 1 Δ = 6

Min r-u1 path: 5


> 7

r

t

41


Same children → Extra paths are costly

4

1

1

2

2

v1

v2 q

v3 u3

z* = 1 Δ = 6 r

t

42


Bottom-up comparison of completions

4

1

1

2

2

v1 u1

v2 q u2

v3 u3 u3 → Min extra v3-t path: 2

r

t

43



4

1

1

2

2

v1 u1

v2 q u2


r

t

u2 → Min extra v2-t path: 2

44



4

1

1

2

2

v1 u1

v2 q u2


r

t



45

How well it works Small problems

46


47


48

How well it works Large problems

49


50


51

What we learned Sound relaxations disguise suboptimal solutions

52


If Δ=0, there are no deviations

and no forbidden arcs

53


If Δ=0, there are no deviations

and no forbidden arcs

When sound-reducing u into v,

min r-u path > min r-v path

54

What we learned Sound-reduction converges to minimum diagram

55


Equivalent diagrams → Same completion sets

56


Equivalent diagrams → Same completion sets

{φ} {1} {0,1}

57


Sound diagrams → Essential completion sets

{00,10} {00,01}

v1 u1

v2 q u2

v3 u3

r

t

{000,010} {100,101}

58

What we learned What if the diagram is exact for [z*, z*+ Δ]?

59


Decision diagram Generalized sound (z* = 1 Δ = 0) relaxation

1

1

60



1

1

61


Generalizing the relaxation makes it NP-hard


1

1

62

What we learned What if we approximate or go heuristic?

63


64


65


Every step may be co-NP-complete

66

What comes next Branch less often

67


We are still generating all tree nodes

68


We are still generating all tree nodes

69

What comes next Identify nodes with equivalent states

70


71


x1 = 0 x1 = 1

72


x1 = 0 x1 = 1

x2 = 0 x2 = 0 x2 = 1 x2 = 1

73


x1 = 0 x1 = 1

x2 = 0 x2 = 0 x2 = 1 x2 = 1

74


Same RHS → Same completions

75


Same RHS → Same completions

Saturated constraints can be ignored

76


Resulting diagram may be a relaxation

77



z* = 7, Δ = 1

78



z* = 7, Δ = 1

r

5 p1 p2 6

79



z* = 7, Δ = 1

r

5 p1 p2 6

2 s1 s2 3 t

80



z* = 7, Δ = 1

c([p1 s1]) = 7

c([p1 s2]) = 8

c([p2 s1]) = 8

c([p2 s2]) = 9

r

5 p1 p2 6

2 s1 s2 3 t

81

How far we are Small problems

82

How far we are Large problems

82

DECISION DIAGRAMS SAVE SPACE

Takeaway 1

RELAXED BRANCHING DOES LESS WORK

Takeaway 2

QUESTIONS?

Concise Decision Diagrams for Branching and Storing �Near-Optimal SolutionsIf finding one is easy,�what about all?We care about that�even in fictionSlide Number 4One optimal solution�MAY not BE enough Using and relaxing decision diagramsFrom trees to diagramsFrom trees to diagramsFrom trees to diagramsFrom trees to diagramsFrom trees to diagramsFrom trees to diagramsFrom trees to diagramsSound relaxationsSound relaxationsSound relaxationsSound relaxationsSound relaxationsSound relaxationsHow it worksHow it worksHow it worksHow it worksHow it worksHow it worksHow it worksHow it worksHow it worksHow it worksHow it worksHow it worksHow it worksHow it worksHow it worksHow it worksHow it worksHow it worksHow it worksHow well it worksHow well it worksHow well it worksHow well it worksHow well it worksHow well it worksWhat we learnedWhat we learnedWhat we learnedWhat we learnedWhat we learnedWhat we learnedWhat we learnedWhat we learnedWhat we learnedWhat we learnedWhat we learnedWhat we learnedWhat we learnedWhat we learnedWhat we learnedWhat comes nextWhat comes nextWhat comes nextWhat comes nextWhat comes nextWhat comes nextWhat comes nextWhat comes nextWhat comes nextWhat comes nextWhat comes nextWhat comes nextWhat comes nextWhat comes nextWhat comes nextHow far we areHow far we areDecision diagrams SAVE SPACERelaxed branching Does less WORKQuestions?

concise decision diagrams for branching and storing … · 2017. 9. 5. · from trees to diagrams....

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