Exploring Higher Dimensions: Experiments with Extended Capacity Cuts

1
Exploring Higher Dimensions Experiments with
Extended Capacity Cuts

• Eduardo Uchoa, Engenharia de Produção, UFF,
  Brazil.
• Ricardo Fukasawa, ISYE, Georgia Tech, USA.
• Jens Lysgaard, Aarhus School of Business,
  Denmark.
• Artur Pessoa, Engenharia de Produção, UFF,
  Brazil.

2
Initial Motivation
3
The Capacitated Minimum Spanning Tree Problem
(CMST)

• Given
• Undirected graph G(V,E),
• Edge Costs ce,
• Vertex demands dv,
• Capacity C,
• A root vertex.
• Find
• A MST where the total demand of the vertices in
  each subtree hanging from the root does not
  exceeds C.

4
The Capacitated Minimum Spanning Tree Problem
(CMST)

• Example (unit demands, C10)

5
The Capacitated Minimum Spanning Tree Problem
(CMST)

• Example (unit demands, C10)

6
CMST Arc Formulation

• Replace edges (i,j) by pairs of arcs (i,j) and
  (j,i), with cost cij cji ce
• Let 0 be the root node
• Let V1,...,n
• Define binary variables
• Xa 1 if arc a belongs to an arborescence rooted
  at 0.

7
CMST Arc Formulation
All vertices have exactly one incident arc
Capacity Cuts Eliminate cycles and enforce
maximum tree demand
8
Violated Capacity Cuts
C2, unit demands, root 0
0
0
S
S
2
1
2
1
3
3
9
Root Cutset Cuts
C3, unit demands, root0
0
S
2
4
1
A0, B4
3
VIOLATED
10
Other Known Cuts

• Many families of cuts (multistars) are known for
  the CMST (Araque, Hall, Magnanti, Gouveia).
• Those more complex cuts have little impact on
  bounds.

11
The Capacity-Indexed Formulation (CIF) for the
CMST Gouveia (1995)
12
The CIF for the CMST

• O(mC) variables, but only 2n constraints.
• Weak linear relaxation, but the additional
  cap-indexed vars suggested the use of new kinds
  of cuts in a robust branch-cut-and-price context
  (FPPU03).

13
Extended Capacity Cuts
14
The Fixed Charge Network Flow Problem
Suppose d and u integral
15
FCNF Capacity-Indexed Reformulation

16
FCNF Capacity-Indexed Reformulation

• Same value of LP relaxation.
• Models several classical problems, including
• CMST itself,
• Many VRP, facility location, bin packing and lot
  sizing variants,
• The Steiner Tree problem.

17
The Balance Equalities for CI-FCNF

• For each vertex i in V
• Summing over a set S in V

18
Extended Capacity Cuts (ECCs)

• An ECC over a set S is any inequality valid for

19
Extended Capacity Cuts (ECCs)

• Many known cuts for the previously mentioned
  problems can be shown to be equivalent or
  dominated by ECCs.
• In particular, CMST Capacity Cuts, Root Cutsets
  and Multistars are eq. or dom. by ECCs obtained
  by integral rounding.

20
Homogeneous ECCs

• Given a set S, define
• HECCs are cuts obtained from polyhedra

21
Facet HECCs

• When C is small, we can use a package (PPL) to
  enumerate all facets of P(C,D), for each value of
  D.

22
P(5,6)
C5, d(S)6
23
P(5,6)
24
HECCs obtained by rounding

• For larger values of C, the integer rounding of
• with multipliers r a/b, a and b in the range
  1...C, already gives a reasonable family of cuts.

25
Computational Results on the CMST

• A routine for separation of HECCs was included in
  a branch-cut-and-price code for the CMST.
• Enumeration of all sets S up to size 10
• Facet HECC separation for C lt 10
• Rounded HECC separation for larger C.
• Extensive computational experiments available at
  U., Fukasawa, Lysgaard, Pessoa, Poggi de Aragão
  and Andrade (2006).

26
Bounds and times on formerly open instance te80-2
C10

• GM (2005) 1609.95, 4039s (AMD
  1GHz)
• BCP q-treesArc Cuts 1609.23, 33s
• BCP q-trees HECCs 1624.78, 378s
• Opt 1639, 29134s 7,5h (1105 nodes)
• Facet HECCs with D up to 10.
• Pentium 3GHz

27
Bounds and times on formerly open instance
cm50-r1 C200

• BC (Similar to Hall95) 1039.9,
  3s
• BCP q-arbs Arc Cuts 1093.4, 65s
• BCP q-trees HECCs 1097.8, 148s
• Opt 1098, 153s (2 nodes)
• Rounded HECCs with D up to 6

28
Bounds and times on formerly open instance
cm100-r1 C200

• BC (Similar to Hall95) 465.9,
  11s
• BCP q-arbs Arc Cuts 501.8, 341s
• BCP q-trees HECCs 506.7, 1480s
• Opt 509 (Previous Known516), 17227s 5h (64
  nodes)
• The ECCs always led to substantive gap redutions,
  allowing the solution of many open instances.

29
Breaking News

• Fukasawa, Dash and Gunluk characterized the
  facets of polyhedra P(C,D) (if D lt C), what they
  call the generalized Master polyhedra
• This may lead to facet HECC separation also for
  large values of C.

30
Trying to understand the results and the
potential of cuts over extended spaces
31
First Observation

• Fractional solutions over the extended vars can
  be much richer in terms of information content.
  This may allow detecting inconsistencies by only
  looking at a small part of the instance.
• A related observation is that a set of simple and
  general cuts over the extended vars may replace
  complex and specific cuts over the original vars.

32
Part of a typical CMST fractional solution
4
2
1
3
1
1
3
1
1
2
2
All vars in the with value 0.5
33
Looking over the extended space
Capacity-Indices
4
2
1
3
1
1
3
1
1
2
2
All vars with value 0.5
34
Looking over the extended space
4
2
1
3
1
1
3
1
1
2
2
q-arb A with value 0.5
35
Looking over the extended space
4
2
1
3
1
1
3
1
1
2
2
q-arb B with value 0.5
36
Looking over the extended space
4
2
1
3
1
1
3
1
1
2
2
q-arb C with value 0.5
37
Looking only at the vars entering or leaving set
S
4
2
3
2
2
What the ECC actually sees
38
Looking only at vars entering or leaving set S
4(a)
2(b)
3
2(c)
2(d)
4x4a 2x2b 2x2c -2x2d 3 Has no 0-1 solution
39
Looking only at vars entering or leaving set S
4(a)
2(b)
3
2(c)
2(d)
4x4a 2x2b 2x2c -2x2d 3 Has no 0-1 solution gt
There must be a violated cut
40
Looking only at vars entering or leaving set S
4(a)
2(b)
3
2(c)
2(d)
4x4a 2x2b 2x2c -2x2d 3 Has no 0-1 solution gt
There must be a violated cut 2x4a x2b x2c -x2d
gt 2
41
No cut would be possible looking only at the
original space
3
All arcs have value 0.5 This is convex
combination of valid solutions over the arc space
42
No cut would be possible looking only at the
original space
3
There must be some violated cut over the arc
space ! But such cut certainly includes
variables outside S and is likely to be much
harder to identify and separate
43
Steiner Tree Problem

• The Steiner Tree problem can be reformulated as a
  FCNF over the capacity indexed vars. The value of
  C is the number of terminals -1.
• Steiner dicuts correspond to Capacity Cuts.

44
Directed Steiner Tree Odd Wheel Configuration
45
Directed Steiner Tree Odd Wheel Configuration
0.5
0.5
Extreme point of the dicut formulation
0.5
Cut by Odd Wheel ineq. (Chopra and Rao) x04
x05 x41 x42 x51 x53 x16 x62 x63 gt
5
0.5
0.5
0.5
0.5
Separation requires partitioning the Graph into 7
components ! Seems very hard, never implemented.
0.5
0.5
46
Directed Steiner Tree Odd Wheel Configuration
Introduce cap-indexed Vars x04 x041 x042
x043 x05 x051 x052 x053 . . . and Balance
Equations x411 2 x412 3 x413 x511 2
x512 3 x513 - x161 - 2 x162 -3 x163 1 x041
2 x042 3 x043 - x411 - 2 x412 - 3 x413 - x421
x422 x423 0 . . .
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
47
Directed Steiner Tree Odd Wheel Configuration
There are values for the new vars that does not
require changes in the original vars.
0.5
0.5
1
1
0.5
2
0.5
0.5
1
1
0.5
0.5
2
2
3
3
0.5
0.5
48
Directed Steiner Tree Odd Wheel Configuration
There are values for the new vars that does not
require changes in the original vars. But,
something is clearly wrong with terminal 1. The
value of its adjacent vars is not a convex
combination of 0-1 solutions of its balance
equation, the ECC x162 lt x423 x513 can be
derived .
6
0.5
0.5
2
3
1
1
0.5
2
0.5
0.5
1
1
1
0.5
0.5
2
2
4
5
3
3
0.5
0.5
0
49
Directed Steiner Tree Odd Wheel Configuration
There are still values for the new vars that does
not require changes in the original vars. Now,
something is wrong with non-terminals 4 and 5.
ECCs like x043 lt x412 x413 x422
x423 can be derived . Adding 4 such cuts leads
to integral solutions !
6
0.5
0.5
2
3
1
1
0.5
2
0.5
0.5
1
1
1
0.25
0.25
1
1
0.25
0.25
3
4
5
3
3
3
0.5
0.5
0
50
Directed Steiner Tree Odd Wheel Configuration
Cutting over the original Vars requires Steiner
Knowledge and a global view of the fractional
solution. In this case, the same results are
obtained by cutting on the extended space using
general principles and only a local view of the
fractional solution. Drawback More Vars, More
Cuts.
6
1.0
1.0
2
3
1
1
1.0
2
1
1.0
3
4
5
3
1.0
0
51
Second Observation

• Any known cut over the original vars can still be
  used in the extended formulation.
• However, it is often possible to improve
  coefficients in the extended space, even the cut
  is tight in the original space.

52
CMST Root Cutset
C100
Set A - Vertices with demand lt 20
Set B - Vertices with demand gt 20
7
23
12
180
31
Root
53
Strengthening a Root Cutset
C100
A simple observation shows that most cap-indexed
variables from the A set may have its coefficient
improved
7
23
12
180
31
Root
54
General Integer Programming
x2

• Max x1 x2
• S.t. x1 2x2 lt 5
• x1 lt 2
• x1, x2 ? Z

2
1
x1
1
2
55
General Integer Programming
x2
2
1
x1
1
2
56
General Integer Programming
x2
2
1
x1
1
2
57
General Integer Programming
x2
2
1
x1
1
2
58
General Integer Programming
x2
2
1
x1
1
2
59
General Integer Programming
x2
2
1
x1
1
2
60
General Integer Programming
x2
2
1
x1
1
2
61
General Integer Programming

• Extended Dimensions
• x1 y1 2z1 y1z1 lt 1 y1, z1 ? 0,1
• x2 y2 2z2 y2z2 lt 1 y2, z2 ? 0,1
• Max y1 2z1 y2 2z2
• S.t. y1 2z1 2y2 4z2 lt 5
• y1 2z1 lt 2
• y1z1 lt 1
• y2z2 lt 1

62
General Integer Programming

• Integer Solutions (y1, z1, y2, z2)
• (0,0,0,0) (1,0,0,0) (0,1,0,0)
• (0,0,1,0) (1,0,1,0) (0,1,1,0)
• (0,0,0,1) (1,0,0,1)
• Strengthening y1 2z1 2y2 4z2 lt 5

63
General Integer Programming

• Integer Solutions (y1, z1, y2, z2)
• (0,0,0,0) (1,0,0,0) (0,1,0,0)
• (0,0,1,0) (1,0,1,0) (0,1,1,0)
• (0,0,0,1) (1,0,0,1)
• Strengthening y1 2z1 2y2 4z2 lt 5

64
General Integer Programming

• Integer Solutions (y1, z1, y2, z2)
• (0,0,0,0) (1,0,0,0) (0,1,0,0)
• (0,0,1,0) (1,0,1,0) (0,1,1,0)
• (0,0,0,1) (1,0,0,1)
• Strengthening y1 2z1 2y2 4z2 lt 5
• To y1 3z1 2y2 4z2 lt 5

65
General Integer Programming

• x1 2x2 lt 5
• Equivalent to y1 2z1 2y2 4z2 lt 5
• Strengthened to y1 3z1 2y2 4z2 lt 5
• Equivalent to x1 2x2 lt 5 z1

66
General Integer Programming

• Strengthened cut
• x1 2x2 lt 5 z1
• x1 y1 2z1 and y1 z1 lt 1 ? z1 gt x1 1
• Hence 2x1 2x2 lt 6
• z1 gt 0
• Hence x1 2x2 lt 5

Both inequalities are implied!
67
General Integer Programming

• Strengthened Formulation
• Max x1 x2
• S.t. x1 2x2 lt 5 z1
• x1 lt 2
• x1, x2 ? Z

x2
2
1
x1
1
2
68
General Integer Programming

• Tight inequalities (even facets) in the original
  space may be strengthened in the extended space.
• Such improved inequality in the extended space
  may project into several inequalities in the
  original space.

69
Overall Conclusions

• Cuts over extended spaces, like ECCs, hold
  promise to yield substantially stronger bounds on
  many problems.
• One possibility for tackling the resulting large
  number of variables is by branch-cut-and-price
  schemes.
• Many open issues and questions

70
Obrigado !