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Introduction ILP formulation Valid inequalities Semidefinite relaxation Computational results Further work A traveling salesman problem with quadratic cost structure Anja Fischer, Christoph Helmberg Chemnitz University of Technology

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Introduction ILP formulation Valid inequalities Semidefinite relaxation Computational results Further work

A traveling salesman problem with quadratic cost structure

Anja Fischer, Christoph Helmberg

Chemnitz University of Technology

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Introduction ILP formulation Valid inequalities Semidefinite relaxation Computational results Further work

Outline

Introduction ILP formulation Valid inequalities Semidefinite relaxation Computational results Further work

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Introduction ILP formulation Valid inequalities Semidefinite relaxation Computational results Further work

Outline

Introduction ILP formulation Valid inequalities Semidefinite relaxation Computational results Further work

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Introduction ILP formulation Valid inequalities Semidefinite relaxation Computational results Further work

Problem description of QSTSP

Given:

  • undirected complete 2-graph G = (V , E), V = {1, . . . , n},

V 2 := {{i, j}: i, j ∈ V , i = j} (write ij) – arcs V 3 := {i, j, k = k, j, i: i, j, k ∈ V , |{i, j, k}| = 3} (write ijk) – 2-arcs

  • cost function c : V 3 → R+ with cijk – costs of path i − j − k,

⇒ quadratic cost structure Goal: find tour T = (i1, . . . , in, i1) minimizing

n−2

  • k=1

cik ik+1ik+2 + cin−1ini1 + cini1i2 1 4 5 3 2 costs c2,1,4

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Introduction ILP formulation Valid inequalities Semidefinite relaxation Computational results Further work

Introduction

Problem introduced by G. J¨ ager and P. Molitor (2008): Complexity and Algorithms for the Traveling Salesman Problem and the Assignment Problem of Second Order

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Introduction ILP formulation Valid inequalities Semidefinite relaxation Computational results Further work

Introduction

Problem introduced by G. J¨ ager and P. Molitor (2008): Complexity and Algorithms for the Traveling Salesman Problem and the Assignment Problem of Second Order Application Biology: recognition of transcription factor binding sites in gene regulation – Leibniz Institute of Plant Genetics and Crop Plant Research (Ivo Grosse, Jens Keilwagen)

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Introduction ILP formulation Valid inequalities Semidefinite relaxation Computational results Further work

Introduction

Problem introduced by G. J¨ ager and P. Molitor (2008): Complexity and Algorithms for the Traveling Salesman Problem and the Assignment Problem of Second Order Application Biology: recognition of transcription factor binding sites in gene regulation – Leibniz Institute of Plant Genetics and Crop Plant Research (Ivo Grosse, Jens Keilwagen) Special case of QSTSP Angular-Metric TSP, see Aggarwal et al. (1997) given points in the plane – find tour minimizing total direction changes applications in robotic

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Introduction

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

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Introduction ILP formulation Valid inequalities Semidefinite relaxation Computational results Further work

Introduction

Problem introduced by G. J¨ ager and P. Molitor (2008): Complexity and Algorithms for the Traveling Salesman Problem and the Assignment Problem of Second Order Application Biology: recognition of transcription factor binding sites in gene regulation – Leibniz Institute of Plant Genetics and Crop Plant Research (Ivo Grosse, Jens Keilwagen) Special case of QSTSP Angular-Metric TSP, see Aggarwal et al. (1997) given points in the plane – find tour minimizing total direction changes applications in robotic Complexity NP-complete, even the corresponding cycle cover problem is NP-complete

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Introduction ILP formulation Valid inequalities Semidefinite relaxation Computational results Further work

Outline

Introduction ILP formulation Valid inequalities Semidefinite relaxation Computational results Further work

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Introduction ILP formulation Valid inequalities Semidefinite relaxation Computational results Further work

Integer Linear Program

Linearization of the following quadratic integer model min

  • ijk∈V 3

cijk xijxjk

  • yijk

s.t.

  • ij∈V 2

xij = 2, i ∈ V (degree)

  • ij∈V 2

i∈S,j∈V \S

xij ≥ 2, S ⊂ V , 2 ≤ |S| ≤ n − 2 (subtour) xij ∈ {0, 1}, ij ∈ V 2

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Introduction ILP formulation Valid inequalities Semidefinite relaxation Computational results Further work

Integer Linear Program

Linearization of the following quadratic integer model min

  • ijk∈V 3

cijk xijxjk

  • yijk

s.t.

  • ij∈V 2

xij = 2, i ∈ V (degree)

  • ij∈V 2

i∈S,j∈V \S

xij ≥ 2, S ⊂ V , 2 ≤ |S| ≤ n − 2 (subtour) xij ∈ {0, 1}, ij ∈ V 2 xij =

  • ijk∈V 3

yijk, ij ∈ V 2 (flow) xij =

  • kij∈V 3

ykij, ij ∈ V 2 (flow) yijk ∈ [0, 1], ijk ∈ V 3

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Introduction ILP formulation Valid inequalities Semidefinite relaxation Computational results Further work

Dimension of the polytope PQSTSP

variables: 3 n

3

  • +

n

2

  • equality constraints: n + 2

n

2

  • = n2

Observation

The constraint matrix of the QSTSP (degree, flow) has full row rank.

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Dimension of the polytope PQSTSP

variables: 3 n

3

  • +

n

2

  • equality constraints: n + 2

n

2

  • = n2

Observation

The constraint matrix of the QSTSP (degree, flow) has full row rank. Regard the Quadratic Symmetric Cycle Cover Polytope PQSCCPn (subtours are allowed).

Lemma

The dimension of PQSCCPn equals 3 n

3

  • +

n

2

  • − n2 for n ≥ 7.
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Introduction ILP formulation Valid inequalities Semidefinite relaxation Computational results Further work

Dimension of the polytope PQSTSP

variables: 3 n

3

  • +

n

2

  • equality constraints: n + 2

n

2

  • = n2

Observation

The constraint matrix of the QSTSP (degree, flow) has full row rank. Regard the Quadratic Symmetric Cycle Cover Polytope PQSCCPn (subtours are allowed).

Lemma

The dimension of PQSCCPn equals 3 n

3

  • +

n

2

  • − n2 for n ≥ 7.

Conjecture

The dimension of PQSTSPn equals 3 n

3

  • +

n

2

  • − n2 for n ≥ 7.
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Introduction ILP formulation Valid inequalities Semidefinite relaxation Computational results Further work

Dimension of the polytope PQSTSP

variables: 3 n

3

  • +

n

2

  • equality constraints: n + 2

n

2

  • = n2

Observation

The constraint matrix of the QSTSP (degree, flow) has full row rank. Regard the Quadratic Symmetric Cycle Cover Polytope PQSCCPn (subtours are allowed).

Lemma

The dimension of PQSCCPn equals 3 n

3

  • +

n

2

  • − n2 for n ≥ 7.

Conjecture

The dimension of PQSTSPn equals 3 n

3

  • +

n

2

  • − n2 for n ≥ 7.

For the STSP: Gr¨

  • tschel and Padberg used an arc-disjoint Hamiltonian cycle

decomposition of the complete graph Gn.

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Dimension of the polytope PQSTSP

Question

Is there a 2-arc-disjoint Hamiltonian. cycle decomposition of the complete 2-graph Gn, n ≥ 3? Related to an open question (Bailey, Stevens) concerning the decomposition of complete uniform hypergraphs into arc-disjoint Hamiltonian cycles.

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Dimension of the polytope PQATSP resp. PQACCPn

  • (j,i)∈V 2

x(j,i) =

  • (i,j)∈V 2

x(i,j) = 1, i ∈ V , x(i,j) =

  • (i,j,k)∈V 3

y(i,j,k) =

  • (k,i,j)∈V 3

y(k,i,j), (i, j) ∈ V 2,

  • (i,j)∈V 2 :

i∈S,j∈V \S

x(i,j) ≥ 1, S ⊂ V , 1 ≤ |S| ≤ n − 1, x(i,j) ∈ {0, 1}, y(i,j,k) ∈ [0, 1], (i, j) ∈ V 2, (i, j, k) ∈ V 3 V 2 = {(i, j): i, j ∈ V , i = j}, V 3 = {(i, j, k): i, j, k ∈ V , |{i, j, k}| = 3}

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Dimension of the polytope PQATSP resp. PQACCPn

  • (j,i)∈V 2

x(j,i) =

  • (i,j)∈V 2

x(i,j) = 1, i ∈ V , x(i,j) =

  • (i,j,k)∈V 3

y(i,j,k) =

  • (k,i,j)∈V 3

y(k,i,j), (i, j) ∈ V 2,

  • (i,j)∈V 2 :

i∈S,j∈V \S

x(i,j) ≥ 1, S ⊂ V , 1 ≤ |S| ≤ n − 1, x(i,j) ∈ {0, 1}, y(i,j,k) ∈ [0, 1], (i, j) ∈ V 2, (i, j, k) ∈ V 3 V 2 = {(i, j): i, j ∈ V , i = j}, V 3 = {(i, j, k): i, j, k ∈ V , |{i, j, k}| = 3}

Lemma

The dimension of PQACCPn equals n(n − 1)2

  • # variables

− (2n2 − 1 − n)

  • rank of constr. matrix

for n ≥ 7.

Conjecture

The dimension of PQATSPn equals n(n − 1)2 − (2n2 − 1 − n) for n ≥ 7.

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Outline

Introduction ILP formulation Valid inequalities Semidefinite relaxation Computational results Further work

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Introduction ILP formulation Valid inequalities Semidefinite relaxation Computational results Further work

Valid inequalities

  • forbid certain triangles

ykij + yijk ≤ xij xij + xik + xjk ≤ 2

  • · xij

lift the constraint i j k strengthen triangle inequalities of Boolean Quadratic Polytope (Padberg, 1989)

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Valid inequalities

  • forbid certain triangles

ykij + yijk ≤ xij xij + xik + xjk ≤ 2

  • · xij

lift the constraint i j k strengthen triangle inequalities of Boolean Quadratic Polytope (Padberg, 1989)

  • i

j k Equivalent to: xij + xik + xjk − yijk − yikj − yjik ≤ 1 see triangle inequalities of Boolean Quadratic Polytope

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conflicting arcs inequalities

At most one of the following variables can be 1:

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conflicting arcs inequalities

At most one of the following variables can be 1:

xij

j i S T

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Introduction ILP formulation Valid inequalities Semidefinite relaxation Computational results Further work

conflicting arcs inequalities

At most one of the following variables can be 1:

yikj, k ∈ S

j i S T

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Introduction ILP formulation Valid inequalities Semidefinite relaxation Computational results Further work

conflicting arcs inequalities

At most one of the following variables can be 1:

ykil, k, l ∈ T

j i S T

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conflicting arcs inequalities

xij +

  • ikj∈V 3

k∈S

yikj +

  • lim∈V 3

l,m∈T

ylim ≤ 1 for all i, j ∈ V , i = j, and S, T ⊂ V \ {i, j}, S ∩ T = ∅ j i S T

Lemma

The conflicting arcs inequalities can be separated in polynomial time.

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conflicting arcs inequalities

xij +

  • ikj∈V 3

k∈S

yikj +

  • lim∈V 3

l,m∈T

ylim ≤ 1 for all i, j ∈ V , i = j, and S, T ⊂ V \ {i, j}, S ∩ T = ∅ j i S T

Lemma

The conflicting arcs inequalities can be separated in polynomial time. Proof: Transformation to Maximal Weight Independent Set Problem in bipartite graphs. 1 2 3 4 {1, 2} {1, 3} {1, 4} {2, 3} {2, 4} {3, 4} S T n = 6, i = 5, j = 6 : S = {1}, T = {2, 3, 4}

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strengthened conflicting arcs constraints

Case |T| = 2 At most one of the following variables can be 1: j i S T l m

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strengthened conflicting arcs constraints

Case |T| = 2 At most one of the following variables can be 1: j i S T l m xij +

  • ikj∈V 3 :

k∈S

yikj + ylim + yljm ≤ 1 for all i, j ∈ V , i = j, and S, T ⊂ V \ {i, j}, S ∩ T = ∅, T = {l, m}

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Subtour elimination constraints

  • ij∈V 2 :

i∈S,j∈V \S

xij ≥ 2, for all S ⊂ V , 2 ≤ |S| ≤ n − 2

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Subtour elimination constraints

  • ij∈V 2 :

i∈S,j∈V \S

xij ≥ 2, for all S ⊂ V , 2 ≤ |S| ≤ n − 2 Using only y-variables

  • ijk∈V 3 :

i∈S,j,k∈V \S

yijk + 2 ·

  • ijk∈V 3 :

i,k∈S,j∈V \S

yijk ≥ 2, for all S ⊂ V , 2 ≤ |S| ≤ n − 2

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Subtour elimination constraints

  • ij∈V 2 :

i∈S,j∈V \S

xij ≥ 2, for all S ⊂ V , 2 ≤ |S| ≤ n − 2 Using only y-variables

  • ijk∈V 3 :

i∈S,j,k∈V \S

yijk + 2 ·

  • ijk∈V 3 :

i,k∈S,j∈V \S

yijk ≥ 2, for all S ⊂ V , 2 ≤ |S| ≤ n − 2 ∀ S ⊂ V , 2 ≤ |S| < n 2 :

  • ijk∈V 3 :

i∈S,j,k∈V \S

yi,j,k ≥ 2, (1) S

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Subtour elimination constraints

  • ij∈V 2 :

i∈S,j∈V \S

xij ≥ 2, for all S ⊂ V , 2 ≤ |S| ≤ n − 2 Using only y-variables

  • ijk∈V 3 :

i∈S,j,k∈V \S

yijk + 2 ·

  • ijk∈V 3 :

i,k∈S,j∈V \S

yijk ≥ 2, for all S ⊂ V , 2 ≤ |S| ≤ n − 2 ∀ S ⊂ V , 2 ≤ |S| < n 2 :

  • ijk∈V 3 :

i∈S,j,k∈V \S

yi,j,k ≥ 2, (1) S

Lemma

The separation problem for inequalities (1) is NP-complete.

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Subtour elimination constraints

  • ij∈V 2 :

i∈S,j∈V \S

xij ≥ 2, for all S ⊂ V , 2 ≤ |S| ≤ n − 2 Using only y-variables

  • ijk∈V 3 :

i∈S,j,k∈V \S

yijk + 2 ·

  • ijk∈V 3 :

i,k∈S,j∈V \S

yijk ≥ 2, for all S ⊂ V , 2 ≤ |S| ≤ n − 2 ∀ S ⊂ V , 2 ≤ |S| < n 2 :

  • ijk∈V 3 :

i∈S,j,k∈V \S

yi,j,k ≥ 2, (1) S Case |S| ≥ n

2:

  • ij∈V 2 :

i∈S,j∈V \S

xij − 2

  • ikj∈V 3 :

i,j∈S,k∈T

yi,k,j ≥ 2, for all S, T ⊂ V , S ∩ T = ∅, |S| + |T| = n − 1

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Outline

Introduction ILP formulation Valid inequalities Semidefinite relaxation Computational results Further work

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Semidefinite relaxation

semidefinite relaxation bases on QIP: min

  • ijk∈V 3

ci,j,k · xijxjk s.t. x ∈ TSP(n) (TSP-polytope) with x = (x12, x13, . . . , xn−1,n)T Construction of rank-one matrix: X = 1 x 1 x T Notation: xij,kl ˆ = xij · xkl

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Semidefinite relaxation

min

  • ijk∈V 3

cijkxij,jk s.t.

  • ij∈V 2

xij = 2, i ∈ V xij =

  • k∈V \{i,j}

xij,jk, ij ∈ V 2 xij =

  • k∈V \{i,j}

xki,ij, ij ∈ V 2 X1,1 = 1 X1,i = Xi,i, ∀ i = 2, . . . , n + 1 (xij = xij,ij) 0 ≤ X ≤ E rank(X) = 1 X 0

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Semidefinite relaxation

min

  • ijk∈V 3

cijkxij,jk s.t.

  • ij∈V 2

xij = 2, i ∈ V xij =

  • k∈V \{i,j}

xij,jk, ij ∈ V 2 xij =

  • k∈V \{i,j}

xki,ij, ij ∈ V 2 X1,1 = 1 X1,i = Xi,i, ∀ i = 2, . . . , n + 1 (xij = xij,ij) 0 ≤ X ≤ E rank(X) = 1 X 0

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Semidefinite relaxation

min

  • ijk∈V 3

cijkxij,jk s.t.

  • ij∈V 2

xij = 2, i ∈ V xij =

  • k∈V \{i,j}

xij,jk, ij ∈ V 2 xij =

  • k∈V \{i,j}

xki,ij, ij ∈ V 2 X1,1 = 1 X1,i = Xi,i, ∀ i = 2, . . . , n + 1 (xij = xij,ij) 0 ≤ X ≤ E X 0 strengthening: presented cuts, inequalities of the Boolean Quadratic Polytope

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Outline

Introduction ILP formulation Valid inequalities Semidefinite relaxation Computational results Further work

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Computational results – real-world instances

  • 3 instances 5 ≤ n ≤ 80
  • # variables for n = 80: ∼ 500000
  • all instances solved in < 13 minutes with the presented cuts
  • only SEC: running times much higher

0.001 0.01 0.1 1 10 100 1000 10000 100000 10 20 30 40 50 60 70 80 time in seconds size with cuts sec only

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Computational results – random instances

10 instances uni random asymmetric: cijk uniformly at random in {0, 1, . . . , 10000} random angular: points i ∈ V uniformly at random in {0, 1, . . . , 10000}2, cijk =

  • 18000

π arccos vj − vi vj − vi T vk − vj vk − vj

  • uni random symmetric: cijk uniformly at random in {0, 1, . . . , 10000}
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Computational results – random instances

10 instances uni random asymmetric: cijk uniformly at random in {0, 1, . . . , 10000} random angular: points i ∈ V uniformly at random in {0, 1, . . . , 10000}2, cijk =

  • 18000

π arccos vj − vi vj − vi T vk − vj vk − vj

  • uni random symmetric: cijk uniformly at random in {0, 1, . . . , 10000}

Computer: Intel Core i7 CPU 920, 2.67 GHz, 12 GB RAM

  • (MIP): use SCIP, LP with CPLEX 12.1,
  • (SDP): use Matlab and SDPT3
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Computational results – asymmetric random instances

0.001 0.01 0.1 1 10 100 1000 10000 5 10 15 20 25 time in seconds size random

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Computational results – symmetric instances

random and random angular instances 0.001 0.01 0.1 1 10 100 1000 10000 5 10 15 20 25 30 time in seconds size random angular But: for random instances better not to separate additional inequalities

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Computational results – symmetric random instances

Comparison of the value of the gaps [(opt − relax)/relax] · 100% at the root node IP root relaxation of IP SDP1 SDP relaxation, all inequalities only on the y-support SDP2 additional xij,kl ≥ 0 for all matrix entries SDP3 additional triangle inequalities on whole matrix

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Computational results – symmetric random instances

Comparison of the value of the gaps [(opt − relax)/relax] · 100% at the root node IP root relaxation of IP SDP1 SDP relaxation, all inequalities only on the y-support SDP2 additional xij,kl ≥ 0 for all matrix entries SDP3 additional triangle inequalities on whole matrix n IP SDP1 SDP2 SDP3 5 0.00 0.00 0.00 0.00 6 0.00 0.00 0.00 0.00 7 0.00 0.00 0.00 0.00 8 1.74 0.43 0.30 0.00 9 2.73 1.09 0.69 0.02 10 10.35 4.79 2.99 0.76 11 13.58 8.30 5.21 2.63 12 18.60 11.77 8.38 5.32 13 19.05 10.93 6.79 4.17 14 23.18 14.55 9.94 7.48 15 23.61 13.31 8.50 6.45

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Computational results – symmetric random instances

Comparison of the value of the gaps [(opt − relax)/relax] · 100% at the root node IP root relaxation of IP SDP1 SDP relaxation, all inequalities only on the y-support SDP2 additional xij,kl ≥ 0 for all matrix entries SDP3 additional triangle inequalities on whole matrix n IP SDP1 SDP2 SDP3 5 0.00 0.00 0.00 0.00 6 0.00 0.00 0.00 0.00 7 0.00 0.00 0.00 0.00 8 1.74 0.43 0.30 0.00 9 2.73 1.09 0.69 0.02 10 10.35 4.79 2.99 0.76 11 13.58 8.30 5.21 2.63 12 18.60 11.77 8.38 5.32 13 19.05 10.93 6.79 4.17 14 23.18 14.55 9.94 7.48 15 23.61 13.31 8.50 6.45 n IP 16 35.21 17 31.69 18 36.53 19 40.11 20 44.80 21 47.36 22 41.65 23 46.44 24 42.75 25 51.03 30 60.14

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Outline

Introduction ILP formulation Valid inequalities Semidefinite relaxation Computational results Further work

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Further work

  • further polyhedral studies, includes dimension of the corresponding

polytopes

  • separation heuristics for extended SEC
  • use SDP-bounds for Branch-and-Cut
  • use spectral bundle method for SDP
  • Which support extension is useful?
  • Which (in-)equalities should be used?
  • extend neighborhood
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Thank you for your attention. Questions?

The Cluster of Excellence “Energy-Efficient Product and Process Innovation in Production Engineering”(eniPROD R ) is funded by the European Union (European Regional Development Fund) and the Free State of Saxony.