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Subdeterminants and Concave Integer Quadratic Programming Alberto Del Pia, University of Wisconsin-Madison January 10, 2019 23rd Combinatorial Optimization Workshop, Aussois 2019 The problem Separable Concave Integer Quadratic Programming k

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SLIDE 1

Subdeterminants and Concave Integer Quadratic Programming

Alberto Del Pia, University of Wisconsin-Madison January 10, 2019 23rd Combinatorial Optimization Workshop, Aussois 2019

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SLIDE 2

The problem

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SLIDE 3

Separable Concave Integer Quadratic Programming

min

k

∑

i=1

−qix2

i + h⊤x

  • s. t.

Wx ≤ w x ∈ Zn. (IQP)

  • Integral data, q ≥ 0.
  • NP-hard even if k = 0 (ILP).
  • Polynomially solvable in fixed dimension. [Hartmann ’89]
  • We are interested in algorithms in general dimension.

1

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SLIDE 4

ϵ-approximate solution

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SLIDE 5

ϵ-approximate solution

2

Definition Let x∗ be an optimal solution. For ϵ ∈ [0, 1], x⋄ is an ϵ-approximate solution if

  • bj(x⋄) − obj(x∗)
  • bjmax − obj(x∗) ≤ ϵ.
  • obj(·) : objective function value.
  • objmax : maximum value of obj(x) on the feasible region.
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SLIDE 6

ϵ-approximate solution

2

Definition Let x∗ be an optimal solution. For ϵ ∈ [0, 1], x⋄ is an ϵ-approximate solution if

  • bj(x⋄) − obj(x∗)
  • bjmax − obj(x∗) ≤ ϵ.
  • Definition used in the literature from the 80s.
  • Natural choice for unstructured problems:
  • Preserved under dilation and translation of obj.
  • Insensitive to change of basis.
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SLIDE 7

Main results

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SLIDE 8

Main results: general case

3

min

k

∑

i=1

−qix2

i + h⊤x

  • s. t.

Wx ≤ w x ∈ Zn. (IQP) Theorem Denote by ∆ the largest absolute value of the subdeterminants of

  • W. For every ϵ ∈ (0, 1] there is an algorithm that finds an

ϵ-approximate solution by solving ( 3 + ⌈√ k ( (2n∆)2 + 1 ϵ )⌉)k ILPs of the same size of (IQP), and with a constraint matrix with subdeterminants bounded by ∆.

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SLIDE 9

Main results: ∆ = 2

4

min

k

∑

i=1

−qix2

i + h⊤x

  • s. t.

Wx ≤ w x ∈ Zn. (IQP) Using [Artmann Weismantel Zenklusen ’17]: Corollary Assume ∆ = 2. For every ϵ ∈ (0, 1] there is an algorithm that finds an ϵ-approximate solution in a number of operations bounded by ( 3 + ⌈√ k ( (4n)2 + 1 ϵ )⌉)k poly(n, m).

  • For ∆ = 2, this closes the gap between the best known

algorithm for (IQP) and its continuous version. [Vavasis ’92]

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SLIDE 10

The algorithm

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SLIDE 11

Step 1. Feasibility and boundedness

5

min

k

∑

i=1

−qix2

i + h⊤x

  • s. t.

Wx ≤ w x ∈ Zn. (IQP)

  • Feasibility and boundedness of (IQP) can be checked in

polynomial time by solving 2k + 1 ILPs using [Del Pia ’16].

  • Assume now that (IQP) is feasibile and bounded.
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SLIDE 12

Step 2. Decomposition

6

In particular, we obtain finite bounds on the integer hull: li := min{xi : Wx ≤ w, x ∈ Zn}, ∀i = 1, . . . , k ui := max{xi : Wx ≤ w, x ∈ Zn} ∀i = 1, . . . , k.

b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b

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SLIDE 13

Step 2. Decomposition

6

In particular, we obtain finite bounds on the integer hull: li := min{xi : Wx ≤ w, x ∈ Zn}, ∀i = 1, . . . , k ui := max{xi : Wx ≤ w, x ∈ Zn} ∀i = 1, . . . , k.

b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b

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SLIDE 14

Step 2. Decomposition

6

We need the following. Width assumption: ui − li ≥ ⌈√ k ( (2n∆)2 + 1 ϵ )⌉ =: g

b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b

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SLIDE 15

Step 2. Decomposition

6

If ∃i ∈ {1, . . . , k} such that ui − li < g, we replace problem (IQP) with all the subproblems obtained by fixing xi to each value li, li + 1, li + 2, . . . , ui.

b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b

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SLIDE 16

Step 2. Decomposition

6

  • Recursively, we decompose (IQP) into subproblems that satisfy

the width assumption.

  • For each subproblem, we obtain an ϵ-approximate solution for

it, and then return the best one.

b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b

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SLIDE 17

Step 2. Decomposition

6

  • Each subproblem has the same form of (IQP).
  • Thus, we now assume that (IQP) satisfies the width assumption.

b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b

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SLIDE 18

Step 3. Mesh partition and linear underestimators

7

We partition the box into gk boxes.

b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b

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SLIDE 19

Step 3. Mesh partition and linear underestimators

7

For each box, we construct the affjne function µ(x) that coincides with the quadratic objective on the vertices of the box, and solve the ILP: min µ(x)

  • s. t.

Wx ≤ w x ∈ box x ∈ Zn.

b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b

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SLIDE 20

Step 3. Mesh partition and linear underestimators

7

The best solution x⋄ among the gk obtained ones (one per box) is an ϵ-approximate solution to (IQP).

b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b

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SLIDE 21

Correctness

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SLIDE 22

Correctness of the algorithm

To prove that x⋄ is an ϵ-approximate solution we need to show

  • bj(x⋄) − obj(x∗)
  • bjmax − obj(x∗) ≤ ϵ.

We need two bounds:

  • obj(x⋄) − obj(x∗) ≤ UB :

x⋄ is close to optimal.

  • objmax − obj(x∗) ≥ LB :

∃ x+ feasible far from optimal. Let’s see how to construct x .

8

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SLIDE 23

Correctness of the algorithm

To prove that x⋄ is an ϵ-approximate solution we need to show

  • bj(x⋄) − obj(x∗)
  • bjmax − obj(x∗) ≤ ϵ.

We need two bounds:

  • obj(x⋄) − obj(x∗) ≤ UB :

x⋄ is close to optimal.

  • objmax − obj(x∗) ≥ LB :

∃ x+ feasible far from optimal. Let’s see how to construct x+.

8

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SLIDE 24

A solution far from optimal

9

Wlog x1 is the most concave direction, i.e, max{qi(ui − li)2 : i ∈ {1, . . . , k}} = q1(u1 − l1)2 =: γ.

b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b

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SLIDE 25

A solution far from optimal

9

Let xl, xu be feasible with xl

1 = l1, xu 1 = u1.

b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b

xl xu

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SLIDE 26

A solution far from optimal

9

The vector x◦ = 1

2xl + 1 2xu is far from optimal:

  • bj(x◦) − obj(x∗) ≥ γ

4.

b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b

xl xu x◦

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SLIDE 27

A solution far from optimal

9

Define the box D := x◦ + [−n∆, n∆]k.

b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b

xl xu x◦

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SLIDE 28

A solution far from optimal

9

Using subdeterminants, there exist feasible x−, x+ ∈ D with x◦ = x− + x+ 2 .

b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b

xl xu x◦ x+ x−

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SLIDE 29

A solution far from optimal

9

The width assumption implies that one of the vectors x−, x+, say x+, has obj(x+) close to obj(x◦):

  • bj(x◦) − obj(x+) ≤ γk(n∆)2

g2 .

b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b

xl xu x◦ x+ x−

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SLIDE 30

A solution far from optimal

9

Thus also x+ is far from optimal:

  • bj(x+) − obj(x∗) ≥ γ(g2 − k(2n∆)2)

4g2 .

b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b

xl xu x◦ x+

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SLIDE 31

Correctness of the algorithm

  • We have obtained a feasible point x+ far from optimal:
  • bj(x+) − obj(x∗) ≥ γ(g2 − k(2n∆)2)

4g2 .

  • Using underestimator µ(x), it follows that x⋄ is close to optimal:
  • bj(x⋄) − obj(x∗) ≤ γk

4g2 x is an -approximate solution to ( ) provided that x x x x k 4g2 4g2 g2 k 2n

2

Just choose g k 2n

2

1

10

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SLIDE 32

Correctness of the algorithm

  • We have obtained a feasible point x+ far from optimal:
  • bj(x+) − obj(x∗) ≥ γ(g2 − k(2n∆)2)

4g2 .

  • Using underestimator µ(x), it follows that x⋄ is close to optimal:
  • bj(x⋄) − obj(x∗) ≤ γk

4g2 x⋄ is an ϵ-approximate solution to (IQP) provided that

  • bj(x⋄) − obj(x∗)
  • bj(x+) − obj(x∗) ≤ ✁

γk ✚ ✚ 4g2 · ✚ ✚ 4g2 ✁ γ(g2 − k(2n∆)2) ≤ ϵ. Just choose g := ⌈√ k ( (2n∆)2 + 1 ϵ )⌉ . □

10

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SLIDE 33

Total unimodularity

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SLIDE 34

Total unimodularity

11

min

k

∑

i=1

−qix2

i + h⊤x

  • s. t.

Wx ≤ w x ∈ Zn. (IQP)

  • If W is TU, (IQP) is polynomially equivalent to its continuous

version.

  • Still NP-hard, as it contains as a special case the minimum

concave-cost network flow problem with quadratic costs (MCCNFP) [Guisewite Pardalos ’90].

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SLIDE 35

Total unimodularity

11

min

k

∑

i=1

−qix2

i + h⊤x

  • s. t.

Wx ≤ w x ∈ Zn. (IQP) Theorem Assume W is TU. For every ϵ ∈ (0, 1] there is an algorithm that finds an ϵ-approximate solution by solving ( 3 + ⌈√ k ( 1 + 1 ϵ )⌉)k LPs of the same size of (IQP), and with a TU constraint matrix.

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SLIDE 36

Total unimodularity

  • In the TU case, the algorithm is the same, except that we replace

g = ⌈√ k ( (2n∆)2 + 1 ϵ )⌉ with g := ⌈√ k ( 1 + 1 ϵ )⌉ .

  • The only difgerence in the proof lies in the construction of the

feasible point x+ far from optimal.

12

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SLIDE 37

A solution far from optimal

13

Wlog x1 is the most concave direction, i.e, max{qi(ui − li)2 : i ∈ {1, . . . , k}} = q1(u1 − l1)2 =: γ.

b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b

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SLIDE 38

A solution far from optimal

13

Let xl, xu be feasible with xl

1 = l1, xu 1 = u1.

b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b

xl xu

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SLIDE 39

A solution far from optimal

13

The vector x◦ = 1

2xl + 1 2xu is far from optimal:

  • bj(x◦) − obj(x∗) ≥ γ

4.

b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b

xl xu x◦

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SLIDE 40

A solution far from optimal

13

Define the box D := [⌊x◦

1 ⌋, ⌈x◦ 1 ⌉] × · · · × [⌊x◦ k⌋, ⌈x◦ k⌉].

b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b

xl xu x◦

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SLIDE 41

A solution far from optimal

13

Since W is TU, the vector x◦ lies in the convex hull of the feasible vectors in D.

b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b

xl xu x◦

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SLIDE 42

A solution far from optimal

13

The width assumption implies that one of the feasible vectors in D, say x+, has obj(x+) close to obj(x◦):

  • bj(x◦) − obj(x+) ≤ γk

4g2 .

b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b

xl xu x◦ x+

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SLIDE 43

A solution far from optimal

13

Thus also x+ is far from optimal:

  • bj(x+) − obj(x∗) ≥ γ(g2 − k)

4g2 .

b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b

xl xu x◦ x+

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SLIDE 44

Correctness of the algorithm

  • We have obtained a feasible point x+ far from optimal:
  • bj(x+) − obj(x∗) ≥ γ(g2 − k)

4g2 .

  • Using underestimator µ(x), it follows that x⋄ is close to optimal:
  • bj(x⋄) − obj(x∗) ≤ γk

4g2 x is an -approximate solution to ( ) provided that x x x x k 4g2 4g2 g2 k Just choose g k 1 1

14

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SLIDE 45

Correctness of the algorithm

  • We have obtained a feasible point x+ far from optimal:
  • bj(x+) − obj(x∗) ≥ γ(g2 − k)

4g2 .

  • Using underestimator µ(x), it follows that x⋄ is close to optimal:
  • bj(x⋄) − obj(x∗) ≤ γk

4g2 x⋄ is an ϵ-approximate solution to (IQP) provided that

  • bj(x⋄) − obj(x∗)
  • bj(x+) − obj(x∗) ≤ ✁

γk ✚ ✚ 4g2 · ✚ ✚ 4g2 ✁ γ(g2 − k) ≤ ϵ. Just choose g := ⌈√ k ( 1 + 1 ϵ )⌉ . □

14

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SLIDE 46

Open questions

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SLIDE 47

Open questions

15

min

k

∑

i=1

−qix2

i + h⊤x

  • s. t.

Wx ≤ w x ∈ Zn. (IQP) Computational complexity

  • 1. (IQP) and its continuous version, if we assume that k is fixed

and that the subdeterminants of W are bounded by either 1 or 2 in absolute value?

  • 2. MCCNFP with quadratic costs, if we assume that the number k of

nonlinear arc costs is fixed?

  • 3. Questions open even for k = 1.
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SLIDE 48

Open questions

15

min

k

∑

i=1

−qix2

i + h⊤x

  • s. t.

Wx ≤ w x ∈ Zn. (IQP) Approximation algorithm

  • 1. (IQP) with general separable quadratic objective function.
  • Even if W is TU and number of nonlinear/concave variables fixed.
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SLIDE 49

Subdeterminants and Concave Integer Quadratic Programming

Alberto Del Pia, University of Wisconsin-Madison January 10, 2019 23rd Combinatorial Optimization Workshop, Aussois 2019