The distribution of the proximity function Timm Oertel Joseph Paat + - - PowerPoint PPT Presentation

The distribution of the proximity function

Timm Oertel∗ Joseph Paat+ Robert Weismantel+

∗Cardiff University, +ETH Z¨

urich

Aussois 2019

Oertel, P., Weismantel ETH Z¨ urich The distribution of the proximity function

The proximity question: Given an optimal LP solution, how close is an optimal IP solution? max{c⊺x : Ax = b, x ∈ Rn

≥0/Zn ≥0}

c Q: Given (A, c), how does proximity depend on b?

Oertel, P., Weismantel ETH Z¨ urich The distribution of the proximity function

Fix c ∈ Zn and A ∈ Zm×n with rank(A) = m. For b ∈ Zm, LP(b) := max{c⊺x : Ax = b, x ∈ Rn

≥0}

IP(b) := max{c⊺x : Ax = b, x ∈ Zn

≥0}.

The proximity function is π(b) := min{x∗ − z∗1 : z∗ optimal for IP(b)}. If IP(b) is infeasible, then π(b) = ∞. Assume: LP(b) has a unique vertex optimal solution x∗ = x∗(b).

Oertel, P., Weismantel ETH Z¨ urich The distribution of the proximity function

Fix c ∈ Zn and A ∈ Zm×n with rank(A) = m. For b ∈ Zm, LP(b) := max{c⊺x : Ax = b, x ∈ Rn

≥0}

IP(b) := max{c⊺x : Ax = b, x ∈ Zn

≥0}.

The proximity function is π(b) := min{x∗ − z∗1 : z∗ optimal for IP(b)}. If IP(b) is infeasible, then π(b) = ∞. Assume: LP(b) has a unique vertex optimal solution x∗ = x∗(b). Q: Given (A, c), how does proximity depend on b? Given (A, c), how is π(·) distributed?

Oertel, P., Weismantel ETH Z¨ urich The distribution of the proximity function

The proximity function is π(b) := min{x∗ − z∗1 : z∗ optimal for IP(b)} Example max{x1 − 2x3 : 3x1 + 2x2 + x3 = b, x ∈ R3

≥0/Z3 ≥0}

1 2 3 4 5 6 7 8 9 10

5 3 10 3

π(·)

Oertel, P., Weismantel ETH Z¨ urich The distribution of the proximity function

Q: Why study proximity?

• A. Faster algorithms for IP

(Eisenbrand, Weismantel’18)

Algorithms for LP with support constraints

(Del Pia, Dey, Weismantel’18) (Cook, Gerards, Schrijver, Tardos, Blair, Jeroslow, Weltge...)

Oertel, P., Weismantel ETH Z¨ urich The distribution of the proximity function

Q: Why study proximity?

• A. Faster algorithms for IP

(Eisenbrand, Weismantel’18)

Algorithms for LP with support constraints

(Del Pia, Dey, Weismantel’18) (Cook, Gerards, Schrijver, Tardos, Blair, Jeroslow, Weltge...)

Q: Why study the distribution of π(·)?

• A. Understand what makes π(·) large.

Make average/probabilistic statements about π(·).

Oertel, P., Weismantel ETH Z¨ urich The distribution of the proximity function

Q: How is π(·) distributed?

• If π(b) < ∞, then π(·) ≤ (4m + 2)m∆

(Eisenbrand, Weismantel’18)

• ∃ (A, c) with max π(·) = ∆ + 1, if m = 1

(Aliev et al.’17)

∆ := max{|δ| : δ is an m × m minor of A}.

Oertel, P., Weismantel ETH Z¨ urich The distribution of the proximity function

Q: How is π(·) distributed?

• If π(b) < ∞, then π(·) ≤ (4m + 2)m∆

(Eisenbrand, Weismantel’18)

• ∃ (A, c) with max π(·) = ∆ + 1, if m = 1

(Aliev et al.’17)

∆ := max{|δ| : δ is an m × m minor of A}. Q: How do we move away from the worst case?

Oertel, P., Weismantel ETH Z¨ urich The distribution of the proximity function

For D ⊆ Zm, Pr(D) := lim

t→∞

|{b ∈ D : b∞ ≤ t, π(b) < ∞}| |{b ∈ Zm : b∞ ≤ t, π(b) < ∞}|

Oertel, P., Weismantel ETH Z¨ urich The distribution of the proximity function

For D ⊆ Zm, Pr(D) := lim

t→∞

|{b ∈ D : b∞ ≤ t, π(b) < ∞}| |{b ∈ Zm : b∞ ≤ t, π(b) < ∞}|

• Q. What does Pr(D) = 1 mean?

t ≈ 0 t > 0 t → ∞

Oertel, P., Weismantel ETH Z¨ urich The distribution of the proximity function

Q: How is π(·) distributed?

• If π(b) < ∞, then π(·) ≤ (4m + 2)m∆

(Eisenbrand, Weismantel’18)

• ∃ (A, c) with max π(·) = ∆ + 1, if m = 1

(Aliev et al.’17)

∆ := max{|δ| : δ is an m × m minor of A}

• Thm. (O., P., W.)

Pr( π(·) ≤ ∆ · (m + 1) ) = 1.

• Thm. (O., P., W.) If B ⊆ col(A) is an optimal LP basis

for some b ∈ Zm, then π(·) is ‘eventually’ periodic in cone(B).

Oertel, P., Weismantel ETH Z¨ urich The distribution of the proximity function

Q: How is π(·) distributed?

• If π(b) < ∞, then π(·) ≤ (4m + 2)m∆

(Eisenbrand, Weismantel’18)

• ∃ (A, c) with max π(·) = ∆ + 1, if m = 1

(Aliev et al.’17)

∆ := max{|δ| : δ is an m × m minor of A}

• Thm. (O., P., W.)

Pr( π(·) ≤ ∆ · (m + 1) ) = 1.

• Thm. (O., P., W.) If the optimal LP bases B ⊆ col(A)

partition cone(A), then π(·) is ‘eventually’ periodic in cone(B).

Oertel, P., Weismantel ETH Z¨ urich The distribution of the proximity function

• Q. What does “π(·) is ‘eventually’ periodic in cone(B)” mean?

· Let B ⊆ col(A) be an optimal LP basis. ‘Eventually Periodic’ If b ≡ b′ and b, b′ ∈ b∗ + cone(B), where b∗ := B · ∆✶m, then π(b) = π(b′). B and col(A) ⊆ Zm

Oertel, P., Weismantel ETH Z¨ urich The distribution of the proximity function

• Q. What does “π(·) is ‘eventually’ periodic in cone(B)” mean?

· Let B ⊆ col(A) be an optimal LP basis. · B divides Zm into equivalence classes { , , } ‘Eventually Periodic’ If b ≡ b′ and b, b′ ∈ b∗ + cone(B), where b∗ := B · ∆✶m, then π(b) = π(b′). B and col(A) ⊆ Zm

Oertel, P., Weismantel ETH Z¨ urich The distribution of the proximity function

• Q. What does “π(·) is ‘eventually’ periodic in cone(B)” mean?

· Let B ⊆ col(A) be an optimal LP basis. · B divides Zm into equivalence classes { , , } ‘Eventually Periodic’ If b ≡ b′ and b, b′ ∈ b∗ + cone(B), where b∗ := B · ∆✶m, then π(b) = π(b′). B and col(A) ⊆ Zm b∗ b b′

Oertel, P., Weismantel ETH Z¨ urich The distribution of the proximity function

Conclusions

• Studying π(·) lets us make worst and average case statements.
• π(·) is eventually periodic.

The distribution of other IP functions Sparsity of IP solutions depends on the minimum minor.

(Oertel., P., Weismantel)

Oertel, P., Weismantel ETH Z¨ urich The distribution of the proximity function

Conclusions

• Studying π(·) lets us make worst and average case statements.
• π(·) is eventually periodic.

The distribution of other IP functions Sparsity of IP solutions depends on the minimum minor.

(Oertel., P., Weismantel)

Thank you

Oertel, P., Weismantel ETH Z¨ urich The distribution of the proximity function