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 ∈ R n ≥ 0 / Z n ≥ 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 ∈ Z n and A ∈ Z m × n with rank( A ) = m . For b ∈ Z m , LP( b ) := max { c ⊺ x : Ax = b , x ∈ R n ≥ 0 } IP( b ) := max { c ⊺ x : Ax = b , x ∈ Z n ≥ 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 ∈ Z n and A ∈ Z m × n with rank( A ) = m . For b ∈ Z m , LP( b ) := max { c ⊺ x : Ax = b , x ∈ R n ≥ 0 } IP( b ) := max { c ⊺ x : Ax = b , x ∈ Z n ≥ 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 ) } max { x 1 − 2 x 3 : 3 x 1 + 2 x 2 + x 3 = b , x ∈ R 3 ≥ 0 / Z 3 Example ≥ 0 } 10 3 π ( · ) 5 3 0 1 2 3 4 5 6 7 8 9 10 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 ( · ) ≤ (4 m + 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 ( · ) ≤ (4 m + 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

|{ b ∈ D : � b � ∞ ≤ t , π ( b ) < ∞}| For D ⊆ Z m , Pr( D ) := lim |{ b ∈ Z m : � b � ∞ ≤ t , π ( b ) < ∞}| t →∞ Oertel, P., Weismantel ETH Z¨ urich The distribution of the proximity function

|{ b ∈ D : � b � ∞ ≤ t , π ( b ) < ∞}| For D ⊆ Z m , Pr( D ) := lim |{ b ∈ Z m : � b � ∞ ≤ t , π ( b ) < ∞}| t →∞ 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 ( · ) ≤ (4 m + 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 ∈ Z m , 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 ( · ) ≤ (4 m + 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 , 0 then π ( b ) = π ( b ′ ). B and col( A ) ⊆ Z m 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 Z m into equivalence classes { , , } ‘Eventually Periodic’ If b ≡ b ′ and b , b ′ ∈ b ∗ + cone( B ), where b ∗ := B · ∆ ✶ m , 0 then π ( b ) = π ( b ′ ). B and col( A ) ⊆ Z m 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 Z m into equivalence b ′ classes { , , } b ‘Eventually Periodic’ b ∗ If b ≡ b ′ and b , b ′ ∈ b ∗ + cone( B ), where b ∗ := B · ∆ ✶ m , 0 then π ( b ) = π ( b ′ ). B and col( A ) ⊆ Z m 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