joint work with Kevin Cheung and Daniel Steffy VIPR: Verifying Integer Programming Results Ambros Gleixner, Zuse Institute Berlin 21st Combinatorial Optimization Workshop · Aussois · January 9, 2017
1. Motivation and background ∙ van der Waerden numbers ∙ Frankl’s conjecture ∙ floating-point/exact integer programming 2. Verification of integer programming results ∙ verification in SAT solving ∙ tree-less branch-and-cut certificates ∙ computational experiments 3. Conclusion A. Gleixner, Zuse Institute Berlin Verifying Integer Programming Results 1/15 outline
1. Motivation and background ∙ van der Waerden numbers ∙ Frankl’s conjecture ∙ floating-point/exact integer programming ∙ verification in SAT solving ∙ tree-less branch-and-cut certificates ∙ computational experiments 3. Conclusion A. Gleixner, Zuse Institute Berlin Verifying Integer Programming Results 1/15 outline 2. Verification of integer programming results
1. Motivation and background ∙ van der Waerden numbers ∙ Frankl’s conjecture ∙ floating-point/exact integer programming 2. Verification of integer programming results ∙ verification in SAT solving ∙ tree-less branch-and-cut certificates ∙ computational experiments A. Gleixner, Zuse Institute Berlin Verifying Integer Programming Results 1/15 outline 3. Conclusion
∙ Recent success using MIP solvers: Theorem [Pulaj 2015] Proof: 7-coloring of 257 Verifying Integer Programming Results A. Gleixner, Zuse Institute Berlin . 2 m m m without monochromatic 2/15 1 2 258. W 7 3 lower bounds / show infeasibility (cannot color non-monochromatically) ∙ 7 non-trivial numbers known, mostly by SAT solvers: motivation: extremal combinatorics van der Waerden numbers ∙ W(r,k) := smallest M such that any r -coloring of { 1 , 2 , . . . M } contains a monochromatic arithmetic progression of k integers
∙ 7 non-trivial numbers known, mostly by SAT solvers: lower bounds / show infeasibility (cannot color non-monochromatically) A. Gleixner, Zuse Institute Berlin Verifying Integer Programming Results 2/15 motivation: extremal combinatorics van der Waerden numbers ∙ W(r,k) := smallest M such that any r -coloring of { 1 , 2 , . . . M } contains a monochromatic arithmetic progression of k integers ∙ Recent success using MIP solvers: Theorem [Pulaj 2015] W ( 7 , 3 ) ⩾ 258. Proof: 7-coloring of { 1 , 2 , . . . 257 } without monochromatic { m , m + ℓ, m + 2 ℓ } .
Proposition [Pulaj 2016] c i x S 2 n 2 1 c n 2 1 2 1 Proof: Rewrite Poonen’s Theorem (1992). 3/15 T n x T x U S S n U U x S 0 1 S n A. Gleixner, Zuse Institute Berlin Verifying Integer Programming Results S 1 T x S contained in at least half its sets. is Frankl-complete if weights c n 0 , not all zero, FC S i S c i i S 1 x T x U motivation: frankl’s conjecture Frankl’s Conjecture (1979) Any family of union-closed sets F ⊆ 2 { 1 ,..., n } has an element i ∈ { 1 , . . . , n } Line of attack: Frankl-complete families A : ⇔ conjecture true for all F ⊇ A .
3/15 0 , not all zero, Verifying Integer Programming Results contained in at least half its sets. A. Gleixner, Zuse Institute Berlin c i motivation: frankl’s conjecture Frankl’s Conjecture (1979) Any family of union-closed sets F ⊆ 2 { 1 ,..., n } has an element i ∈ { 1 , . . . , n } Line of attack: Frankl-complete families A : ⇔ conjecture true for all F ⊇ A . Proposition [Pulaj 2016] A is Frankl-complete if ∃ weights c ∈ N n ( ∑ ) ∑ ∑ c i − x S ⩽ − 1 S ∈ 2 [ n ] i ∈ S ∈ S i / ∀ T ∪ U = S ∈ 2 { 1 ,..., n } x T + x U ⩽ 1 + x S FC ( A , c ) n := = ∅ . ∀ S ∈ A , S ∪ T = U ∈ 2 { 1 ,..., n } x T ⩽ x U ∀ S ∈ 2 { 1 ,..., n } x S ∈ { 0 , 1 } Proof: Rewrite Poonen’s Theorem (1992).
∙ tree of dual bounds from LP solutions ∙ floating-point errors tolerances for feasibility and numeric comparisons ∙ aggressive problem modification during the algorithm exact solution approximate solution invalid model strengthening A. Gleixner, Zuse Institute Berlin Verifying Integer Programming Results 4/15 floating-point mip solvers Mixed-Integer Linear Programming minimize c T x subject to Ax ⩽ b and x i ∈ Z for some i LP-based branch-and-cut
∙ aggressive problem modification during the algorithm ∙ tree of dual bounds from LP solutions exact solution approximate solution invalid model strengthening A. Gleixner, Zuse Institute Berlin Verifying Integer Programming Results 4/15 floating-point mip solvers Mixed-Integer Linear Programming minimize c T x subject to Ax ⩽ b and x i ∈ Z for some i LP-based branch-and-cut ∙ floating-point errors ⇝ tolerances for feasibility and numeric comparisons
∙ aggressive problem modification during the algorithm ∙ tree of dual bounds from LP solutions exact solution approximate solution invalid model strengthening A. Gleixner, Zuse Institute Berlin Verifying Integer Programming Results 4/15 floating-point mip solvers Mixed-Integer Linear Programming minimize c T x subject to Ax ⩽ b and x i ∈ Z for some i LP-based branch-and-cut ∙ floating-point errors ⇝ tolerances for feasibility and numeric comparisons
But… ∙ LP: Qsopt_ex [Espinoza 2006; Applegate, Cook, Dash, Espinoza 2007] ∙ LP: SoPlex [G. 2015; G., Steffy, Wolter 2016] ∙ MIP: SCIP [Cook, Koch, Steffy, Wolter 2013] ff Math. Prog. Comput. , 5(3):305–344, 2013 A. Gleixner, Zuse Institute Berlin Verifying Integer Programming Results 5/15 exact integer programming over q Exact rational solvers exist
5/15 ff ∙ LP: Qsopt_ex [Espinoza 2006; Applegate, Cook, Dash, Espinoza 2007] ∙ LP: SoPlex [G. 2015; G., Steffy, Wolter 2016] ∙ MIP: SCIP [Cook, Koch, Steffy, Wolter 2013] Verifying Integer Programming Results A. Gleixner, Zuse Institute Berlin Math. Prog. Comput. , 5(3):305–344, 2013 exact integer programming over q Exact rational solvers exist But… A Hybrid Branch-and-Bound Approach for Exact Rational Mixed-Integer Programming William Cook · Thorsten Koch · Daniel E. Steffy · Kati Wolter 2 Of course, even with a very carful implementation and extensive testing, a certain risk of an imple- mentation error remains (also in the underlying exact LP solver and the software package for rational arithmetic). So, the exact objective values reported here come with no warranty.
1. Motivation and background ∙ van der Waerden numbers ∙ Frankl’s conjecture ∙ floating-point/exact integer programming ∙ verification in SAT solving ∙ tree-less branch-and-cut certificates ∙ computational experiments 3. Conclusion A. Gleixner, Zuse Institute Berlin Verifying Integer Programming Results 6/15 outline 2. Verification of integer programming results
Verification in MIP ∙ many SAT solvers produce a trace: certificate file for infeasible problems ∙ DRAT-trim: formally verified trace checker [Welzer, Heule, Hunt 2014] Kullmann, Marek 2016] ∙ MIP solvers do not output an optimality certificate, but: ∙ verification of a TSP tour through 85,900 cities [Applegate et al. 2009] ∙ formal verification of LP-based proof of Kepler’s Conjecture [Flyspeck: Obua 2008, Hales et al. 2015] A. Gleixner, Zuse Institute Berlin Verifying Integer Programming Results 7/15 status quo Verification in UNSAT ∙ May 2016: 200 TB proof for the Pythagorean Triples problem [Heule,
∙ many SAT solvers produce a trace: certificate file for infeasible problems ∙ DRAT-trim: formally verified trace checker [Welzer, Heule, Hunt 2014] ∙ May 2016: 200 TB proof for the Pythagorean Triples problem [Heule, Kullmann, Marek 2016] ∙ MIP solvers do not output an optimality certificate , but: ∙ verification of a TSP tour through 85,900 cities [Applegate et al. 2009] ∙ formal verification of LP-based proof of Kepler’s Conjecture [Flyspeck: Obua 2008, Hales et al. 2015] A. Gleixner, Zuse Institute Berlin Verifying Integer Programming Results 7/15 status quo Verification in UNSAT Verification in MIP
∙ many SAT solvers produce a trace: certificate file for infeasible problems ∙ DRAT-trim: formally verified trace checker [Welzer, Heule, Hunt 2014] ∙ May 2016: 200 TB proof for the Pythagorean Triples problem [Heule, Kullmann, Marek 2016] ∙ MIP solvers do not output an optimality certificate, but: ∙ verification of a TSP tour through 85,900 cities [Applegate et al. 2009] ∙ formal verification of LP-based proof of Kepler’s Conjecture [Flyspeck: Obua 2008, Hales et al. 2015] A. Gleixner, Zuse Institute Berlin Verifying Integer Programming Results 7/15 status quo Verification in UNSAT Verification in MIP
Goals of our certificate format ∙ MIP solvers use complex and diverse algorithms ∙ formal verification involving rational arithmetic is prohibitively slow ∙ certificate form unclear: tree, cuts, superadditive function, … ∙ expressivity : encode all (or most important) MIP techniques ∙ simplicity : allow verification by short checker code A. Gleixner, Zuse Institute Berlin Verifying Integer Programming Results 8/15 status quo Difficulties
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