Honors Combinatorics CMSC-27410 = Math-28410 ∼ CMSC-37200 Instructor: Laszlo Babai University of Chicago Week 4, Tuesday, April 28, 2020 CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
Today’s material Linear Program (LP) Primal/Dual LP , Duality Theorem Integer Linear Program (ILP), integrality gap Hypergraph cover and matching Hypergraph fractional cover and fractional matching Lovász’s greedy vs. fractional cover theorem CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
System of linear equations k linear equations in n unknowns a 11 x 1 + a 12 x 2 + . . . + a 1 n x n = b 1 a 21 x 1 + a 22 x 2 + . . . + a 2 n x n = b 2 . . . . . . . . . a k 1 x 1 + a k 2 x 2 + . . . + a kn x n = b k CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
System of linear equations k linear equations in n unknowns a 11 x 1 + a 12 x 2 + . . . + a 1 n x n = b 1 a 21 x 1 + a 22 x 2 + . . . + a 2 n x n = b 2 . . . . . . . . . a k 1 x 1 + a k 2 x 2 + . . . + a kn x n = b k Ax = b CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
System of linear equations Ax = b a 11 a 12 . . . a 1 n x 1 b 1 a 21 a 22 . . . a 2 n x 2 b 2 A = x = b = . . . . . . . . . . . . . . . a k 1 a k 2 . . . a kn x n b k CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
System of linear inequalities k linear inequatlities in n unknowns a 11 x 1 + a 12 x 2 + . . . + a 1 n x n ≤ b 1 a 21 x 1 + a 22 x 2 + . . . + a 2 n x n ≤ b 2 . . . . . . . . . a k 1 x 1 + a k 2 x 2 + . . . + a kn x n ≤ b k CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
System of linear inequalities k linear inequatlities in n unknowns a 11 x 1 + a 12 x 2 + . . . + a 1 n x n ≤ b 1 a 21 x 1 + a 22 x 2 + . . . + a 2 n x n ≤ b 2 . . . . . . . . . a k 1 x 1 + a k 2 x 2 + . . . + a kn x n ≤ b k Ax ≤ b CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
System of linear inequalities partially ordering the vectors: coordinatewise u 1 v 1 u 2 v 2 ≤ . . . . . . u k v k if ( ∀ i )( u i ≤ v i ) Same for matrices: A ≤ B if ( ∀ i , j )( a ij ≤ b ij ) DO: A ≤ B ∧ C ≥ 0 = ⇒ AC ≤ BC CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
System of linear inequalities Ax ≤ b feasible system: solution exists CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
System of linear inequalities Ax ≤ b feasible system: solution exists Example of infeasible system: x 1 ≤ 2 x 1 ≥ 3 CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
System of linear inequalities Ax ≤ b feasible system: solution exists Example of infeasible system: x 1 ≤ 2 ≤ x 1 2 equivalently x 1 ≥ 3 − x 1 ≤ − 3 CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
Linear Program (LP) A ∈ R k × n , b ∈ R k , c ∈ R n Input: Ax ≤ b , x ≥ 0 Constraints: max ← c · x Objective: CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
Linear Program (LP) A ∈ R k × n , b ∈ R k , c ∈ R n Input: Ax ≤ b , x ≥ 0 Constraints: max ← c · x Objective: max ← { c · x | Ax ≤ b , x ≥ 0 } Goal: maximize objective function subject to constraints CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
Feasible LP max ← { c · x | Ax ≤ b , x ≥ 0 } x ∈ R n that feasible solution: satisfies the constraints: Ax ≤ b , x ≥ 0 feasible LP: ∃ feasible solution, i.e., ( ∃ x ∈ R n )( Ax ≤ b , x ≥ 0 ) CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
Linear Program (LP) A ∈ R k × n , b ∈ R k , c ∈ R k Input: Constraints: Ax ≤ b , x ≥ 0 Objective: max ← c · x CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
Linear Program (LP) A ∈ R k × n , b ∈ R k , c ∈ R k Input: Constraints: Ax ≤ b , x ≥ 0 Objective: max ← c · x max ← { c · x | Ax ≤ b , x ≥ 0 } CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
Linear Program (LP) A ∈ R k × n , b ∈ R k , c ∈ R k Input: Constraints: Ax ≤ b , x ≥ 0 Objective: max ← c · x max ← { c · x | Ax ≤ b , x ≥ 0 } feasible LP: constraints feasible ( ∃ x ∈ R n )( Ax ≤ b , x ≥ 0 ) CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
Primal/Dual Linear Program INPUT: A ∈ R k × n , b ∈ R k , c ∈ R n UNKNOWNS: primal variables x ∈ R n , dual variables y ∈ R k PRIMAL LP: max ← { c · x | Ax ≤ b , x ≥ 0 } CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
Primal/Dual Linear Program INPUT: A ∈ R k × n , b ∈ R k , c ∈ R n UNKNOWNS: primal variables x ∈ R n , dual variables y ∈ R k PRIMAL LP: max ← { c · x | Ax ≤ b , x ≥ 0 } DUAL LP: min ← { b · y | A T y ≥ c , y ≥ 0 } CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
Primal/Dual Linear Program A ∈ R k × n , b ∈ R k , c ∈ R n INPUT: PRIMAL: max ← { c · x | Ax ≤ b , x ≥ 0 } min ← { b · y | A T y ≥ c , y ≥ 0 } DUAL: Lemma. If x 0 is a feasible solution to the Primal and y 0 a feasible solution to the Dual then c · x 0 ≤ b · y 0 Gives upper bound for MAX lower bound for MIN CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
Primal/Dual Linear Program PRIMAL: max ← { c · x | Ax ≤ b , x ≥ 0 } min ← { b · y | A T y ≥ c , y ≥ 0 } DUAL: Lemma. If x 0 is a feasible solution to the Primal and y 0 a feasible solution to the Dual then c · x 0 ≤ b · y 0 (b) ( AB ) T = B T A T Proof. Note: (a) a · b = a T b CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
Primal/Dual Linear Program PRIMAL: max ← { c · x | Ax ≤ b , x ≥ 0 } min ← { b · y | A T y ≥ c , y ≥ 0 } DUAL: Lemma. If x 0 is a feasible solution to the Primal and y 0 a feasible solution to the Dual then c · x 0 ≤ b · y 0 (b) ( AB ) T = B T A T Proof. Note: (a) a · b = a T b y T 0 ( Ax 0 ) ≤ y T 0 b = y 0 · b = b · y 0 CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
Primal/Dual Linear Program PRIMAL: max ← { c · x | Ax ≤ b , x ≥ 0 } min ← { b · y | A T y ≥ c , y ≥ 0 } DUAL: Lemma. If x 0 is a feasible solution to the Primal and y 0 a feasible solution to the Dual then c · x 0 ≤ b · y 0 (b) ( AB ) T = B T A T Proof. Note: (a) a · b = a T b y T 0 ( Ax 0 ) ≤ y T 0 b = y 0 · b = b · y 0 ( y T 0 A ) x 0 = ( A T y 0 ) T x 0 ≥ c T x 0 = c · x 0 QED CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
Duality Theorem of Linear Programming PRIMAL: max ← { c · x | Ax ≤ b , x ≥ 0 } min ← { b · y | A T y ≥ c , y ≥ 0 } DUAL: Lemma. If x 0 is a feasible solution to the Primal and y 0 a feasible solution to the Dual then c · x 0 ≤ b · y 0 Max PRIMAL Min DUAL CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
Duality Theorem of Linear Programming PRIMAL: max ← { c · x | Ax ≤ b , x ≥ 0 } min ← { b · y | A T y ≥ c , y ≥ 0 } DUAL: Lemma. If x 0 is a feasible solution to the Primal and y 0 a feasible solution to the Dual then c · x 0 ≤ b · y 0 Max PRIMAL ≤ Min DUAL CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
Duality Theorem of Linear Programming PRIMAL: max ← { c · x | Ax ≤ b , x ≥ 0 } min ← { b · y | A T y ≥ c , y ≥ 0 } DUAL: Lemma. If x 0 is a feasible solution to the Primal and y 0 a feasible solution to the Dual then c · x 0 ≤ b · y 0 Max PRIMAL ≤ Min DUAL LP Duality Theorem If Primal feasible and range of obj functn bounded from above then Dual feasible and the two attain equal optima: ( ∃ feasible x ∈ R n , y ∈ R k )( c · x = b · y ) CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
Duality Theorem of Linear Programming PRIMAL: max ← { c · x | Ax ≤ b , x ≥ 0 } min ← { b · y | A T y ≥ c , y ≥ 0 } DUAL: Lemma. If x 0 is a feasible solution to the Primal and y 0 a feasible solution to the Dual then c · x 0 ≤ b · y 0 Max PRIMAL ≤ Min DUAL LP Duality Theorem If Primal feasible and range of obj functn bounded from above then Dual feasible and the two attain equal optima: ( ∃ feasible x ∈ R n , y ∈ R k )( c · x = b · y ) Max PRIMAL = Min DUAL CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
Program checking Untrusted party wishes to sell us a cloud service to solve huge LPs Money-back Guarantee: gives optimal solution whenever one exists Can we catch them at cheating without running a trusted LP solver on our problems? We can check that the solution they bring is feasible . Can we check it is optimal? CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
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