On Strong NP-completeness of Rational Problems Dominik Wojtczak University of Liverpool CSR 2018
Motivation Dominik Wojtczak On Strong NP-completeness of Rational Problems 2/18
What we found A rather subtle point is the question of rational coefficients. In- deed, most textbooks get rid of this case, where some or all input values are non-integer, by the trivial statement that multiplying with a suitable factor, e.g. with the smallest common multiple of the denominators, if the values are given as fractions or by a suitable power of 10, transforms the data into integers. Clearly, this may transform even a problem of moderate size into a rather unpleasant problem with huge coefficients. — Hans Kellerer, Ulrich Pferschy, and David Pisinger. Knapsack problems . Springer, 2004. Dominik Wojtczak On Strong NP-completeness of Rational Problems 3/18
Definitions (1) A rational number is given as (numerator, denominator) written in unary. Dominik Wojtczak On Strong NP-completeness of Rational Problems 4/18
Definitions (1) A rational number is given as (numerator, denominator) written in unary. Definition ( Knapsack problems) Assume there are n items whose non-negative rational weights and profits are given as a list L = { ( w 1 , v 1 ) , . . . , ( w n , v n ) } . Let the capacity be W ∈ Q ≥ 0 and the profit threshold be V ∈ Q ≥ 0 . Dominik Wojtczak On Strong NP-completeness of Rational Problems 4/18
Definitions (1) A rational number is given as (numerator, denominator) written in unary. Definition ( Knapsack problems) Assume there are n items whose non-negative rational weights and profits are given as a list L = { ( w 1 , v 1 ) , . . . , ( w n , v n ) } . Let the capacity be W ∈ Q ≥ 0 and the profit threshold be V ∈ Q ≥ 0 . 0-1 Knapsack : Is there a subset of L whose total weight does not exceed W and total profit is at least V ? Dominik Wojtczak On Strong NP-completeness of Rational Problems 4/18
Definitions (1) A rational number is given as (numerator, denominator) written in unary. Definition ( Knapsack problems) Assume there are n items whose non-negative rational weights and profits are given as a list L = { ( w 1 , v 1 ) , . . . , ( w n , v n ) } . Let the capacity be W ∈ Q ≥ 0 and the profit threshold be V ∈ Q ≥ 0 . 0-1 Knapsack : Is there a subset of L whose total weight does not exceed W and total profit is at least V ? Unbounded Knapsack : Is there a list of non-negative integers ( q 1 , . . . , q n ) such that n n � � q i · w i ≤ W and q i · v i ≥ V ? i =1 i =1 (Intuitively, q i denotes the number of times the i -th item in A is chosen.) Dominik Wojtczak On Strong NP-completeness of Rational Problems 4/18
Definitions (2) Definition ( Subset Sum problems) Assume we are given a list of n items with rational non-negative weights A = { w 1 , . . . , w n } and a target total weight W ∈ Q ≥ 0 . 0-1 Subset Sum : Does there exists a subset B of A such that the total weight of B is equal to W ? Unbounded Subset Sum : Does there exist a list of non-negative integer quantities ( q 1 , . . . , q n ) such that n � q i · w i = W ? i =1 (Intuitively, q i denotes the number of times the i -th item in A is chosen.) Dominik Wojtczak On Strong NP-completeness of Rational Problems 5/18
Definitions (3) Definition ( Partition problem) Assume we are given a list of n items with non-negative rational weights A = { w 1 , . . . , w n } . Can the set A be partitioned into two sets with equal total weights? Dominik Wojtczak On Strong NP-completeness of Rational Problems 6/18
Definitions (3) Definition ( Partition problem) Assume we are given a list of n items with non-negative rational weights A = { w 1 , . . . , w n } . Can the set A be partitioned into two sets with equal total weights? Example Problem Dominik Wojtczak On Strong NP-completeness of Rational Problems 6/18
Money in 18th century England Dominik Wojtczak On Strong NP-completeness of Rational Problems 7/18
The Reductions Dominik Wojtczak On Strong NP-completeness of Rational Problems 8/18
The Actual Reductions One-in-Three-SAT for 3-CNF ≤ p m One-in-Three-SAT for 3-CNF ≤ 4 ≤ p m All-the-Same-SAT for 3-CNF ≤ 4 ≤ p m Unbounded Subset Sum ≤ p m Unbounded Knapsack All-the-Same-SAT for 3-CNF ≤ 4 ≤ p m Partition All-the-Same-SAT for 3-CNF ≤ 4 ≤ p m Subset Sum ≤ p m Knapsack Dominik Wojtczak On Strong NP-completeness of Rational Problems 9/18
In the Pursuit of Satisfaction The One-in-Three-SAT problem for 3-CNF formulae asks for an truth assignment that makes exactly one literal in each clause true. Dominik Wojtczak On Strong NP-completeness of Rational Problems 10/18
In the Pursuit of Satisfaction The One-in-Three-SAT problem for 3-CNF formulae asks for an truth assignment that makes exactly one literal in each clause true. 3-CNF ≤ 4 is the set of 3-CNF formulae that use each variable at most four times. Theorem The One-in-Three-SAT problem for 3-CNF ≤ 4 is NP -complete. Dominik Wojtczak On Strong NP-completeness of Rational Problems 10/18
In the Pursuit of Satisfaction The One-in-Three-SAT problem for 3-CNF formulae asks for an truth assignment that makes exactly one literal in each clause true. 3-CNF ≤ 4 is the set of 3-CNF formulae that use each variable at most four times. Theorem The One-in-Three-SAT problem for 3-CNF ≤ 4 is NP -complete. We define All-the-Same-SAT for 3-CNF formulae to be a problem of checking for a valuation that makes exactly the same number of literals true in every clause (this may be zero). Dominik Wojtczak On Strong NP-completeness of Rational Problems 10/18
In the Pursuit of Satisfaction The One-in-Three-SAT problem for 3-CNF formulae asks for an truth assignment that makes exactly one literal in each clause true. 3-CNF ≤ 4 is the set of 3-CNF formulae that use each variable at most four times. Theorem The One-in-Three-SAT problem for 3-CNF ≤ 4 is NP -complete. We define All-the-Same-SAT for 3-CNF formulae to be a problem of checking for a valuation that makes exactly the same number of literals true in every clause (this may be zero). Theorem The All-the-Same-SAT problem for 3-CNF ≤ 4 formulae is NP -complete. Dominik Wojtczak On Strong NP-completeness of Rational Problems 10/18
Prime Suspects (1) Theorem (Rosser (1962)) π i < i (log i + log log i ) for i ≥ 6 Dominik Wojtczak On Strong NP-completeness of Rational Problems 11/18
Prime Suspects (1) Theorem (Rosser (1962)) π i < i (log i + log log i ) for i ≥ 6 Corollary The total size of the first n prime numbers, when written down in unary, is O ( n 2 log n ) . Furthermore, they can be computed in polynomial time. Dominik Wojtczak On Strong NP-completeness of Rational Problems 11/18
Prime Suspects (2) Lemma Let ( p 1 , . . . , p n ) be a list of n different prime numbers. Dominik Wojtczak On Strong NP-completeness of Rational Problems 12/18
Prime Suspects (2) Lemma Let ( p 1 , . . . , p n ) be a list of n different prime numbers. Let ( a 0 , a 1 , . . . , a n ) and ( b 0 , b 1 , . . . , b n ) be two lists of integers such that | a i − b i | < p i holds for all i = 1 , . . . , n. Dominik Wojtczak On Strong NP-completeness of Rational Problems 12/18
Prime Suspects (2) Lemma Let ( p 1 , . . . , p n ) be a list of n different prime numbers. Let ( a 0 , a 1 , . . . , a n ) and ( b 0 , b 1 , . . . , b n ) be two lists of integers such that | a i − b i | < p i holds for all i = 1 , . . . , n. We then have a 0 + a 1 + . . . + a n = b 0 + b 1 + . . . + b n p 1 p n p 1 p n if and only if a i = b i for all i = 0 , . . . , n . Dominik Wojtczak On Strong NP-completeness of Rational Problems 12/18
All-the-Same-SAT ≤ p m Unbounded Subset Sum Assume we are given a 3-CNF ≤ 4 formula φ = C 1 ∧ C 2 ∧ . . . ∧ C m with m clauses C 1 , . . . , C m and n propositional variables x 1 , . . . , x n , Dominik Wojtczak On Strong NP-completeness of Rational Problems 13/18
All-the-Same-SAT ≤ p m Unbounded Subset Sum Assume we are given a 3-CNF ≤ 4 formula φ = C 1 ∧ C 2 ∧ . . . ∧ C m with m clauses C 1 , . . . , C m and n propositional variables x 1 , . . . , x n , where C j = a j ∨ b j ∨ c j for j = 1 , . . . , m , Dominik Wojtczak On Strong NP-completeness of Rational Problems 13/18
All-the-Same-SAT ≤ p m Unbounded Subset Sum Assume we are given a 3-CNF ≤ 4 formula φ = C 1 ∧ C 2 ∧ . . . ∧ C m with m clauses C 1 , . . . , C m and n propositional variables x 1 , . . . , x n , where C j = a j ∨ b j ∨ c j for j = 1 , . . . , m , each a j , b j , c j is a literal equal to x i or ¬ x i for some i . Dominik Wojtczak On Strong NP-completeness of Rational Problems 13/18
All-the-Same-SAT ≤ p m Unbounded Subset Sum Assume we are given a 3-CNF ≤ 4 formula φ = C 1 ∧ C 2 ∧ . . . ∧ C m with m clauses C 1 , . . . , C m and n propositional variables x 1 , . . . , x n , where C j = a j ∨ b j ∨ c j for j = 1 , . . . , m , each a j , b j , c j is a literal equal to x i or ¬ x i for some i . For a literal l , we write that l ∈ C j iff l is equal to a j , b j or c j . Dominik Wojtczak On Strong NP-completeness of Rational Problems 13/18
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