
Recent Posts
 Linear Bounded Automaton Acceptance November 21, 2017
 Consistency of Database Frequency Tables November 15, 2017
 Safety of Database Transaction Systems November 7, 2017
 Serializability of Database Histories October 31, 2017
 NonCircular Satisfiability October 27, 2017
Recent Comments
Archives
 November 2017
 October 2017
 September 2017
 August 2017
 July 2017
 June 2017
 May 2017
 April 2017
 March 2017
 February 2017
 January 2017
 December 2016
 November 2016
 October 2016
 September 2016
 August 2016
 July 2016
 June 2016
 May 2016
 April 2016
 March 2016
 February 2016
 January 2016
 December 2015
 November 2015
 October 2015
 September 2015
 August 2015
 July 2015
 June 2015
 May 2015
 April 2015
 March 2015
 February 2015
 January 2015
 December 2014
 November 2014
 October 2014
 September 2014
 August 2014
 July 2014
 June 2014
Categories
Meta
Monthly Archives: July 2014
Protected: Partition Into Paths of Length 2
Enter your password to view comments.
Posted in Chapter 3 Exercises
Tagged 3DM, Difficulty 7, PPL2, reductions, uncited reduction
Protected: Set Splitting
Enter your password to view comments.
Posted in Chapter 3 Exercises
Tagged 3sat, Difficulty 5, Hypergraph 2colorability, NAE3SAT, reductions, Set Splitting, SP4, uncited reduction
Not All Equal 3SAT
This is jumping ahead a little, but this is a variant of 3SAT that will be helpful in the future (actually, for the next problem)
The problem: NotAllEqual3SAT (NAE3SAT). This is problem LO3 in the appendix.
The description: Just like 3SAT, except instead of requiring at least one of the literals in each clause to be true, we’re requiring at least one literal to be true and at least one literal to be false. So we’re removing the case where all three literals can be true.
Notice that if we have an assignment of variables that is a NAE3SAT solution, then if we negate all of the variables, we still have a NAE3SAT solution
The reduction: From regular 3SAT. We’re given a formula that is a collection of 3literal clauses. For each clause (x_{1} ∨ x_{2} ∨ x_{3}) we create a new variable c_{i} (for clause i) and add 2 clauses to our NAE3SAT instance:
(x_{1}∨ x_{2} ∨ c_{i}) and (x_{3} ∨ ~c_{i} ∨ F) (Where “F” is the constant value false. We’ll talk more about this later)
The idea is that c_{i} is there to “fix” the clause. If the original clause was satisfiable because of x_{1} or x_{2}, then c_{i} can be false, and its negation can make the second clause true. If the original clause was satisfiable because of x_{3}, we can make c_{i} true. If the original clause had all three literals satisfiable, we can make c_{i} false, and become an acceptable NAE3SAT solution.
Note that because the definition of NAE3SAT (and, for that matter, regular 3SAT) is defined in terms of collections of 3 variables, the constant value false we used above isn’t strictly legal. We’ve really just shown “NAE3SAT with some constants replacing variables” is NPHard. To show “normal” NAE3SAT is NPHard, we need to reduce the fixed constant version into the normal version:
Given an instance of NAE3SAT with fixed constants, we create 2 new variables x_{T} and x_{F}. We replace all instances of the constant value true with x_{T}, and all instances of the constant value false with x_{F}. We also add one additional clause:
(x_{T} ∨ x_{T} ∨ x_{F}).
Note that this new clause is only NAESatisfiable if x_{T} != x_{F}. We can’t directly bind x_{T} to true and x_{F} to false because if you take a NAE3SAT solution and negate all literals, you get another NAE3SAT solution. But because of that, this is NAESatisfiable if and only if the original formula is.
Difficulty: It depends on what you give them. If you make them do everything I did above, I’d say it’s an 8. If you let them stop at the NAE3SAT with fixed constants, then maybe a 7.
Posted in AppendixLogic
Tagged 3sat, Difficulty 8, LO3, NAE3SAT, reductions, uncited reduction
Protected: Regular Expression Inequivalence
Enter your password to view comments.
Posted in Chapter 3 Exercises
Tagged 3sat, AL9, Difficulty 9, reductions, Regular Expression Inequivalence, StarFree Regular Expression Inequivalence
Protected: Steiner Tree in Graphs
Enter your password to view comments.
Posted in Chapter 3 Exercises
Tagged Difficulty 7, ND12, reductions, Steiner Tree, uncited reduction, X3C
Protected: Dominating Set
Enter your password to view comments.
Posted in Chapter 3 Exercises
Tagged Difficulty 4, Dominating Set, GT2, No G&J reference, reductions, uncited reduction, Vertex Cover
Protected: Exact Cover By 4Sets
Enter your password to view comments.
Posted in Core Problems
Tagged Difficulty 5, reductions, uncited reduction, X3C, X4C
Protected: Feedback Vertex Set
Enter your password to view comments.
Posted in Chapter 3 Exercises
Tagged Difficulty 3, Fedback Vertex Set, GT7, reductions, uncited reduction, Vertex Cover
Protected: Minimum Sum of Squares
Enter your password to view comments.
Posted in Appendix Sets and Partitions, Chapter 3 Exercises
Tagged Difficulty 3, MSS, No G&J reference, Part, reductions, SP19, uncited reduction
Protected: Largest Common Subgraph
Enter your password to view comments.
Posted in Chapter 3 Exercises
Tagged Clique, Difficulty 3, GT49, Largest Common Subgraph, No G&J reference, reductions, uncited reduction