Difference between revisions of "Chapter 11"

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===Transformations and Satisfiability===
 
===Transformations and Satisfiability===
  
:[[11.1]]
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:[[11.1]]. Give the 3-SAT formula that results from applying the reduction of SAT to 3-SAT for the formula:
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:::<math> (x\or y \or \overline z \or w \or u \or \overline v) \and (\overline x \or \overline
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y \or z \or \overline w \or u \or v) \and (x \or \overline y \or \overline z \or w \or u \or \overline v)\and (x \or \overline y) </math>
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[[11.1|Solution]]
  
  
:11.2
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:11.2. Draw the graph that results from the reduction of 3-SAT to vertex cover for the expression
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:::<math>(x \or \overline y \or z) \and (\overline x \or y \or \overline z) \and(\overline x \or y \or z) \and (x \or \overline y \or \overline x) </math>
  
  
:[[11.3]]
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:[[11.3]]. Prove that 4-SAT is NP-hard.
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[[11.3|Solution]]
  
  
:11.4
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:11.4. ''Stingy'' SAT is the following problem: given a set of clauses (each a disjunction of literals) and an integer <math>k</math>, find a satisfying assignment in which at most <math>k</math> variables are true, if such an assignment exists. Prove that stingy SAT is NP-hard.
  
  
:[[11.5]]
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:[[11.5]]. The ''Double SAT'' problem asks whether a given satisfiability problem has '''at least two different satisfying assignments'''. For example, the problem <math>{{v1, v2}, {v_1, v_2}, {v_1, v_2}}</math> is satisfiable, but has only one solution <math>(v_1 =F, v_2 = T)</math>. In contrast, <math>{{v_1, v_2}, {v_1, v_2}}</math> has exactly two solutions. Show that Double-SAT is NP-hard.
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[[11.5|Solution]]
  
  
:11.6
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:11.6. Suppose we are given a subroutine that can solve the traveling salesman decision problem on page 357 in (say) linear time. Give an efficient algorithm to find the actual TSP tour by making a polynomial number of calls to this subroutine.
  
  
:[[11.7]]
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:[[11.7]]. Implement a SAT to 3-SAT reduction that translates satisfiability instances into equivalent 3-SAT instances.
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[[11.7|Solution]]
  
  
:11.8
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:11.8. Design and implement a backtracking algorithm to test whether a set of clause sets is satisfiable. What criteria can you use to prune this search?
  
  
:[[11.9]]
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:[[11.9]]. Implement the vertex cover to satisfiability reduction, and run the resulting clauses through a satisfiability solver code. Does this seem like a practical way to compute things?
 
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[[11.9|Solution]]
  
 
===Basic Reductions===
 
===Basic Reductions===

Revision as of 21:32, 10 September 2020

NP-Completeness

Transformations and Satisfiability

11.1. Give the 3-SAT formula that results from applying the reduction of SAT to 3-SAT for the formula:

Solution


11.2. Draw the graph that results from the reduction of 3-SAT to vertex cover for the expression


11.3. Prove that 4-SAT is NP-hard.

Solution


11.4. Stingy SAT is the following problem: given a set of clauses (each a disjunction of literals) and an integer , find a satisfying assignment in which at most variables are true, if such an assignment exists. Prove that stingy SAT is NP-hard.


11.5. The Double SAT problem asks whether a given satisfiability problem has at least two different satisfying assignments. For example, the problem is satisfiable, but has only one solution . In contrast, has exactly two solutions. Show that Double-SAT is NP-hard.

Solution


11.6. Suppose we are given a subroutine that can solve the traveling salesman decision problem on page 357 in (say) linear time. Give an efficient algorithm to find the actual TSP tour by making a polynomial number of calls to this subroutine.


11.7. Implement a SAT to 3-SAT reduction that translates satisfiability instances into equivalent 3-SAT instances.

Solution


11.8. Design and implement a backtracking algorithm to test whether a set of clause sets is satisfiable. What criteria can you use to prune this search?


11.9. Implement the vertex cover to satisfiability reduction, and run the resulting clauses through a satisfiability solver code. Does this seem like a practical way to compute things?

Solution

Basic Reductions

11.10


11.11


11.12


11.13


11.14


11.15


11.16


11.17


11.18


11.19


11.20


11.21


Creatvie Reductions

11.22


11.23


11.24


11.25


11.26


11.27


11.28


11.29


11.30


Algorithms for Special Cases

11.31


11.32


11.33


11.34


11.35


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