-lalitha pragada. - mississippi state university
TRANSCRIPT
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- Lalitha Pragada.
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Proposition 8.1:
Vertex Cover remains NP-Complete when limited to graphs of degree 5.to graphs of degree 5.
Restriction to planar graphs.
Proof of NP-Completeness: By reduction from one of the versions of 3SAT.
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Constructions from 3SAT:
1. A part( 1 fragment per variable) that ensures legal truth assignments.
2. 2. A part ( 1 fragment per clause) that ensures 2. 2. A part ( 1 fragment per clause) that ensures satisfying truth assignments.
3. 3. A part that ensures consistency of truth assignments among clauses and variables.
� Planarity typically lost in the third part.
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� The planar satisfiability problem is the satisfiability problem restricted to planar satisfiability problem restricted to planar instances. An instances of SAT is deemed planar if its graph representation is planar.
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The simplest way to define a graph representation for an instance of SATISFIABLITY is to set up a vertex for each variable, a vertex for each clause and an edge SATISFIABLITY is to set up a vertex for each variable, a vertex for each clause and an edge between a variable vertex and a clause whenever the variable appears in the clause.
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With the representations defined above, the polar and non-polar versions of Planar Three –and non-polar versions of Planar Three –Satisfiabilty are NP-Complete.
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� Corollary 8.1: Planar Vertex Cover is NP-Complete.Complete.
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3SAT uses a clause piece that can be assimilated to a single vertex in terms of planarity and does not connect clause pieces. not connect clause pieces.
Proposition 8.1 and Corollary 8.1 should not be combined for the conclusion- “ Vertex Cover remains NP-Complete “ !
A planar version of (3,4)- SAT is needed to draw the conclusion.
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Planar 1in3SAT is also NP-Complete, however, Planar NAE3SAT is in P in both polar and Planar NAE3SAT is in P in both polar and nonpolar versions.
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� The (Semi)generic approach: The problem is used in reduction for proving the general version to be NP-hard may have a known NP-Complete special case that, when used in the reduction , produces only the type of instance needed.
� The ad hoc approach : Usage of a reduction from the general version of the problem to its special case requires one or more gadgets.
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The ad hoc approach is combined with the generic approach when the generic approach generic approach when the generic approach restricted the instances to a subset of the general problem but a superset of your problem.
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The Minimal Research Program problem is NP-Complete!!
An instance of this problem is given by a set of An instance of this problem is given by a set of unclassified problem S, a partial order on S denoted <, and a bound B.
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� A subset S’C S, with S< B, and a complexity classification function c: S -> { hard, easy} such that c can be extended to a total function on S.that c can be extended to a total function on S.
� c can be extended on S by applying the two rules:
i . x<y and c(y) = easy =>c(x) = easy;
ii . X<y and c(x) = hard => c(y) = hard.
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� All the restrictions so far have been reasonable restrictions.
� They are characterized by easily verifiable features.
� Only such restrictions fit within the framework developed previously.developed previously.
Restrictions of NP-Complete problems must be verified in polynomial time
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� Perfect Graphs- Important example of such an unreasonable restriction
� A graph is perfect iff the chromatic number of � A graph is perfect iff the chromatic number of every subgraph equals the size of largest clique of the subgraph.
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� Several problems that are NP-Hard on general graphs are solvable in polynomial time on perfect graphs.
� Promise Problem: A regular problem with the addition of a predicate defined on instances-the promise.
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Uniquely Promised SAT cannot be solved in Uniquely Promised SAT cannot be solved in polynomial time unless RP equals NP.
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Verifying the promise of uniqueness is generally hard for hard problems.hard for hard problems.
Compare : Uniquely Promised SAT and Unique Satisfiability
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Thank you!!Thank you!!