- Lalitha Pragada.
Proposition 8.1:
Vertex Cover remains NP-Complete when limited
to graphs of degree 5.
Restriction to planar graphs.
Proof of NP-Completeness: By reduction from
one of the versions of 3SAT.
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
satisfying truth assignments.
3. 3. A part that ensures consistency of truth
assignments among clauses and variables.

Planarity typically lost in the third part.

The planar satisfiability problem is the
satisfiability problem restricted to planar
instances. An instances of SAT is deemed
planar if its graph representation is planar.
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
between a variable vertex and a clause
whenever the variable appears in the clause.
With the representations defined above, the polar
and non-polar versions of Planar Three –
Satisfiabilty are NP-Complete.

Corollary 8.1: Planar Vertex Cover is NPComplete.
3SAT uses a clause piece that can be assimilated
to a single vertex in terms of planarity and does
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.
Planar 1in3SAT is also NP-Complete, however,
Planar NAE3SAT is in P in both polar and
nonpolar versions.


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.
The ad hoc approach is combined with the
generic approach when the generic approach
restricted the instances to a subset of the
general problem but a superset of your
problem.
The Minimal Research Program problem is NPComplete!!
An instance of this problem is given by a set of
unclassified problem S, a partial order on S
denoted <, and a bound B.


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.
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.



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.
Restrictions of NP-Complete problems must be
verified in polynomial time


Perfect Graphs- Important example of such an
unreasonable restriction
A graph is perfect iff the chromatic number of
every subgraph equals the size of largest clique
of the subgraph.


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 instancesthe promise.
Uniquely Promised SAT cannot be solved in
polynomial time unless RP equals NP.
Verifying the promise of uniqueness is generally
hard for hard problems.
Compare : Uniquely Promised SAT and Unique
Satisfiability
Thank you!!