COMPLEXITY THEORY IN PRACTICE - Lalitha Pragada.

Circumscribing Hard Problems: 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: A part( 1 fragment per variable) that ensures legal truth assignments. 2. A part ( 1 fragment per clause) that ensures satisfying truth assignments. 3. A part that ensures consistency of truth assignments among clauses and variables. Planarity typically lost in the third part.

Definition 8.1: 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.

Theorem 8.5: 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 NP-Complete.

Proof: 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.

Further Inferences!! 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.

Theorem 8.6: The Minimal Research Program problem is NP-Complete!! 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.

8.1.2 Promise Problems

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 instances- the promise.

Theorem 8.7: 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!!