Complexity Theory in Practice
-Benjamin Daggolu
Introduction
Knowing that a problem is NP-hard or worse
does not disappear the problem, we need to
design an algorithm to solve it.
Approximation algorithm- runs quickly and
returns good results.
Probabilistic approach- uses efficient algorithms
that returns optimal results in most cases but
may fail miserably in some cases.
Very little can be said about which technique
can be used to solve a specific problem.
The complexity theory helps the algorithms
designer in assessing hard problems bases on
their complexity.
8.1 Circumscribing Hard Problems
It is not fully clear that a program is correct,
under certain inputs it may fail.
Most of the instances of the problems are easily
solvable i.e. the solution algorithm runs
quickly on most or all instances to which it
applies.
• It is not possible to measure the time required
by an algorithm on a single instance in usual
terms like polynomial or exponential as these
terms are defined only for infinite classes of
instances.
• Hence we consider only some restricted
versions of hard problems to examine their
complexity
Restrictions of Hard Problems
SAT problem is NP complete and it remains so
even if it is restricted to instances where each
clause contains exactly three literals.
SAT problem becomes tractable when restricted
to instances where each clause contains at
most two literals.
In terms of the number of literals per clause
different variants of the problem can be
classified.
Other dimensions:
We can consider the number of times a variable
may appear among all clauses.
A satisfiability problem with instances
containing k literals and each variable appears
at most l times among all clauses is called as
k,l-SAT
3,4-SAT is NP-complete.
(Page no-288)
It contains 3 literals and each variable appears at
most 4 times among all clauses
• Binpacking problem:
Instance of this problem is given by a set ‘S’ of
elements each with size s:S->N, and with a bin
size ‘B’. The goal of this problem is to pack all
the elements into the smallest number of
bins.
• Restricted Binpacking problem:
Let the problem be restricted only to the
instances where all the elements have sizes at
least equal to a third of the bin size.
Now the restricted problem can be claimed to
be solved in polynomial time.
• The solution for the restricted problem
involves preprocessing of identifying the
elements which are at least a third of the bin
size which takes at most linear time.
• Identify all possible pairs of elements that can
fit together in a bin, which takes at most
quadratic time.
• Elect the largest subset of pairs that do not
share any element which is a matching
problem which can be solved in polynomial
time.
• Overall the run time of the algorithm is
dominated by the run time of the matching
algorithm which is a low polynomial.
• Hence the binpacking algorithm is optimal.
Planar G3C is NP-complete
Two crossing edges cannot share an endpoint;
hence they are independent of each other
from the point of view of coloring.
The crossing gadget must replace the two edges
in such a way that
-the gadget is planar and the three colorable
-the coloring of original edge’s endpoint do not
effect the coloring of the other edge
-two endpoints of an original edge cannot be
given the same color.
• Embedding an arbitrary graph in a plane
detecting all edge crossings and replacing
each crossing with the gadget are all easily
done in polynomial time.
• Planarity is not the only reasonable parameter
involved in the graph. Other such parameter is
the maximum degree of its vertices.
G3C is NP-complete even when restricted to
instances where no vertex degree may exceed
four.
Replace any vertex of degree larger than four
with a gadget such that:
-the gadget is three colorable
-there is connection for each vertex with the
original vertex
-all attaching points are colored identically.
Contd..
-Thank you!