CSE680-17NP-Complete.pptx

Introduction to Algorithms
NP-Complete
CSE 680
Prof. Roger Crawfis

Polynomial Time
 Most (but not all) of the algorithms we have
studied so far are easy, in that they can be solved
in polynomial time, be it linear, quadratic, cubic,
etc.
 Cubic may not sound very fast, and isn’t when
compared to linear, but compared to exponential
we have seen that it has a much better asymptotic
behavior.
 There are many algorithms which are not
polynomial. In general, if the space of possible
solutions grows exponentially as n increases, then
we should not hope for a polynomial time
algorithm. These are the exception rather than the
rule.

Hard Problems
 Problems with no known polynomial solution are
called Hard problems.
 Another class of problems (NP) seem like they
should be easy. They have the following
property:
1. A solution can be verified in polynomial time.
 Consider the following very common example
called Satisfiability: (SAT)
 Input: A Boolean expression, F, over n variables
(x1,. ..,xn)
 Output: 1 if the variables of F can be fixed to
values which make F equal true; 0 otherwise.

NP Problems
 NP stands for non-deterministic
polynomial time.
 The basic premise is we could guess at
a solution and then test whether we
guessed right or not easily. Guessing is
of course non-deterministic!
 So, think of this as guessing in
polynomial time. No guarantee that you
will ever guess correctly though.

Polynomial Time Reductions
 The SAT problem is a circuit, so it is very
useful and interesting. Can we write a
program to do X? This and many other
problems have been studied at length.
 There are many mappings (called
reductions) that have been discovered that
map one problem to the other, so if we
solve one we can solve the other.
 Moreover, these mappings are polynomial
time mappings.

NP-Complete
 The set of NP problems that can be mapped to each other in
polynomial time is called NP-Complete.
 It is difficult to prove that something can not be done.
 Let me repeat that, it is difficult to prove that something can not
be done.
 So, while we know how to solve many of these algorithms in
exponential time, whose to say that one of you bright students
won’t come up with a clever polynomial time algorithm!
 NP-Complete is useful from that standpoint, if any of them turns
out to have a polynomial time algorithm, they all do (hence
P=NP).
 Of course, us old professors feel that these problems have been
looked at from every angle and no such solution will ever be
found (hence, P ≠ NP).

NP-Complete
 It is also useful to know these algorithms,
as they occur frequently in real
applications and tackling them in a brute
force fashion may be disastrous.
 SAT problem
 Traveling Salesman problem
 Knapsack Problem
 Longest Path
 Graph Clique

TSP
 For each two cities, an integer cost is given to travel from
one of the two cities to the other. The salesperson wants
to make a minimum cost circuit visiting each city exactly
once.
3
1
1
1
2
2
3
4
1
2 2
1
2
2
4
4 1
5
1
i = 23
2

Circuit-SAT
Logic Gates
NOT
AND
OR 1
1
1
0
0
0
1
1
1
1
1
0
0
 Take a Boolean circuit with a single output node
and ask whether there is an assignment of
values to the circuit’s inputs so that the output is
“1”

Knapsack
 Given s and w can we translate a subset
of rectangles to have their bottom edges
on L so that the total area of the
rectangles touching L is at least w?
s
L
1
2
3
4
5 6
7

Definitions
Matching
Free Vertex

Definitions
 Maximum Matching: matching with the
largest number of edges

Definition
 Note that maximum matching is not
unique.

Intuition
 Let the top set of vertices be men
 Let the bottom set of vertices be women
 Suppose each edge represents a pair of
man and woman who like each other
 Maximum matching tries to maximize the
number of couples!

Graph Clique
 Set of vertices
such that they
all connect
to each
other.

This graph contains a 4-clique
K-Cliques
 A K-clique is a set of K nodes with all
K(K-1)/2 possible edges between them

K-Cliques
 Given: (G, k)
Question: Does G contain a k-clique?
 BRUTE FORCE:
 Try out all {n choose k} possible locations
for the k clique

NP-Complete
 It is also interesting to look at the
relationship between some easy
problems and hard ones:
Hard Problems
(NP-Complete)
Easy Problems
(in P)
SAT, 3SAT 2SAT
Traveling Salesman Problem Minimum Spanning Tree
3D Matching Bipartite Matching
Knapsack Fractional Knapsack

CSE680-17NP-Complete.pptx

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