IN210 − lecture 8
Strategy: Understand two classes in-depth.
N PC
P
formal
languages
Next:
• Extend the “map of classes”.
• Show how to use it for organizing our views
on
— problems
— solutions
— other issues
Autumn 1999
1 of 14
IN210 − lecture 8
Other classes
Def. 1 (Co-N P) A language L is in class
Co-N P if its complement Lc is in N P.
Note: The complement of a formal language
L is
• formally all the strings (over the given
alphabet) that are not in the set L.
• informally the reverse property, i.e. all the
instances that don’t have the property
corresponding to language L.
Co
NP
NP
P
Example: N ON -M ATCHING and
N ON -H AMILTONICITY are in Co-N P.
Autumn 1999
2 of 14
IN210 − lecture 8
Lemma 1 A language L is Co-N P-complete if
its complement Lc is N P-complete.
Proof:
A reduction R from L0c to Lc is also a
reduction from L0 to L.
Co-N P
L0
R
L
NP
L0c
R
c
L
0c
L
0
L
R
R
Σ*
Lc
L
Σ*
Example:
• N ON -H AMILTONICITY
• N ON -S ATISFIABILITY
• N ON -C LIQUE
• N ON - . . .
are Co-N P-complete problems. They don’t
seem to have “ID’s” (efficient membership
proofs).
Autumn 1999
3 of 14
IN210 − lecture 8
Def. 2 (Polynomial Space, PSPACE)
[
PSPACE =
SPACE (nk )
k
(Note: Space complexity is measured as number of tape
squares used on (the work tape of) a TM.)
Lemma 2 NTIME f (n) ⊆ SPACE f (n)
Proof: A TM moves its head one square at a
time.
Corollarly 1 N P, Co-N P ⊆ PSPACE
Examples of PSPACE-complete problems:
• QUANTIFIED B OOLEAN F ORMULAS
(∃x1)(∀x2)(∃x3) · · · (Qxn)B
• Generalized games (n × n boards)
Def. 3 (Exponential time, EXP)
k
[
EXP =
TIME 2n
k
f (n)
Lemma 3 SPACE f (n) ⊆ TIME k
for
some constant k.
Proof (idea): The maximum number of
distinct configuration with f (n) tape squares
used is |Q| · f (n) · |Γ|f (n).
Corollarly 2 PSPACE ⊆ EXP
Autumn 1999
4 of 14
IN210 − lecture 8
..
Map of classes
Not acceptable
...
Undecidable
EXP TIME
a cut!
PSPACE
Co
NP
NP
= complete or "hardest"
problems in a class
P
Are inclusions proper (()?
• P ( N P?
• P ( PSPACE?
Autumn 1999
5 of 14
IN210 − lecture 8
Do complexity classes really exist?
Theorem 1 (Time Hierachy Theorem) If
f (n) ≥ n is a proper complexity
function
then
TIME f (n) ( TIME (f (2n + 1))3 .
Basic message: Given more time we can
(provably) solve more problems.
Corollarly 3 P ( EXP
In other words: L ∈ EXP-complete ⇒ L 6∈ P.
Autumn 1999
6 of 14
IN210 − lecture 8
Are complexity classes practically
relevant?
Assumptions, abstractions, modeling
simplifications are possible weak points of
thoery as a model:
• Problem
;
;
formal language
;
— search/optimization
• Solution, algorithm
decision
Turing machine
Alternative approaches to computation:
— Analog computation
— Biological computers
— Neural networks
— Quantum computers
— ...
• Real time
;
;
# of TM steps (discrete, finite)
• Real intractability
worst case /
best solution
Alternative approaches to algorithm
design & analysis:
— Approximation
— Average-case analysis
— Randomized algorithms
—
Autumn 1999
...
7 of 14
IN210 − lecture 8
Placing problems on the map
• Problem ↔ complexity
• Complexity = performance of best
possible algorithm
• Lower bound ≤ complexity ≤ upper bound
— Upper bound → algorithm
— Lower bound → reduction, etc.
Autumn 1999
8 of 14
IN210 − lecture 8
Examples (insights)
Finding a path in a graph
• any path: “greedy” algorithm
(linear time)
s
For all v 0
neighbors of v:
Mark v 0
s
t
G
• shortest path: iterative improvement,
Dijkstra (polynomial time)
4
s
6
a
Let v be node with smallest D(v).
1 For all v 0 neighbors of v:
0
0
0
b D(v ) = min[D(v) + d(v, v ), D(v )]
• longest path: NP-complete (exhaustive
search, backtracking, exponential time)
H AMILTONICITY ∝ Longest path
Autumn 1999
9 of 14
IN210 − lecture 8
Matching
1DM
a
b
2DM
a
b
1
2
greedy
(lin. time)
O (n)
3DM
1
2
iterative
improvement
3
O n alg.
exhaustive
search
2n
O 2 alg.
Satisfiability
1SAT
(x1) ∧ · · ·
2SAT
(x1 ∨ x2) ∧ · · ·
3SAT
(x1 ∨ ¬x2 ∨ ¬x3)∧
Generalization: Matroid intersection
1 matroid
Autumn 1999
2 matroids
3 matroids
10 of 14
IN210 − lecture 8
Placing real-world problems
on the map
• Decision problems . . . is there a . . .
• Search problems . . . find a . . .
• Optimization problems . . . find the
largest/smallest . . .
Def. 4 Problem L is N P-easy if there exists a
problem L0 ∈ N P such that L ∝ L0.
Note: We generalize the notion of polynomial
reduction (∝) in order to deal with
search/optimization problems:
• We allow a polynomial number of “calls” to
ML0 instead of just 1.
• We allow the output to be a general string
instead of just YES or NO.
(G&J calls this Turing reduction, ∝T )
Note: If L ∈ N P-easy and P=N P
then L ∈ P, i.e. L is not any more
difficult/complex than N P-complete
problems.
Autumn 1999
11 of 14
IN210 − lecture 8
Example (of basic technique)
Theorem 2 Search version of H AMILTONICITY
is N P-easy.
Proof: (Turing) reduction to decision version
of H AMILTONICITY:
G
RT
G0
Y
Hd
alg.
Ham. cycle
in G or
"Doesnt’ exist"
N
Algorithm for search-HAM
a
0
1. G ← G
b
c
e
d
2. Run Hd on G0. If HD (G0) = ’NO’ then output
“Ham. path doesn’t exist” else remove an
edge from G0:
Autumn 1999
12 of 14
IN210 − lecture 8
a
0
0
3. G ← G − {a, b}
b
c
e
d
4. Run Hd on G0. If HD (G0) = ’YES’ then
remove another edge from G0 (goto step 3),
else mark edge {a, b}, return {a, b} to G0
and remove an unmarked edge:
a
0
0
5. G ← G + {a, b} − {e, d}
b
c
e
d
6. . . . eventually the graph will only have n
marked edges left, namely the
Hamiltonian cycle.
Autumn 1999
13 of 14
IN210 − lecture 8
Def. 5 Problem L is N P-equivalent if L is
both N P-hard and N P-easy.
Examples: Search versions of HAM, C LIQUE,
SAT, . . . are N P-equivalent.
Optimization problems – are they
harder?
We say that language (problem) L is in class
DP if it can be defined by L = L1 ∩ L2 where
L1 is an N P language (property) and L2 is a
Co-N P language (property).
Example: Opt-TSP is DP-complete. When
we say
“t is the shortest tour in G and t has length=K”
we say two things:
• There is a tour of length K in G.
(N P-property)
• There is no tour of length K − 1 in G.
(Co-N P-property)
Question: Is Opt-TSP N P-equivalent?
Autumn 1999
14 of 14