Hardness Results for Problems

1
Hardness Results for Problems

• P Class of easy to solve problems
• Absolute hardness results
• Relative hardness results
• Reduction technique

2
Fundamental Setting

• When faced with a new problem P, we alternate
  between the following two goals
• Find a good algorithm for solving P
• Use algorithm design techniques
• Prove a hardness result for problem P
• No good algorithm exists for problem P

3
Complexity Class P

• P is the set of problems that can be solved using
  a polynomial-time algorithm
• Sometimes we focus only on decision problems
• The task of a decision problem is to answer a
  yes/no question
• If a problem belongs to P, it is considered to be
  efficiently solvable
• If a problem is not in P, it is generally
  considered to be NOT efficiently solvable
• Looking back at previous slide, our goals are to
• Prove that P belongs to P
• Prove that P does not belong to P

4
Hardness Results for Problems

• P Class of easy to solve problems
• Absolute hardness results
• Relative hardness results
• Reduction technique

5
Absolute Hardness Results

• Fuzzy Definition
• A hardness result for a problem P without
  reference to another problem
• Examples
• Solving the clique problem requires W(n) time in
  the worst-case
• Solving the clique problem requires W(2n) time in
  the worst-case.
• The clique problem is not in P.

6
Proof Techniques

• Diagonalization
• We dont cover, but can be used to prove
  superpolynomial times required for some problems
• Information Theory argument
• W(nlog n) lower bound for sorting
• Typically not a superpolynomial lower bounds
• Size of input argument
• Prove that solving the graph connectivity problem
  requires W(V2) time
• Prove that solving the maximum clique problem
  requires W(V2) time
• Typically not a superpolynomial lower bound

7
Status

• Many natural problems can be shown to be in P
• Graph connectivity
• Shortest Paths
• Minimum Spanning Tree
• Very few natural problems have been proven to NOT
  be in P
• Variants of halting problem are one example
• Many natural problems cannot be placed in or out
  of P
• Satisfiability
• Clique Problem
• Hamiltonian Path
• Traveling Salesperson

8
Hardness Results for Problems

• P Class of easy to solve problems
• Absolute hardness results
• Relative hardness results
• Reduction technique

9
Relative Hardness Results

• Fuzzy Definition
• A hardness result for a problem P without
  reference to another problem
• Examples
• Satisfiability is at least as hard as Hamiltonian
  Path to solve
• If Satisfiability is unsolvable, then Hamiltonian
  Path is unsolvable.
• If Satisfiability is in P, then Hamiltonian Path
  is in P
• If Hamiltonian Path is not in P, then
  Satisfiability is not in P

10
Important Observation

• We are interested in relative hardness results
  BECAUSE of our inability to prove absolute
  hardness results
• That is, if we could prove strong absolute
  hardness results, we would not be as interested
  in relative hardness results
• Example
• If I could prove Satisfiability is not in P,
  then I would be less interested in proving If
  Hamiltonian Path is not in P, then Satisfiability
  is not in P.

11
Relative Hardness Proof Technique

• We show that P2 is at least as hard as P1 in the
  following way
• Informal We show how to solve problem P1 using a
  procedure P2 that solves P2 as a subroutine

12
Examples

• Multiplication and Squaring
• square(x) mult(x,x)
• Proves multiplication is at least as hard as
  squaring
• mult(x,y) (square(xy) square(x-y))/4
• Prove squaring is at least as hard as
  multiplication
• Assumes that addition, subtraction, and division
  by 4 can be done with no substantial increase in
  complexity
• Specific complexity of multiplication may be
  higher as there are two calls to square, but the
  difference is polynomially bounded

13
Hardness Results for Problems

• P Class of easy to solve problems
• Absolute hardness results
• Relative hardness results
• Reduction technique

14
Decision Problems

• We now restrict our attention to decision
  problems
• Key characteristic 2 types of inputs
• Yes input instances
• No input instances
• Almost all natural problems can be converted into
  an equivalent decision problem without changing
  the complexity of the problem
• One technique add an extra input variable that
  represents the solution for the original problem

15
No loss of complexity

• Example using clique problem
• Non-decision Problems
• Input Graph G(V,E)
• Task Output size of maximum clique in G
• Task 2 Output a maximum sized clique of G
• Decision Problems
• Input Graph G(V,E), integer k lt V
• Y/N Question Does G contain a clique of size k?
• Your task
• Show that if we can solve decision clique in
  polynomial-time, then we can solve the
  non-decision clique problems in polynomial-time.

16
Reduction Technique

• In CSE 460, I use the terminology
  Answer-preserving input transformation
• A many-one reduction R is a computable function
  mapping inputs of problem P1 to inputs of problem
  P2
• R is computable by some program/algorithm
• x is a yes input to P1 ?? R(x) is a yes input to
  P2
• Notation P1 ltm P2

17
Pictoral Representation of Reduction
18
Solvability Implications
Yes Input to P2
Yes Input to P1
Yes
No Input to P2
No
No Input to P1
If many-one reduction R exists, then If P2 is
solvable, then P1 is solvable If P1 is not
solvable, then P2 is not solvable
19
Relative hardness properties

• Suppose P1 ltm P2
• Consequences
• If P2 is solvable, then P1 is solvable
• If P1 is not solvable, then P2 is not solvable
• Thus, in some sense, P1 is no harder than P2
• However, for all solvable problems P1 and P2, P1
  ltm P2 and P2 ltm P1, so this does not help us
  define the relative hardness of solvable problems

20
Polynomial-time Reductions

• A polynomial-time many-one reduction R is a
  computable function mapping inputs of problem P1
  to inputs of problem P2
• R is computable by some program/algorithm in
  polynomial-time
• x is a yes input to P1 ?? R(x) is a yes input to
  P2
• Notation P1 ltp P2

21
In P Implications
Yes Input to P2
P2 solves P2 in poly time
Yes Input to P1
Yes
R
No Input to P2
No
No Input to P1
P1 solves P1 in polynomial-time
If polynomial time many-one reduction R exists,
then If P2 is in P, then P1 is in P If P1 is not
in P, then P2 is not in P
22
Relative hardness properties

• Suppose P1 ltp P2
• Consequences
• If P2 is in P, then P1 is in P
• If P1 is not in P, then P2 is not in P
• Thus, in exactly the sense we want, P1 is no
  harder than P2

23
Showing P1 lt P2

• For any x input for P1, specify what R(x) will be
• Show that R(x) has polynomial size relative to x
• You should show that R runs in polynomial time I
  only require the size requirement above
• Show that if x is a yes instance for P1, then
  R(x) is a yes instance for P2
• Show that if x is a no instance for P1, then R(x)
  is a no instance for P2
• Often done by showing that if R(x) is a yes
  instance for P2, then x must have been a yes
  instance for P1

24
Example HP ltp HC

• Hamiltonian Path
• Input Undirected Graph G (V,E)
• Y/N Question Does G contain a Hamiltonian Path?
• Hamiltonian Cycle
• Input Undirected Graph G (V,E)
• Y/N Question Does G contain a Hamiltonian Cycle?

25
Specification of R(x)

• Consider any undirected graph G (V,E) as input
  x
• R(x) will be a graph G (V, E) where
• V V union v where v is not in V
• E E union (v,w) w in V
• Argument that R(x) has polynomial size
• We add exactly 1 node and V edges.

26
x is yes ? R(x) is yes

• Suppose graph G has a Hamiltonian Path
• Let this path be v1, v2, , vn
• We now argue that v1, v2, , vn, v is a
  Hamiltonian Cycle in G
• First, all nodes in V are included exactly once
  above or else v1, v2, , vn would not be a HP in
  G
• Since G has all the edges that G has, (vi,vi1)
  is an edge in E for 1 lt i lt n-1
• Finally, since E contains edge (v,w) for all w
  in V, it must be the case that E contains edges
  (vn, v) and (v,v1).

27
R(x) is yes ? x is yes

• Suppose graph G has a Hamiltonian Cycle
• Let this cycle be v1, v2, , vn, v
• We now argue that v1, v2, , vn is a Hamiltonian
  Path in G
• First, all nodes in V are included exactly once
  above or else v1, v2, , vn, v would not be a HC
  in G
• Since the only extra edges in E compared to E
  are edges involving node v, it must be the case
  that E contains edge (vi,vi1) for 1 lt i lt n-1

28
Turing Reducibility

• Phil made a suggestion for an alternate
  reduction.
• Given graph G, output Q(n2) graphs Gv,w
  (V,Ev,w) where
• Ev,w E union (v,w) where v,w are nodes in V
• This is not a polynomial-time reduction because
  we are outputting Q(n2) graphs.
• However, this idea can be used to show that if HC
  can be solved in polynomial time, then HP can be
  solved in polynomial time.
• Run each graph Gv,w through our procedure that
  solves HC.
• If HC says yes for any one of these graphs,
  return yes.
• Otherwise return no.
• This more general reduction is often called a
  Turing reduction.
• We allow ourselves to use the procedure that
  solves HC (or P2) a polynomial number of times
  rather than just once.