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Jan 05, 2020

Nathan Brunelle

Department of

Computer Science

University of Virginia

www.cs.virginia.edu/~njb2b/theory

Theory of Computation CS3102 – Spring 2014

A tale of computers, math, problem solving, life, love and tragic death

Complexity Classes

Def: DTIME(t(n))={L | L is decidable within

time O(t(n)) by some deterministic TM}

Def: NTIME(t(n))={L | L is decidable within

time O(t(n)) by some non-deterministic TM}

Def: DSPACE(s(n))={L | L decidable within

space O(s(n)) by some deterministic TM}

Def: NSPACE(s(n))={L | L decidable within

space O(s(n)) by some non-deterministic TM}

Non-Robustness of TM Complexity • Computability: all variations on TMs have the same

computing power – If there is a multi-tape TM that can decide L, there is a

regular TM that can decide L. – If there is a nondeterministic TM that can decide L, there is a

deterministic TM that can decide L.

• Complexity: variations on TM can solve problems in different times – Is a multi-tape TM faster than a regular TM? – Is a nondeterministic TM faster than a regular TM?

Complexity Class P

P = U k

DTIME(nk)

P is the class of languages that can be decided in Polynomial Time on a deterministic, single- tape Turing machine.

Classes in P

a) TIME(n2)

b) TIME(O(n7))

c) TIME(O(2n))

d) Class of languages that can be decided in Polynomial Time by a 2-tape TM

e) Class of languages that can be decided in Polynomial Time by a nondeterministic TM

Unknown! This is the P = NP question. Focus of this class…

Yes! We can simulate each step of a 2- tape TM by making 2 passes over the whole tape ~ 2(n+t(n)) (See Theorem 7.8)

P Examples

• What are some examples of problems in P?

– Arithmetic

– Matrix Multiplication

– Matrix Inversion

– Sorting

Non-deterministic polynomial time

• Deterministic Polynomial Time: TM takes at most O(nc) steps to accept a string of length n

• Non-deterministic Polynomial Time: The TM takes at most O(nc) steps on some path to accept a string of length n

Complexity Class NP

NP = U k

NTIME(nk)

P is the class of languages that can be decided in Polynomial Time on a non-deterministic Turing Machine.

Alternative Definition • A language is in NP if some execution path answers in

polynomial time

• What if we had a “list of directions”?

• Some information that told us which path accepted quickly

• L is in NP if there is a “verifier” which given a string x in L and a polynomial length “witness” it can verify x is a member of L in polynomial time

• NP = “easy to verify” problems

Neil Immerman

Alternate Definition of NP Using Descriptive complexity

Argue why these are in NP • Sets of integers which sum to 0

– Add them together, check if sum is 0

• Sets of integers which have a subset with sum 0 – Given a “witness” w (a particular subset which sums to 0) we can:

• Verify w is indeed a subset of the input • Verify that w sums to 0

• Sets of integers which have a subset of size 3 which sums to 0 – Try all subsets of size 3, check if any sum to 0 – Called “3-Sum”

• Primality testing: Given an integer, is it prime? – Given a “witness” (a pair of numbers) we can:

• Verify the product of these numbers gives the input (just 1 multiplication) • Surprisingly, this can be solved in Polynomial time (AKS, 2002)

The Class P and the Class NP

• P = { L | L is accepted by a deterministic Turing Machine in polynomial time }

• NP = { L | L is accepted by a non-deterministic Turing Machine in polynomial time }

• They are sets of languages

Most important CS Problem

• Does P = NP?

• Are all “easily verifiable problems” also “easily computable”?

• How much more efficient are non- deterministic Turing Machines?

Progress • P = NP if every NP problem has a deterministic

polynomial time algorithm • We could find an algorithm for every NP problem • Seems… hard…

• Study an “archetypical” example

– An NP problem which is in P iff P=NP – Called Complete

• We need: At least one such problem

NP

NP Hardness & Completeness Def: A problem L’ is NP-hard if:

(1) Every L in NP reduces to L’ in polynomial time.

Def: A problem L’ is NP-complete if:

(1) L is NP-hard; and

(2) L is in NP.

If one NPC problem is in P P=NP

P

NP-Hard

NP Completeness Benefits 1. Saves time & effort of trying to solve intractable problems

efficiently;

2. Saves money by not separately working to efficiently solve different problems;

3. Helps systematically build on & leverage the work (or lack of progress) of others;

4. Transformations can be used to solve new problems by reducing them to known ones;

5. Illuminates the structure & complexity of seemingly unrelated problems;

NP Completeness Benefits 6. Informs as to when we should use approximate solutions vs.

exact ones;

7. Helps understand the ubiquitous concept of parallelism (via non- determinism);

8. Enabled vast, deep, and general studies of other “completeness” theories;

9. Helps explain why verifying proofs seems to be easier than constructing them;

10. Illuminates the fundamental nature of algorithms and computation;

NP Completeness Benefits 11. Gave rise to new and novel mathematical approaches, proofs,

and analyses;

12. Robustly decouples / abstracts complexity from underlying computational models;

13. Gives disciplined techniques for identifying “hardest” problems / languages;

14. Forged new unifications between computer science, mathematics, and logic;

15. NP-Completeness is interesting and fun!

Our “Favorite” NP Complete Problem Satisfiability (SAT): Given a Boolean expression in conjuctive

normal form, is there some assignment of T/F to its variables such that the expression resolves to “True”?

Note: we may restrict each disjunction to have 3 variables (3-SAT)

Examples:

¬𝑎 ∨ 𝑏 ∨ ¬𝑐 ∧ 𝑎 ∨ 𝑑 ∨ 𝑐 ∧ (𝑑 ∨ 𝑏 ∨ ¬𝑒)

Satisfied if: 𝑎 = 𝐹, 𝑏 = 𝐹, 𝑐 = 𝑇, 𝑑 = 𝑇 𝑎 ∨ 𝑏 ∧ ¬𝑎 ∨ ¬𝑏 ∧ (¬𝑎 ∨ 𝑏) ∧ (𝑎 ∨ ¬𝑏) Can’t be Satisfied!

Stephen Cook

Leonid Levin

The Cook/Levin Theorem Theorem [Cook/Levin, 1971]: SAT is NP-complete. Proof idea: given a non-deterministic polynomial time TM M and input w, construct a CNF formula that is satisfiable iff M accepts w.

Create boolean variables:

q[i,k] at step i, M is in state k

h[i,k] at step i, M’s RW head scans tape cell k

s[i,j,k] at step i, M’s tape cell j contains symbol Sk

M halts in polynomial time p(n)

total # of variables is polynomial in p(n)

Qk

Stephen Cook

Leonid Levin

Add clauses to the formula to enforce necessary restrictions on how M operates / runs:

• At each time i:

M is in exactly 1 state

r/w head scans exactly 1 cell

All cells contain exactly 1 symbol

• At time 0 M is in its initial state

• At time P(n) M is in a final state

• Transitions from step i to i+1 all obey M's transition function

Resulting formula is satisfiable iff M accepts w!

Qk

The Cook/Levin Theorem

“Guess and Verify” Approach Note: SAT NP.

Idea: Nondeterministically “guess” each Boolean variable value, and then verify the guessed solution.

polynomial-time nondeterministic algorithm NP

This “guess & verify” approach is general.

Idea: “Guessing” is usually trivially fast ( NP)

NP can be characterized by the “verify” property:

NP set of problems for which proposed solutions can be quickly verified

set of languages for which string membership can be quickly tested.

Historical Note The Cook/Levin theorem was independently proved by Stephen Cook

and Leonid Levin

• Denied tenure at Berkeley (1970) • Invented NP completeness (1971) • Won Turing Award (1982)

• Student of Andrei Kolmogorov • Seminal paper obscured by Russian, style, and Cold War

An NP-Complete Encyclopedia Classic book: Garey & Johnson, 1979

• Definitive guide to NP-completeness

• Lists hundreds of NP-complete problems

• Gives reduction types and refs

Michael Garey David Johnson

NP-Completeness Proof Method

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