Reductions, Algorithmic Bounds, NP Completeness

HugPlant

Kolmogorov Complexity (Language Independence)

• Random bitstream → unable to deterministically produce it in much shorter program
• Some bitstreams = special, can be produced by all programming languages in compact amount of space

Kolmogorov Complexity (Uncomputability)

• Impossible to write program that calculates/computes Kolmogorov Complexity of any bitstream → corollary = impossible to write "perfect" compression algorithm
  • No way to do it in reverse, impossible b/c if possible to produce smallest program could just see how long it is (to obtain Kolmogorov Complexity) → violates uncomputability of Kolmogorov Complexity

Problem Difficulty

• Hardest = impossible

Independent Set Problem

• Worst case each red vertex has $N$ neighbors → $\forall$ red vertex, check that neighbors are all white → $O(k \cdot N)$
• Computing next coloring fast → enumerate binary number (bitshifting for all possible permutations)

Reductions & "Cracking"

• Sorting = tool to solve median finding problem
• $Y$ = tool to solve $X$

The Class P

• Decision problem = problem w/ yes/no (true/false) answer
• $\Theta(N \log{N}) \in O(N^{k})$ where $k = 2$ → sorting is in complexity class P
• Solvable quickly
• Matters b/c fast

The Class NP

• NP problem = yes/no (true/false), & checkable/verifiable quickly

Why Does The Complexity Class NP Matter?

• Computation may be hard
• NP = yes/true answers verifiable quickly
• co-NP = no/false answers verifiable quickly

Independent Set Difficulty

• $P \subset NP$

Problem Difficulty

• Good Compression: Given bitstream, is there sequence of bits $<$ 1 MB able to solve it? → NP problem

P = NP?

• If P = NP → P = NP = NP Complete
  • Easily verifiable = easily solvable

Even More Impressive Consequences

• Program output by NP solver should contain structure (hierarchical, with functions/ideas)
• Compression provides understanding → know underlying pattern/structure