16.2 NP-Completeness and Cook's Theorem

NP-completeness is a crucial concept in computational complexity. It represents the hardest problems in the NP class, marking the boundary between tractable and intractable problems. If any NP-complete problem is solved efficiently, it would revolutionize computer science.

Cook's Theorem kickstarted NP-completeness theory by proving the Boolean Satisfiability Problem is NP-complete. This led to identifying other NP-complete problems like the Traveling Salesman Problem and Graph Coloring, which are important in various fields but lack efficient solutions.

Understanding NP-Completeness

Definition of NP-completeness

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• NP-completeness encompasses hardest problems in NP class, subset of Nondeterministic Polynomial time problems
• Represents boundary between tractable and intractable problems, if any NP-complete problem solved in polynomial time, then P = NP
• Provides framework for classifying problem difficulty, characterizes decision problems verifiable in polynomial time
• At least as hard as any problem in NP, examples include (Boolean Satisfiability, Traveling Salesman)

Cook's Theorem and NP-completeness

• Cook's Theorem (Cook-Levin Theorem) states Boolean Satisfiability Problem (SAT) is NP-complete
• Proved SAT is in NP and showed all problems in NP can be reduced to SAT in polynomial time
• Provided starting point for proving other problems NP-complete, established foundation for NP-completeness theory
• Implications include identifying computational complexity of various problems (Graph Coloring, Hamiltonian Cycle)

Proving and Identifying NP-Completeness

Polynomial-time reductions for NP-completeness

• Steps to prove NP-completeness:

1. Show problem is in NP
2. Choose known NP-complete problem
3. Construct polynomial-time reduction from known problem to given problem

• Polynomial-time reductions transform instances of one problem to another in polynomial time, preserve yes/no answers
• Demonstrate solution to new problem implies solution to known NP-complete problem
• Ensure reduction computable in polynomial time, examples include (SAT to 3-SAT, 3-SAT to Clique)

Famous NP-complete problems

• Boolean Satisfiability Problem (SAT) determines if Boolean formula has satisfying assignment, first proven NP-complete
• Traveling Salesman Problem (TSP) finds shortest route visiting each city once, applications in logistics and planning
• Graph Coloring Problem assigns colors to vertices so no adjacent vertices share same color
• Hamiltonian Cycle Problem seeks cycle visiting each vertex exactly once in undirected graph
• Subset Sum Problem finds subset of integers summing to target value
• Clique Problem finds complete subgraph of given size in larger graph
• Share characteristics: exponential time algorithms in worst case, no known polynomial-time algorithms, significant practical importance in various fields (optimization, network design, scheduling)