NP-Completeness, Reducibility

(NP-完全性、帰着可能性)

Data Structures and Algorithms

14th lecture, January 16, 2020

http://www.sw.it.aoyama.ac.jp/2019/DA/lecture14.html

Martin J. Dürst

AGU

© 2009-20 Martin J. Dürst 青山学院大学

Today's Schedule

 

Remaining Schedule

 

Questions about Final Exam

 

Summary of Last Lecture

 

Today's Goal

 

Example Problems from Last Lecture

 

Homework: Find Commonalities in Problems

 

Problem Shape

 

Decision Problem

 

Polynomial Problems and Exponential Problems

(the time complexity of a problem is the lowest time complexity among the algorithms that solve the problem, or the lowest theoretical time complexity)

 

Properties of Polynomial Time

(or why is polynomial time so special)

The set of problems that can be solved in polynomial time is denoted as P

 

NP Problems

 

Comparing Problems in NP

 

An Example of Reduction

Solving a 3-SAT problem by converting (reducing) it to an independent set problem

 

Overview of Reduction

 

NP-Completeness

 

Computational Complexity Theory

Reference: Complexity Zoo

 

Post's Correspondence Problem

(Emil Post, 1946)

 

Summary

 

Student Survey

(授業改善のための学生アンケート)

WEB Survey

お願い: 自由記述に必ず良かった点、問題点を具体的に書きましょう

(悪い例: 発音が分かりにくい; 良い例: さ行が濁っているかどうか分かりにくい)

 

Glossary

polynomial problem
多項式問題
tractable problem
手に負える問題
exponential problem
指数的問題
intractable problem
手に負えない問題
ring of polynomials
多項式環
nondeterministic polynomial time
非決定性多項式時間
function problem
関数問題
optimization problem
最適化問題
decision problem
決定問題
NP-hard
NP 困難
reduction
帰着
reducibility
帰着可能性
NP-completeness
NP 完全性
graph isomorphism
グラフ同型
Computational Complexity Theory
計算複雑性理論
Post's correspondence problem
ポストの対応問題
approximation algorithms
近似アルゴリズム