CSE6140 NP问题初步框架

CSE6140 NP问题初步框架

NP1:

• why should we care about NP problems?
• optimization and decision problems?
• class P problem?
• tractable and intractable problems?
• class NP problems?
• nondeterministic algorithm and what is NP algorithms?
• the difference between solving a problem and verifying a candidate solution? (using hamiltonian cycle as an example)
• is P = NP?
• NPC?
• reductions and polynomial reductions?
• The implications of polynomial time reductions
• Three ways of reduction strategies?（reduction by simple equivalence, reduction from special case to general case, reduction by encoding with gadgets）

NP2:

• vertex cover reduces to set cover

NP3:

• NPC and how to establishing NPC.
• prove that if Y is NPC, iff P=NP so Y can be solve in poly-time
• P and 相关follow up被证明是NPC的.
• 展示NPC问题的广泛性和影响深远

NP4:

• satisfiability problem(SAT) CNF satisfiability, 3SAT, 3CNF
• showing that 3SAT is NPC by reduction
• the relations of some NPC problems(including set cover, vertex cover, maximum cut, 3SA and so on)
• six basic genres of NPC problems and paradigmatic examples(so far we have covered three of them)

NP5:

• four basic reduction strategies(we are gonna to discuss the fourth strategy of this which is 3SAT ${\le }_p$ INDEPENDENT SET)
• approvement of 3SAT ${\le }_p$ INDEPENDENT SET using reduction by encoding with gadgets.
• 证明某问题是NPC问题需要三步，下面以vertex cover decision problem为例 and prove it is NPC
• 3SAT reduces to vertex cover
• Clique和vertex cover相互之间的 reduce to 的证明
• directed hamiltonian cycle reduce to hamiltonian cycle 的证明。
• 3SAT reduces to directed hamiltonian cycle的证明
• ham-cycle reduce to TSP问题的证明
• 3SAT reduce to longest-path问题的证明
• directed hamiltonian cycle reduces to longest path的证明

NP6:

• 3SAT reduce to 3-color问题证明

Coping with NPC-1:

• 应对NPC问题的六种常见思想： branch&bound, approximation, local search, restriction, randomization, parameterized algs. 这些方法或多或少牺牲了一些方面去得到合适的解。
• NPC问题具体三种解决方法：exhaustive search, backtracking, branch&bound. 以及这三种方法解决具体问题示例。

Coping with NPC-2:(branch and bound)

• 详细叙述了TSP的branch&bound解

Coping with NPC-3:(branch and bound/local search)

• Heuristic/Local search
• Local Search algs

Coping with NPC-4:(local search/empirical analysis of algorithms)

• Symm TSP问题的解题策略以及其优化版本（hill climbing algorithm）
• 优化版本的优缺点
• stochastic local search algs 及其优缺点, stochastic local search for walkSAT.
• Simulated annealing(SA)及其伪代码. SA for TSP
• Tabu Search. tabu search for SAT
• Iterate local search. iterate local search for TSP

Coping with NPC-5:(local search/approximation)

• TSP construction: nearest neighborhood, closest insertion, farthest insertion, cheapest insertion, random insertion.
• approximation algorithms: performance bounds, categories of approximability(3)
• approximation algorithms examples: load balancing

Coping with NPC-6 (approximation algorithms):

• approximation algorithms examples NO.2: clustering(center selection)

Coping with NPC-7:(empirical analysis)

• empirical analysis of algorithms(what’s the difference between theoretical and empirical analysis)
• what’s the goals of empirical analysis? what’s the issues about empirical analysis?
• what are the benchmark of selection?
• what is the difference between CPU time and elementary operations?(how to measure run time and what are the issues?)
• examples: Las Vegas algorithms.
• two definitions of Run time distribution(RTD)
• How to empirically measuring RTDs?
• how to do optimization on RTDs?(after optimal, it called Qualified RTDs)
• what is SQDs?(solution quality distributions)
• and others, we will talk about it later