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The np-completeness of the knapsack problem and its approximability. Definitions, facts, propositions, and algorithms related to the knapsack problem and its optimization and decision versions. It also mentions the relationship between knapsack and other np-complete problems such as subset sum and vertex cover.
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CMPSCI 601: Recall From Last Time Lecture 21
To prove
is NP-complete:
The following problems are NP-Complete:
Knapsack
Given objects:
object weight value = max weight I can carry in my knapsack.
Optimization Problem:
choose
to maximize (^)
such that (^)
Decision Problem:
Given! "!
$# , can I get total value # while total
?
CMPSCI 601: Approximability Lecture 21
Fact: NP-complete decision problems are all equiva- lent.
Belief: NP-complete problems require exponential time in the worst case.
Fact: Difficulty of NP Approximation problems varies widely.
Definition 21.
is an NP-optimization problem iff For each instance , , is the
. Each has a cost Z. The cost is computable in P.
For minimization problems,
OPT
For maximization problems,
OPT
Four Classes of NP Optimization Problems
NP
APPROX
exists PTIME -approx alg no PTIME -approx alg if P
NP
PTAS exists PTIME -approx alg
FPTAS exists uniform -approx alg running in time poly( )
(F)PTAS stands for (Fully) Polynomial-Time Approxima- tion Scheme.
APPROX
P
FPTAS (^) Knapsack
PTAS ETSP
Clique TSP
VertexCover MAX SAT^ ∆TSP
INAPPROX
Better: Find a Maximal Matching
) do
The edges picked are a maximal matching , a disjoint set of edges to which we can’t add another disjoint edge. If there are edges in this matching, we’ve used nodes in but any algorithm would have to use at least .
opt Best known approx ratio
that starts and ends at some vertex and visits every other vertex exactly once.
HC
has a Hamilton Circuit
Fact 21.5 HC is NP -Complete. (Nicest proof is in Sipser.)
TSP #
has a HC of weight
# ,
, let # ,
if
otherwise
HC
TSP
, then P = NP. Thus, TSP
INAPPROX_._
i k
j TSP: TSP where
Claim 21.10 Minimum Spanning Tree is a lower bound for
TSP MST
TSP_._
Proof: Visualize optimal tour:
Delete one edge and we have a spanning tree.
Theorem 21.11 c(MST)
TSP c(MST)
Proof: The multigraph MST, made by taking two copies of each edge in the tree, is connected and all its nodes have even degree.
8
(^1 2 ) 1
4
2 3 (^5465)
6
7
8
3
Thus it has an Euler’s tour, providing an approxi- mation algorithm.
Christofides Algorithm (1976)
In the MST, only worry about the odd degree nodes.
There are an even number of vertices of odd degree.
In polynomial time we can find a minimum weight per-
MST TSP; TSP.
Thus, we get a tour at most 1.5 times optimal.
Euclidean TSP
ETSP: Euclidean distance in plane:
ETSP has a Polynomial-Time Approximation Scheme (PTAS) [Arora 1997].
ing conditions hold:
, there exists a proof
, for every proof
Any decision problem
NP has a deterministic, polynomial- time verifier satisfying Definition ??.
By adding randomness to the verifier, we can greatly re- strict its computational power and the number of bits of that it needs to look at, while still enabling it to accept all of NP.
of its proof,
.
The proof of this theorem is pretty messy, certainly more than we can deal with here. But we can look at the appli- cations of the PCP Theorem to approximation problems.
