The document provides an overview of linear programming and its usage in approximation algorithms for NP-hard optimization problems. It discusses linear programming formulations, the complexity classes P and NP, approximation algorithms, and two case studies on the minimum weight vertex cover problem and the MAXSAT problem. Randomized rounding techniques are used to generate approximation algorithms for these problems from their linear programming relaxations.
A Numerical Analytic Continuation and Its Application to Fourier TransformHidenoriOgata
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A Numerical Analytic Continuation and Its Application to Fourier TransformHidenoriOgata
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This file contains the concepts of Class P, Class NP, NP- completeness, Travelling Salesman Person problem, Clique Problem, Vertex cover problem, Hamiltonian problem, FFT and DFT.
NP completeness. Classes P and NP are two frequently studied classes of problems in computer science. Class P is the set of all problems that can be solved by a deterministic Turing machine in polynomial time.
Kernelization algorithms for graph and other structure modification problemsAnthony Perez
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Jury:
Stéphane Bessy, Bruno Durand, Frédéric Havet, Rolf Niedermeier, Christophe Paul & Ioan Todinca.
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International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Lecture 1 from https://irdta.eu/deeplearn/2022su/
Covers concepts from Part 1 of my new book, https://meyn.ece.ufl.edu/2021/08/01/control-systems-and-reinforcement-learning/
It presents various approximation schemes including absolute approximation, epsilon approximation and also presents some polynomial time approximation schemes. It also presents some probabilistically good algorithms.
Exact Matrix Completion via Convex Optimization Slide (PPT)Joonyoung Yi
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- Code: https://github.com/JoonyoungYi/MCCO-numpy
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for some positive numerical constant C, then with very high probability, most n×n matrices of rank r can be perfectly recovered by solving a simple convex optimization program. This program finds the matrix with minimum nuclear norm that fits the data. The condition above assumes that the rank is not too large. However, if one replaces the 1.2 exponent with 1.25, then the result holds for all values of the rank. Similar results hold for arbitrary rectangular matrices as well. Our results are connected with the recent literature on compressed sensing, and show that objects other than signals and images can be perfectly reconstructed from very limited information.
Low Complexity Secure Code Design for Big Data in Cloud Storage SystemsReza Rahimi
In the era of big data, reducing the computational complexity of servers in data centers will be an important goal. We propose Low Complexity Secure Codes (LCSCs) that are specifically designed to provide information theoretic security in cloud distributed storage systems. Unlike traditional coding schemes that are designed for error correction capabilities, these codes are only designed to provide security with low decoding complexity. These sparse codes are able to provide (asymptotic) perfect secrecy similar to Shannon cipher. The simultaneous promise of low decoding complexity and perfect secrecy make these codes very desirable for cloud storage systems with large amount of data. The design is particularly suitable for large size archival data such as movies and pictures. The complexity of these codes are compared with traditional encryption techniques.
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Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
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The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
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• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
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The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
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Linear Programming and its Usage in Approximation Algorithms for NP Hard Optimization Problems
1. 1
Linear Programming and its Usage in
Approximation Algorithms for NP Hard
Optimization Problems
M. Reza Rahimi,
November 2005.
2. 2
Outline
• Introduction
• Linear Programming Overview
• NP Complexity Class
• Approximation Algorithms
• Case Study1: Minimum Weight Vertex Cover
• Case Study2: MAXSAT Problem
• Conclusion
Linear Programming and its Usage in Approximation Algorithms for NP Hard Optimization Problems
3. 3
Introduction
• One of the most challenging problem in complexity
theory is problem P vs. NP.
NP
• Research on this open problem leads researchers to
think of new methods and different approaches.
• For example PCP, IP, approximation algorithms,...
• For some problems it is proved that there does not
exist any suitable approximation unless P=NP.
Linear Programming and its Usage in Approximation Algorithms for NP Hard Optimization Problems
4. 4
• So it seems that research on approximation
algorithms may lead us to good results about P vs.
NP.
• General Method for Approximation Algorithms of
NP Hard Optimization is Greedy Method.
• But we must search for Suitable Framework for
studying NP problems.
Linear Programming and its Usage in Approximation Algorithms for NP Hard Optimization Problems
5. 5
• Linear Programming was this brake through
method.
• In this talk I focus on LP and its usage in NP
Hard Optimization Problems.
Linear Programming and its Usage in Approximation Algorithms for NP Hard Optimization Problems
6. 6
Linear Programming Overview
• General formulation of linear programming is
presented as follow:
Maximize C1X1 + C2X2 + … + CdXd
Subject to A1,1X1 + … + A1,dXd ≤ b1
A2,1X1 + … + A2,dXd ≤ b2
…
An,1X1 + … + An,dXd ≤ bn
All calculation and Numbers are on Real Numbers.
Linear Programming and its Usage in Approximation Algorithms for NP Hard Optimization Problems
7. 7
• There are several Polynomial Algorithms for this
problem such as Ellipsoid and Interior point
method.
method
• I neglect talking about them.
• For these algorithms please refer to optimization
references.
Linear Programming and its Usage in Approximation Algorithms for NP Hard Optimization Problems
8. 8
NP Complexity Class
• NP Complexity Class is only related to Decision
Problem.
• For example:
SAT = {< Φ >: Φ is a satisfying assignment}.
• So for any optimization problem we consider its
related decision problem.
• This will give us intuition about its hardness.
Linear Programming and its Usage in Approximation Algorithms for NP Hard Optimization Problems
9. 9
• We know the following definition about NP:
L ∈ NP ⇔ ∃V(.,.) ∈ P, ∃P(.), ∀x ∈ Σ∗ ,
1. x ∈ L ⇒ ∃y, y ≤ P( x ) and V(x, y) accepts.
2. x ∉ L ⇒ ∀y, y ≤ P( x ) and V(x, y) rejects.
• We can look at this process like this:
Y
Find Certificate Y y V(X,Y)
X x
X
x
Linear Programming and its Usage in Approximation Algorithms for NP Hard Optimization Problems
10. 10
• Example:
SAT = {< Φ >: Φ is a satisfying assignment}.
SAT ∈ NP Because :
1) if Φ 0 ∈ SAT ⇒ We are given True assignment
, and could check it in polynomial time.
2)if Φ 0 ∉ SAT ⇒ Then there is no true assignment.
• There is one another important concept in
complexity which is Reduction.
Linear Programming and its Usage in Approximation Algorithms for NP Hard Optimization Problems
11. 11
Definition :
L ≤ P L* ⇔ ∃f (Polynomial Time Function) ∀x ∈ L ⇔ f ( x) ∈ L* .
f
L L*
f
L L
L*
Fig1: Graphical Diagram of Reduction Concept
Linear Programming and its Usage in Approximation Algorithms for NP Hard Optimization Problems
12. 12
Example :
SAT = {< Φ >| Φ is satisfiable assignment}.
3 - CNF = {< Φ >| Φ is 3 - cnf and satisfiable}.
we show that :
SAT ≤ P 3 − CNF .
(3 − CNF is in this form for example :
(x1 ∨ x 2 ∨ x 3 ) ∧ ( x1 ∨ x 2 ∨ x 3 ) ∧ (x1 ∨ x 2 ∨ x 3 )
Linear Programming and its Usage in Approximation Algorithms for NP Hard Optimization Problems
13. 13
Φ ≡ (¬x1 → x2 ) ↔ ( x3 ∧ x2 )
<->
y1 y2
-> ^
-x1 x2 x3 x4
Φ ≡ ( y1 ↔ (¬x1 → x2 )) ∧ (( x3 ∧ x4 ) ↔ y2 ) ∧ ( y1 ↔ y2 )
Now each paranthesis can be converted into 3 - OR form
using truth table.
Linear Programming and its Usage in Approximation Algorithms for NP Hard Optimization Problems
14. 14
Example :
INTEGER − PROGRAMMING = {< Am×n , bm×1 >| Am×n and bm×1 are integer matrices,
∃X n×1 ∈ Ζ n , Am×n X n×1 ≤ bm×1}.
Obviously INTEGER − PROGRAMMING ∈ NP.
We will show that :
3 − CNF ≤ P INTEGER − PROGRAMMING.
Proof :
¬X 1 ∨ X 2 ∨ X 3 ↔ (1 − X 1 ) + X 2 + X 3
¬X 1 ∨ X 2 ∨ X 3 ≡ 1 ↔ (1 − X 1 ) + X 2 + X 3 ≥ 1
X i ∈ Ζ, 0 ≤ X i ≤ 1
Linear Programming and its Usage in Approximation Algorithms for NP Hard Optimization Problems
15. 15
• We conclude this section by the following theorem:
Every language in NP is polynomial time
reducible to SAT language.
• Language like SAT, INTEGER-PROGRAMMING,
Hamiltonian Cycle, and 3-CNF are also the same
and they belong to NP-Complete set.
Linear Programming and its Usage in Approximation Algorithms for NP Hard Optimization Problems
16. 16
Approximation Algorithms
• The following describes the connection of NP
optimization Complete problem and its related
optimization problem.
Decision Version :
TSP = {< Gn×n , b >| G n×n is complete non negative weighted graph and there
exists Hamiltonian Cycle such that its cost is less than or
equal b}.
Optimization Version :
OP − TSP :
we have complete non negative weighted graph G n×n , find the Hamiltonian Cycle
which its cost is minimun.
Linear Programming and its Usage in Approximation Algorithms for NP Hard Optimization Problems
17. 17
• Bound of Approximation:
• Suppose that problem P has an optimum solution with cost C-
op. Algorithm X solves problem P
with bound ρ(n) if it finds feasible solution with cost C such
as:
max {c/c-op, c-op/c} ≤ ρ(n).
• Note that with this definition we always have ρ(n) ≥1.
• In the following I try to show that sometimes, finding good
approximation for optimization problems is very hard.
Linear Programming and its Usage in Approximation Algorithms for NP Hard Optimization Problems
18. 18
• Theorem:
It does not exist any polynomial time approximation
algorithm with Polynomial Bound for OPT-TSP unless
P=NP.
• Proof:
we prove that if such an algorithm exists then we can Solve
Hamiltonian Cycle in Polynomial Time.
For each graph G with n vertex convert
It into weighted complete graph G1 as G has Hamiltonian Cycle
the following procedure: Iff
4) Assign 1 to each of its edge. OPT-TSP(G1)=n
5) Assign nρ(n) to the other edges.
Linear Programming and its Usage in Approximation Algorithms for NP Hard Optimization Problems
19. 19
• I think that it’s now time to study Randomized
Rounding technique for solving optimization problems.
• I explore it with two examples.
Linear Programming and its Usage in Approximation Algorithms for NP Hard Optimization Problems
20. 20
Case Study1: Minimum Weight Vertex
Cover
• Now we study randomized rounding method in
approximation algorithms with an example.
• Definition:
• Vertex Cover: In undirected graph G=(V,E), Set A of
vertices is said vertex cover if
for every (u,v) ε E then u ε A or v ε A .
• Minimum Weight Vertex Cover: In undirected graph
G=(V,E) which each vertex has w(v) as positive weight Set
A of vertices is said minimum weight vertex cover if for
every (u,v) ε E then u ε A or v ε A and w(A) is the least.
Linear Programming and its Usage in Approximation Algorithms for NP Hard Optimization Problems
21. 21
Fig2: Vertex Cover
Linear Programming and its Usage in Approximation Algorithms for NP Hard Optimization Problems
22. 22
• It is proved that decision version of this
optimization problem is NP Hard.
• At the first step we model it as integer
programming.
x : V → {0,1}
if v ∈ Minimum Weight Vertex Cover Set ⇒ x(v) = 1.
else x(v) = 0.
Integer Programming Method :
min ∑ w(v)x(v)
v∈V
∀u , v ∈ V x(v) + x (u ) ≥ 1
0 ≤ x (v ) ≤ 1
Linear Programming and its Usage in Approximation Algorithms for NP Hard Optimization Problems
23. 23
• As it was known it is believed that there is no
polynomial time algorithm for integer programming.
• We must think about another method.
• We investigate its related linear programming.
x : V → [0,1]
Linear Programming Method :
min ∑ w(v)x(v)
v∈V
∀u , v ∈ V x(v) + x(u ) ≥ 1
0 ≤ x (v ) ≤ 1
Linear Programming and its Usage in Approximation Algorithms for NP Hard Optimization Problems
24. 24
• The Last method is called Relaxation to linear
programming.
programming
• It’s now time to state the approximation
algorithm.
Approximation Min Weight Vertex Cover(G, w)
1) C ← Φ
2) Compute x an optimal solution to linear programming.
3) for each v ∈ V
4) do if x (v) ≥ 0.5 / * Rounding Method * /
5) then C ← C ∪ {v}
6) Return C.
Linear Programming and its Usage in Approximation Algorithms for NP Hard Optimization Problems
25. 25
Proof :
C :: The soulution of algorithm.
C* :: The real optimal set of integer programming.
Z* :: The optimal value of the linear programming.
Obviously we have Z* ≤ w(C* ).
Obviously C is vertex cover.
Z* = ∑ w(v) x(v) ≥ ∑ w(v) x(v)
v∈V v∈V , x ( v ) ≥ 0.5
≥ ∑ w(v)0.5 =∑ w(v)0.5 = 0.5w(C ).
v∈C
v∈V , x ( v ) ≥ 0.5
0.5w(C) ≤ Z* ≤ w(C* ) ⇒ { ρ = 2}
Linear Programming and its Usage in Approximation Algorithms for NP Hard Optimization Problems
26. 26
• Hastad proved that there is no Polynomial time algorithm for
vertex cover that achieves an approximation better that 1.16
unless P=NP (1997).
• Now we consider another example.
Linear Programming and its Usage in Approximation Algorithms for NP Hard Optimization Problems
27. 27
Case Study2: MAXSAT Problem
MAXSAT :
Given K - CNF Formula with each Clause C j has weight Wj ≥ 0.
Find assignment to variables that maximizes :
∑w
C j is true
j
Example :
Φ = (x1 + x 2 + x 3 + x 4 )( x1 + x 2 + x 3 + x 4 )(x1 + x 2 + x 3 + x 4 ).
C1 = (x1 + x 2 + x 3 + x 4 ) → w 1
C 2 = ( x1 + x 2 + x 3 + x 4 ) → w 2
C3 = (x1 + x 2 + x 3 + x 4 ) → w 3
Max ∑w
C j is true
j
Linear Programming and its Usage in Approximation Algorithms for NP Hard Optimization Problems
28. 28
• We state the following randomized algorithm for this
problem. (Johnson 1974).
1) for variables x1 ,...x n randomly assign
to 0 or 1 with probability 0.5.
2) return ∑w
C J istrue
j
Claim :
1 −1
The preceding algorithm has [(1- k
)] approximation bound.
2
Linear Programming and its Usage in Approximation Algorithms for NP Hard Optimization Problems
29. 29
Proof :
Define the following random varibles :
I j = 1 if clause C j satisfiable if not 0.
∑ w =∑ w I
C j is true
j
Cj
j j
1
p(I j = 1 ) = (1 - ).
2k
1
p(I j = 0 ) = ( ).
2k
1
E{ ∑ w j} =E{∑ w jI j} =∑ w jE{I j} =(1-
C j is true Cj Cj
)∑ w j .
2k C j
1 −1
ρ = [(1 - k
)]
2
Linear Programming and its Usage in Approximation Algorithms for NP Hard Optimization Problems
30. 30
• We can extend Johnson algorithm as follow:
1) for variables x1 ,...x n randomly and independently assign
to 1 or 0 with probability p i and 1 - p i
2) return ∑w
C J istrue
j
Claim :
E{ ∑ w } = ∑ w (1 − ∏ (1 − p )∏ p )
C j is true
j
Cj
j
if xi
i i
if xi
The proof is just the same.
Linear Programming and its Usage in Approximation Algorithms for NP Hard Optimization Problems
31. 31
• Now we model it with IP and relax it.
Integer Programming Model for MAXSAT :
Max ∑ w jz j
Relaxation
Cj
∀C j ∑ y + ∑ (1- y ) ≥ z
if x i
i i j
if x i
0 ≤ yi ≤ 1
0 ≤ zj ≤1 yi , z j ∈ Ζ
Relaxation for MAXSAT :
Max ∑ w jz j
Cj
∀C j ∑ y + ∑ (1- y ) ≥ z
if x i
i
if x i
i j
0 ≤ yi ≤ 1
0 ≤ zj ≤1 yi , z j ∈ ℜ
Linear Programming and its Usage in Approximation Algorithms for NP Hard Optimization Problems
32. 32
Algorithm
1) Solve the LP and find vector (y* , z * ).
2) use extended Johnson algorithm with p i = y*i .
3) return
Lemma :
for any feasible solution (y* , z* ) to LP and for any clause C j with k Literals,
we have
1
1 - ∏ (1- y i )∏ (y i ) ≥ (1 − (1 − ) k ) z j
if x i if x i k
Claim :
The above algorithm has (1- 1/e) -1 approximation ratio.
Linear Programming and its Usage in Approximation Algorithms for NP Hard Optimization Problems
33. 33
Proof :
∑ w {1 − ∏ (1 − p )∏ ( p )} =
Cj
j
xi
i i
xi
1
∑ w j {1 − ∏ (1 − yi )∏ ( yi )} ≥ (1 − (1 − ) k )∑ w j z j =
* * *
Cj xi xi k Cj
1 k * 1 k *
(1 − (1 − ) ) Z LP ≥ (1 − (1 − ) ) Z IP
k k
[ρ = (1 − 1 / e) −1 ]
Linear Programming and its Usage in Approximation Algorithms for NP Hard Optimization Problems
35. 35
• Algorithm
2. Compute the value from Johnson
algorithm (w1).
3. Compute the value from LP Based
algorithm (w2).
4. Return MAX(w1,w2).
Linear Programming and its Usage in Approximation Algorithms for NP Hard Optimization Problems
36. 36
Conclusion
• In this talk the randomized rounding method is explored
with two example.
• This new method opens new insight into approximation
algorithms and complexity theory.
The End
Linear Programming and its Usage in Approximation Algorithms for NP Hard Optimization Problems