Combinatorial Problems2

Outline
Introduction
What are combinatorial problems ?
Challenges
Real life Problems
Solving combinatorial problems
Reducing combinatorial problems
Modeling as
Constraint satisfaction problem
Subset generation problem
Satisfaction problem
Error correcting & Cryptology
Hamming Distance Problem
Logical Cryptanalysis of DES

What are combinatorial problems ?
A class of problems that are at least NP:
Enumerative combinatorics
Counting the number of certain combinatorial objects
How many ways are there to make a 2-letter word?
How many ways are there to select 5 integers from {1,..,20} such that the (positive) difference between any 2 of the 5 integers is at least 3?
Design Theory
Study of combinatorial design, which are collection of subsets with certain intersection properties
Ternary Steiner systems
oSelect 3-element subsets from the set {0,1,2,3,4,5,6} such that each object occurs in 3 of the subsets and the intersection of every two subsets has precisely one member.
Graph theory, Order theory, Geometric combinatorics, …etc.

What are combinatorial problems ?
Characteristics & Challenges
Most of these problems are classified as NP problems
Huge search space
Real combinatorial problems
Cryptography / Error Correcting
Finding Shortest round trips (TSP)
Boolean satisfaction problems (SAT)
Planning, scheduling & timetabling
Protein structure prediction
Internet packet data routing , …etc.

Solving combinatorial design problems
Reducing combinatorial design problems
Need to prune search space before actually searching
Get results in reasonable time
How can this be done ?
Constraint programming
Model problem as a constraint satisfaction problem (CSP)
o(V,D,C)
oV is a set of variables taking values from domain D upon which constraints in set C are satisfied.
Constraint solving
oUse clever algorithms to solve a CSP problem
•Node, Arc & Path consistencies
•Constraint propagation

Constraint satisfaction problem (CSP) vs. Generate & Test
15 variables [V1,..,V15] all taking values in the domain {1,..,15}
V1<V2, V2<V3, …, V14<V15
Generate & Test Approach
Enumerating all possible assignments
1515= 437,893,890,380,859,375 possibility
109assignments/second computer would need 14 years to find solution

Constraint satisfaction problem (CSP) vs. Generate & Test
Model as a constraint satisfaction problem
Constraint satisfaction Solving
Interleaving generation and checking
After each choice check if constraints have been violated and then fail early
Propagate & distribute
Making tests active rather than passive
oNot just testing whether constraints are violated or not but making them active and propagate constraints
V1 < V2
oV2 must be 1 greater than V1
oV1 = {1,..15} and V2 = {2,..,15}
Repeat this reasoning until V14 < V15
oV15 = {15}

Combinatorial Design Problems using sets
Combinatorial Design Problems
Study of combinatorial design, which are collection of subsets with certain intersection properties
Approach
Model variables as sets
Solve using set solvers

Select 3-element subsets from the set {0,1,2,3,4,5,6} such that each object occurs in 3 of the subsets and the intersection of every two subsets has precisely one member.
Model
Use intersection constraint to state that the intersection of every two 3- element subsets has precisely one member
Use element constraint to state that each element occurs in 3 of the 3- element subsets
Solving
Set solver

Solving
Set solver
Order in sets doesn’t matter
Symmetry breaking
o{0,1,2}
o={1,0,2}
o={2,1,0}
o={0,2,1}
Solution
{012},{034},{056},{135},{146},{236},{245}

Combinatorial Problems as SAT problems
Boolean satisfaction problems (SAT)
The problem of determining if the variables of a given Boolean formula can be assigned in such a way as to make the formula evaluate to TRUE.
Example
For (A B) (B ¬C) (A ¬B) to be true which variables have to be true to satisfy all clauses ?
Approaches
oExhaustive search in the form of truth table to find the solution
•Number of rows for the truth table is 2# of variables

Combinatorial Problems as SAT problems
Model as a SAT problem and use SAT solvers
SAT solvers
Davis-Putnam Algorithm
Splits the clauses using truth assignments
(A B) (B ¬C) (A ¬B) B (B ¬C) ¬B
B ¬C()
()¬C
 
() A=0
A=1B=0
B=1B=0
B=1C=0
C=1() = Failed Partial Assignment = Successful Partial Assignment

Error correcting & Cryptology
In a good cryptographic system, changing one bit in the cipher text changes enough bits in the corresponding plain text to make it unreadable.
What causes the change/error of the cipher text?
Noise from communication channels
Could be an attack

Error correcting & Cryptology
Holding a conversation in a noisy room becomes more difficult as the noise become louder.
Data transmission becomes difficult as the communication channel gets noisier.
Solution ?
Raise your voice
Repeat yourself
Adding some redundancy to the transmission in order for the recipient to be able to reconstruct the message.
Replace the symbols in the original message by codewords that have some redundancy built into them.
Code words
Block codes (not varying in length)

Hamming Distance Problem
We are trying to find how close the 2 codewords/code vectors are.
The Hamming Distance = d(u,v)
The number of places where the 2 codewords/code vectors differ.
Examples
u = (1,0,1,0,1,0,1,0)
v = (1,0,1,1,1,0,0,0)
d(u,v) = 2

Hamming distance optimization problem
As an optimization problem then, the task is to find (and prove) the maximum size (number of codewords = a(n, d,w)) for a binary error correcting code with given parameters (n, d) and optionally fixed weight (w).
Approach
Trying to find increasingly larger codes and proving optimality by failing to find one larger.

Hamming Distance Optimization problem
How to model this ?
Using set variables (Si) to represent code words (Ci)
The element x is in the set Si iffthe code Cihas a 1 at position x
[0000]->S={}
[1000]->S={1}
[0100]->S={2}
[1100]->S={1,2}

Hamming Distance Optimization problem
Example
a(4,1,1)
How many code words of length 4 , minimal distance of 1 and weight 1 ?
[1000]S = {1}
[0100]S = {2}
[0010]S = {3}
[0001]S = {4}
Answer = 4

Logical Cryptanalysis of DES
What is DES ?
Data encryption standard which is a state of the art cipher system used to encrypt/decrypt text.
Cryptanalysis
Trying to break the cipher system and acquire key
Logical cryptanalysis of DES ?
Modeling the DES algorithm as a SAT problem and therefore explicit representation of Plain bits, Key bits, and Cipher bits.

Logical Cryptanalysis of DES
ε(P,K,C)
which is true if and only if for the corresponding sequences of bits (P0,P1,C0,C1,… etc) we have that C = Ek(P) holds.

Logical Cryptanalysis
Now that we have modeled our problem
We can use the formula to crack the key given a known plain text and cipher text.
We can find out interesting properties of the cipher system
Faithful cipher
for any pair of plaintext and cipher text there is only one key that could have produced them.

Combinatorial Problems2

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