1. A Dating Algorithm
Implementing the Simplex Algorithm
Implementing Euclid’s GCD
Presented in partial fulfillment of the requirements for Project 2
By: David Poeschl & Tim Reynolds

2. A Quick Preview
• The Problem: Weighted Bipartite Matching
– Given a group of n (heterosexual) boys and n (heterosexual)
girls, each of whom has previously assigned values to each
member of the opposite gender, find a matching of boys with
girls which maximizes happiness (which can be defined many
ways). To do this, we designed the…
• Reynolds-Poeschl Matching Algorithm (RPMA)
– Works by converting the data into a linear programming problem
which can then be solved by…
• The Simplex Algorithm
– A well known algorithm for solving linear programming problems.
Since Simplex uses a lot of operations on rational numbers,
we’ve created our own Rational Number Class which
implements
• Euclid’s GCD

3. Obligatory Diagram
Input: Output:
Preferences An optimal matching
LP Problem (only once…) Solution to the LP Problem
Integers x & y (repeatedly…) The GCD of x and y

4. Algorithm – Simplex
• The Simplex Algorithm portion of the project is a stand-
alone application which solves any linear programming
problem with a feasible solution.
– Input: A LP Problem P = (c[ ], b[ ], A[ ][ ]) in n variables
{x1 ,… xn} and m constraints:
Maximize c1x1 + … + cnxn (the objective function) subject to:
A1,1x1 + … + A1,nxn ≤ b1
…
Am,1x1 + … + Am,nxn ≤ bm
– Output: An array representing a setting of the values of
{x1 ,… xn} which maximizes the objective function (if possible).

5. Algorithm – Simplex
• Simplex Algorithm
– If you honestly want us to drag you through the details of the Simplex
Algorithm and you ∈ Class–{Mike, Jarod, Will}, raise your hand…
– No takers?
– Last chance…
– Jarod, put your hand down.
– Seriously, put your hand down.
• Everybody* knows how to implement the Simplex Algorithm by now,
so we’ll move on to the mildly interesting** portion.
* Unless you’re a dirty*** cheater
** Interesting is a relative term
*** Dirty means “of low academic integrity”

6. Algorithm – RPMA
• Input: Two n by n matrices PBG and PGB
– PBGi, j = The value that boy i assigned girl j
– PGBi, j = The value that girl i assigned boy j
• Output: A set of n boy-girl pairs which produce
the optimal happiness.
– Each boy and girl appears exactly once in the set of
pairings.
• The values assigned are positive integers. The
lower a value is, the more (for example) the boy
likes that particular girl.

7. Algorithm – RPMA
• What is happiness?
– The happiness of a complete matching is defined as the PBG Lisa Jess
sum of the happiness of each individual pair. The
happiness of a pair can be defined in several ways. Joe 1 4
Consider the following matrices:
Bob 3 7
• Maximizing Happiness (Higher value = happier)
– Define max_happy as the highest value appearing in either matrix, P GB
Joe Bob
and min_happy similarly. Then…
– Additive Happiness Lisa 2 6
• Happiness(Boy b, Girl g) =
[max_happy + min_happy – PBGb, g ] + Jess 1 5
[max_happy + min_happy – PGBg, b ]
– Multiplicative Happiness
• Happiness(Boy b, Girl g) = Note: Lower values = more attraction
[max_happy + min_happy – P b, g ] *
BG
[max_happy + min_happy – PGBg, b ]
Ex. Multiplicative Happiness (Bob, Lisa)
• Minimizing Unhappiness (Higher value = less happy)
– Additive Unhappiness = [7+1-3] * [7+1-6] = 5 * 2 = 10
• Happiness(Boy b, Girl g) = -1 * (PBGb, g + PGBg, b)
– Multiplicative Unhappiness
• Happiness(Boy b, Girl g) = -1 * (PBGb, g * PGBg, b)

8. Algorithm – RPMA
• We now proceed to creating an LP:
– We have n * n variables {x1 ,… xn*n}
• Variable xi represents whether the boy-girl pair of (Boy (i-1) / n + 1,
Girl [(i -1) mod n]+1) is part of our matching
– We have 2n constraints…
• Constraints 1 through n ensure that each boy is paired with no more than
one girl.
• Constraints n+1 through 2n ensure that each girl is paired with no more than
one boy.
• If we are minimizing unhappiness, then we need one additional constraint
-1*x1 + -1*x2 + … + -1*xn*n ≤ -1*n
– This causes any matching with less than n pairs to be infeasible.
– The objective function
• Happiness(Boy (1-1) / n + 1, Girl [(1-1) mod n]+1)*x1 +
Happiness(Boy (2-1) / n + 1, Girl [(2-1) mod n]+1)*x2 +
…+
Happiness(Boy (n*n-1) / n + 1, Girl [(n*n-1) mod n]+1)*xn*n

9. Algorithm – RPMA
• A (potentially) big problem
– What if somebody assigns their favorite person a value of 1,
but assigns everybody else very large values like 999,937 or
827,399?
• This person is selfishly trying to hack our algorithm to guarantee
they get their number one choice.
– Solution: Convert values to rankings, then calculate
happiness values.
PBG Lisa Jess PBG Lisa Jess
Joe 1 4 Joe 1 2
Bob 3 7 Bob 1 2

10. Illustrative Example
PBG G1 G2 G3 G4 G5 PGB B1 B2 B3 B4 B5
B1 1 2 3 8 10 G1 10 8 3 2 1
B2 1 2 2 2 15 G2 15 2 2 2 1
B3 7 5 3 1 1 G3 5 6 3 1 1
B4 9 2 9 10 4 G4 4 5 12 8 6
B5 7 7 6 2 1 G5 3 12 7 2 9
• Ex. Additive Happiness(2, 3) = (15+1-2) + (15+1-6) = 14 + 10 = 24
• Ex. Multiplicative Unhappiness(2, 3) = -1 * (2 + 6) = -8
MIN MAX MIN MAX MIN MAX MIN MAX
ADD ADD ADD ADD MUL MUL MUL MUL
RANK RANK RANK RANK
(1,3) (1,3) (1,1) (1,3) (1,1) (1,4) (1,5) (1,1)
(2,4) (2,4) (2,4) (2,4) (2,4) (2,2) (2,4) (2,4)
(3,5) (3,5) (3,3) (3,1) (3,5) (3,3) (3,3) (3,3)
(4,2) (4,2) (4,2) (4,2) (4,2) (4,5) (4,2) (4,2)
(5,1) (5,1) (5,5) (5,5) (5,3) (5,1) (5,1) (5,5)

11. Analysis, Testing & Results
• Data Structures
– We used Arrays instead of ArrayLists…
• Because we did not use iterative structures, this
improved our performance.
– The implementation of our Rational Number
Class greatly decreased our performance.
Every time two rational numbers are added,
we actually perform 3 integer multiplications,
then run GCD to reduce the new rational.

12. Analysis, Testing & Results
• Testing the Simplex Algorithm
– A test case for which there is no feasible solution
– A test case for which the solution is unbounded
– A test case for which the initial basic solution is not feasible, but there
exists a feasible solution somewhere
– A large test case with 3600 variables and 121 constraints.
• Testing the RPMA (Matching)
– Wrote a program that, given an integer n, generates random data sets
of for 2*n people
• Two n x n arrays of positive integers in [1, 100]
– We then have n2 variables and 2*n constraints

13. Analysis, Testing & Results
• Results
– Ran timings on random data sets for n boys & n girls
• n = 1, 2, 5, 10, 15, 20, 25, 30, 35, 40, 50, 60.
Time versus n (Logarithmic Scale)
100000
10000
1000
Time (s)
100 RPMA
10
1
0.1
5 10 15 20 25 30 35 40 45 50 55 60
n

14. Extension: The Poeschl-Reynolds Gender
Preference Neutral Matching Algorithm
(or NAMbLA)
• Thus far, we have only considered cases with n boys and n girls, each of
whom is heterosexual…
– But what if some boys like other boys? What if some girls like boys and
girls?
• Consider a new, non-heterocentric problem which considers n
people, each of whom rates everybody else.
– Creates an n x n matrix P
– Lower rating values mean more attraction
– A value of 0 at Pi, j to indicates that person i is not attracted to person j
at all and should not be matched with them.
• We can then support heterosexual, bisexual, or homosexual users.
• Alternatively, if you think homosexuality is immoral, we could be
trying to find an optimal seating of n friends/enemies on a 2 seat
wide rollercoaster train.

15. Extension: The Poeschl-Reynolds Gender
Preference Neutral Matching Algorithm
• This new version will have (n C 2) variables and n+1 constraints. Xi p1 p2 p3 P4
– Variables p1 0 1 2
• Variable X0 represents whether person 1 and 2 are paired together
p2 3 4
• Variable X1 represents whether person 1 and 3 are paired together …
• Variable Xn-2 represents whether person 1 and n are paired together p3 5
• … p4
• Variable Xn-1 represents whether person 2 and 3 are paired together
• Variable Xn represents whether person 2 and 4 are paired together ……
– Constraints
• Constraint i is made up of all 0’s except where person i is part of that pairing
– Constraint 1: 1 1 1 0 0 0 [ p1 is part of pairings 0, 1, and 2]
– Constraint 2: 1 0 0 1 1 0
– Constraint 3: 0 1 0 1 0 1
– Constraint 4: 0 0 1 0 1 1
– Objective Function (Minimizing Additive Unhappiness Only)
• The coefficient on Xi is
– If both people involved in Xi have non-0 entries for each other, -1 * (summed
happiness)
– Else, -1 * (an arbitrarily large value)

16. The Poeschl-Reynolds Gender Preference
Neutral Matching Algorithm - EXAMPLE
Objective Function:
-5X0 -6X1 -5X2 -100000X3 -13X4 -9X5 -6X6 -6X7 -7X8 -4X9
b values: P p1 p2 p3 p4 p5
1 1 1 1 1 -2
p1 0 2 5 1 0
p2 3 0 5 2 1
Normal Constraints
1111000000 p3 1 8 0 4 5
1000111000 p4 4 7 2 0 3
0100100110 p5 8 5 2 1 0
0010010101
0001001011
Results
Constraint to Force Pairings (1, 2), (4, 5), (3)
-1 -1 -1 -1 -1 -1 -1 -1 -1 -1

17. The Poeschl-Reynolds Gender Preference
Neutral Matching Algorithm
• Why is this interesting?
– You mean besides the political statement?
– The constraints in this generalization are much more complicated than in the
heterocentric version
– We also have to consider what happens if a person absolutely cannot be paired with
somebody else. We might end up with several singletons
– It really is an extension of the original problem.
P {Boys} {Girls}
• Given PBG and PGB , create our new matrix P =
• This gives the same solution as our original
algorithm. {Boys} 0…0 PBG
. .
. .
0…0
{Girls} PGB 0…0
. .
. .
0…0

18. Demo in (3-1)D!!!

19. References & Thanks
- Introduction to Algorithms, Second Edition Cormen, Leiserson, Rivest, Stein
- Thanks to Mike & Jarod for fun discussions about rounding errors
- We told you so.
- Wikipedia
- It’s always correct.