## Tuesday, February 1, 2011

### Binary Exclusive Or

Lately I've been playing with a mixed integer programming model that involves Hamming distances between binary vectors.  This involves what amounts to computing the exclusive or between two binary variables.  In a previous post I made a comment about exclusive or being somewhat tricky to implement, but that was in reference to general linear constraints -- requiring that exactly one of two possible linear equalities/inequalities be satisfied.  Fortunately, an exclusive or between to binary variables is easy.

Let $x$ and $y$ be two binary variables, and let $z=x\oplus y$  be the exclusive disjunction of $x$ and $y$, also declared as a binary variable. The truth table is as follows:
$x$$y$$z=x\oplus y$
000
011
101
110

We accomplish this with the following inequalities:\begin{eqnarray*}z\le x + y\\z\le 2 - x - y\\ z\ge x - y\\z \ge y-x.\end{eqnarray*} Verification that this works is left to the reader as an exercise.

1. This is a neat problem. i tried this alternative route by "fitting to the truth table":

z = (x-y)^2 = x + y - 2xy

Apply level-1 RLT on the bilinear term xy ~ w to recover the answer.

z = x + y -2w
w <= x
w <= y
w >= 1 - x - y
0 <= w,z,x,y <= 1
x, y binary

the end result is probably equivalent - their performance under fractional x, y doesn't seem different at first glance.

2. sorry, a boo-boo. constraint 4 would be:
w >= x + y -1

3. Just what I was looking for! Thanks!

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