## Tuesday, April 30, 2024

### From IP to CP

Someone asked on OR Stack Exchange how to convert an integer programming model into a constraint programming model. I think you can reasonably say that it involves a "paradigm shift", for a couple of reasons.

The first paradigm shift has to do with how you frame the problem, mainly in terms of the decision variables. Math programmers are trained to turn discrete decisions with a logical flavor into binary variables. Discrete quantities, such as how many bins of a certain type to use or how many workers to assign to a task, are expressed as general integer variables, but most other things end up turning into a slew of binary variables. The problem being solved in the ORSE question illustrates this nicely.

The problem is as follows. You have $N$ participants in a tournament involving some kind of racing. Importantly, $N$ is guaranteed to be an even number. There is one track with two lanes, and races are spread over $N-1$ days. Every participant races head to head with every other participant exactly once, and nobody races twice in the same day. For whatever reason, the left lane is preferable to the right lane, and so there is a "fairness" constraint that nobody is assigned the left lane on more than $M$ consecutive days. For some reason, the author also imposed a second fairness constraint that nobody be assigned to the right lane on more than $M$ consecutive days. Dimensions for the author's problem were $N=20$ and $M=2.$

The model has to assign participant pairs (races) to days and also make lane assignments. To decide against whom I must race on a given day, someone building an IP model will use binary variables to select my opponent. Similarly, they will use binary variables to select my lane assignment each day. So the author of the question had in his IP model a variable array opp[Competitors][Competitors][Tracks][Days] taking value 1 "if competitor 'c1' races with 'c2' on track 't' on day 'd'".

CP models are more flexible in their use of variables, and in particular general integer variables. So to decide my opponent on a given day, I can just an integer variable array indexed by day where the value is the index number of my opponent on the given day. Similarly, I could (and would) use an integer variable indexed by day to indicate my lane assignment that day, although in this case that variable does turn out to be binary, since there are only two lanes.

The second paradigm shift has to do with constraints, and it ties to what solver you are using. IP models have a very limited constraint "vocabulary". They all understand linear equalities and inequalities, and some understand some combination of SOS1, SOS2, second order cone and implication constraints. That's pretty much it. CP solvers have a richer "vocabulary" of constraints, but with the caveat that not many of those constraints are universal. I would wager that every CP solver has the "all different" constraint, and they must have the usual arithmetic comparisons ($=,\neq,\lt,\le,\gt,\ge$). Beyond that, it pays to check in advance.

I wrote a CP model (in Java) using IBM's CP Optimizer (CPO) to solve the scheduling problem. Details of the model can be sussed out from the Java code, but I will mention a few pertinent details here.

• I did use an integer variable array to determine, for each combination of participant and day, the participant's opponent that day, as well as an integer array giving the lane assignment (0 or 1) for each combination of participant and day.
• To make sure that, on any day, the opponent of X's opponent is X I used CPO's  inverse constraint. The constraint inverse(f, g) says that f[g[x]] = x and g[f[x]] = x for any x in the domain of the inner function.
• To ensure that nobody raced the same opponent twice, I used allDiff, which is CPO's version of the all different constraint.
• We have to do something to force opponents in a race to be in different lanes. Let $x_{i,d}$ and $y_{i,d}$ denote respectively the opponent and lane assignment for participant $i$ on day $d.$ In mathematical terms, the constraint we want is $y_{x_{i,d},d} \neq y_{i,d}.$ Indexing a variable with another variable is impossible in an IP model. In CPO, I used the element constraint to do just that.

I added an objective function, namely to minimize the difference between the most and fewest times any participant gets assigned the preferred left lane. I also added one constraint to mitigate symmetry. Since any solution remains a solution (with the same objective value) under any permutation of the participant indices, I froze the first day's schedule as $1\ v.\ N$, $2\ v.\ N-1$, $3\ v.\ N-2$ etc.

On my decent but not screamingly fast PC, CPO found a feasible solution almost instantly and a solution with objective value 1 in under a second. In that solution, every participant gets the left lane either nine or ten times out of the 19 racing days. It's not hard to prove that 1 is the optimal value (you cannot have everybody get exactly the same number of left lane assignments), but don't tell CPO that -- it was still chugging along trying when it hit my five minute time limit.

My Java code is available from my repository under a Creative Commons 4.0 open source license.