# Challenging MIPs Instances

Today, I share seven challenging MIP instances as .mps files along with the AMPL model and data files I used to generate them. While I like the MIPLIBs, I do prefer problem libraries similar to the CSPLIB where you get both a problem description and a set of data. This allows anyone to try with her new model and/or method.

The MIP instances I propose come from my formulation of the Machine Reassignment Problem proposed for the Roadef Challenge sponsored by Google last year. As I wrote in a previous post, the Challenge had huge instances and a micro time limit of 300 seconds. I said micro because I have in mind exact methods: there is little you can do in 300 seconds when you have a problem with potentially as many as $50000 \times 5000$ binary variables. If you want to use math programming and start with the solution of a linear programming relaxation of the problem, you have to be careful: it might happen that you cannot even solve the LP relaxation at the root node within 300 seconds.

That is why most of the participants tackled the Challenge mainly with heuristic algorithms. The only general purpose solver that qualified for the challenge is Local Solver, which has a nice abstraction (“somehow” similar to AMPL) to well-known local search algorithms and move operators. The Local Solver script used in the qualification phase is available here.

However, in my own opinion, it is interesting to try to solve at least the instances of the qualification phase with Integer Linear Programming (ILP) solvers such as Gurobi and CPLEX. Can these branch-and-cut commercial solvers be competitive on such problems?

## Problem Overview

Consider you are given a set of processes $P$, a set of machines $M$, and an initial mapping $\pi$ of each process to a single machine (i.e., $\pi_p = i$ if process $p$ is initially assigned to machine $i$). Each process consumes several resources, e.g., CPU, memory, and bandwidth. In the challenge, some processes were defined to be transient: they consume resources both on the machine where they are initially located, and in the machine they are going to be after the reassignment. The problem asks to find a new assignment of processes to machines that minimizes a rather involved cost function.

A basic ILP model will have a 0-1 variable $x_{pi}$ equals to 1 if you (re)assign process $p$ to machine $i$. The number of processes and the number of machines give a first clue on the size of the problem. The constraints on the resource capacities yield a multi-dimensional knapsack subproblem for each machine. The Machine Reassignment Problem has other constraints (kind of logical 0-1 constraints), but I do not want to bore you here with a full problem description. If you like to see my model, please read the AMPL model file.

## A first attempt with Gurobi

In order to convince you that the proposed instances are challenging, I report some computational results.

The table below reports for each instance the best result obtained by the participants of the challenge (second column). The remaining four columns give the upper bound (UB), the lower bound (LB), the number of branch-and-bound nodes, and the computation time in seconds obtained with Gurobi 5.0.1, a timeout of 300 seconds, and the default parameter setting on a rather old desktop (single core, 2Gb of RAM).

Instance Best Known UB Upper Bound Lower Bound Nodes Time
a1-1 44,306,501 44,306,501 44,306,501 0 0.05
a1-2 777,532,896 780,511,277 777,530,829 537 -
a1-3 583,005,717 583,005,720 583,005,715 15 48.76
a1-4 252,728,589 320,104,617 242,404,632 24 -
a1-5 727,578,309 727,578,316 727,578,296 221 2.43
a2-1 198 54,350,836 110 0 -
a2-2 816,523,983 1,876,768,120 559,888,659 0 -
a2-3 1,306,868,761 2,272,487,840 1,007,955,933 0 -
a2-4 1,681,353,943 3,223,516,130 1,680,231,407 0 -
a2-5 336,170,182 787,355,300 307,041,984 0 -

Instances a1-1, a1-3, a1-5 are solved to optimality within 300 seconds and hence they are not further considered.

The remaining seven instances are the challenging instances mentioned at the begging of this post. The instances a2-x are embarrassing: they have an UB that is far away from both the best known UB and the computed LB. Specifically, look at the instance a2-1: the best result of the challenge has value 198, Gurobi (using my model) finds a solution with cost 54,350,836: you may agree that this is “slightly” more than 198. At the same time the LB is only 110.

Note that for all the a2-x instances the number of branch-and-bound nodes is zero. After 300 seconds the solver is still at the root node trying to generate cutting planes and/or running their primal heuristics. Using CPLEX 12.5 we got pretty similar results.

This is why I think these instances are challenging for branch-and-cut solvers.

## Search Strategies: Feasibility vs Optimality

Commercial solvers have usually a meta-parameter that controls the search focus by setting other parameters (how they are precisely set is undocumented: do you know more about?). The two basic options of this parameter are (1) to focus on looking for feasible solution or (2) to focus on proving optimality. The name of this parameter is MipEmphasis in CPLEX and MipFocus in Gurobi. Since the LPs are quite time consuming and after 300 seconds the solver is still at the root node, we can wonder whether generating cuts is of any help on these instances.

If we set the MipFocus to feasibility and we explicitly disable all cut generators, would we get better results?

Look at the table below: the values of the upper bounds of instances a1-2, a1-4, and a2-3 are slightly better than before: this is a good news. However, for instance a2-1 the upper bound is worse, and for the other three instances there is no difference. Moreover, the LBs are always weaker: as expected, there is no free lunch!

Instance Upper Bound Lower Bound Gap Nodes
a1-2 779,876,897 777,530,808 0.30% 324
a1-4 317,802,133 242,398,325 23.72% 48
a2-1 65,866,574 66 99.99% 81
a2-2 1,876,768,120 505,443,999 73.06% 0
a2-3 1,428,873,892 1,007,955,933 29.45% 0
a2-4 3,223,516,130 1,680,230,915 47.87% 0
a2-5 787,355,300 307,040,989 61.00% 0

If we want to keep a timeout of 300 seconds, there is little we can do, unless we develop an ad-hoc decomposition approach.

Can we improve those results with a branch-and-cut solver using a longer timeout?

Most of the papers that uses branch-and-cut to solve hard problems have a timeout of at least one hour, and they start by running an heuristic for around 5 minutes. Therefore, we can think of using the best results obtained by the participants of the challenge as starting solution.

So, let us make a step backward: we enable all cut generators and we set all parameters at the default value. In addition we set the time limit to one hour. The table below gives the new results. With this setting we are able to “prove” near-optimality of instance a1-2, and we reduce significantly the gap of instance a2-4. However, the solver never improves the primal solutions: this means that we have not improved the results obtained in the qualification phase of the challenge. Note also that the number of nodes explored is still rather small despite the longer timeout.

Instance Upper Bound Lower Bound Gap Nodes
a1-2 777,532,896 777,530,807 ~0.001% 0
a1-4 252,728,589 242,404,642 4.09% 427
a2-1 198 120 39.39% 2113
a2-2 816,523,983 572,213,976 29.92% 18
a2-3 1,306,868,761 1,068,028,987 18.27% 69
a2-4 1,681,353,943 1,680,231,594 0.06% 133
a2-5 336,170,182 307,042,542 8.66% 187

What if we disable all cuts and set the MipFocus to feasibility again?

Instance Upper Bound Lower Bound Gap Nodes
a1-2 777,532,896 777,530,807 ~0.001% 0
a1-4 252,728,589 242,398,708 4.09% 1359
a2-1 196 70 64.28% 818
a2-2 816,523,983 505,467,074 38.09% 81
a2-3 1,303,662,728 1,008,286,290 22.66% 56
a2-4 1,681,353,943 1,680,230,918 0.07% 108
a2-5 336,158,091 307,040,989 8.67% 135

With this parameter setting, we improve the UB for 3 instances: a2-1, a2-3, and a2-5. However, the lower bounds are again much weaker. Look at instance a2-1: the lower bound is now 70 while before it was 120. If you look at instance a2-3 you can see that even if we got a better primal solution, the gap is weaker, since the lower bound is worse.

## RFC: Any idea?

With the focus on feasibility you get better results, but you might miss the ability to prove optimality. With the focus on optimality you get better lower bounds, but you might not improve the primal bounds.

1) How to balance feasibility with optimality?

To use branch-and-cut solver and to disable cut generators is counterintuitive, but if you do you, you get better primal bounds.

2) Why should I use a branch-and-cut solver then?

Do you have any idea out there?

### Minor Remark

While writing this post, we got 3 solutions that are better than those obtained by the participants of the qualification phase: a2-1, a2-3, and a2-5 (the three links give the certificates of the solutions). We are almost there in proving optimality of a2-3, and we get better lower bounds than those published in [1].

## References

1. Deepak Mehta, Barry O’Sullivan, Helmut Simonis. Comparing Solution Methods for the Machine Reassignment Problem. In Proc of CP 2012, Québec City, Canada, October 8-12, 2012.

# Credits

Thanks to Stefano Coniglio and to Marco Chiarandini for their passionate discussions about the posts in this blog.