Fractional Chromatic Index test fails with GLPK

[Jeroen Demeyer]

The test

            sage: g = graphs.PetersenGraph()
            sage: g.fractional_chromatic_index(solver='GLPK')
            3.0

added in src/sage/graphs/graph.py by #23658 fails with GLPK-4.63 on 32-bit.

As a workaround, we use PPL by default in #24099.

[David Coudert]

I suspect that we need to change if M.solve(log = verbose) <= 1: to if M.solve(log = verbose) <= 1 + tol:, where tol = 0 if solver=='PPL' else 1e-6. I don't like this solution, but I don't know what else we can do.

I don't have access to a 32-bit machine and so cannot test.

[Jeroen Demeyer]

You could also forbid using a non-exact solver for this problem.

[David Coudert]

Sure, we can force PPL, but it is way slower (can sometimes be faster on small graphs).

sage: G = graphs.Grid2dGraph(6,6)
sage: %time G.fractional_chromatic_index(solver='GLPK')
CPU times: user 43.4 ms, sys: 4.9 ms, total: 48.3 ms
Wall time: 52.1 ms
4.0
sage: %time G.fractional_chromatic_index(solver='PPL')
CPU times: user 1min 11s, sys: 256 ms, total: 1min 11s
Wall time: 1min 12s
4

I agree that using a tolerance gap is not a nice solution either.

[David Coudert]

I don't see better solution than making PPL the default solver here.

---

New commits:

​7485007

trac #23798: set PPL has default solver

[Jeroen Demeyer]

"Be aware that this method may loop endlessly when using some non exact solvers on 32-bits". I doubt that this is problem specific to 32 bits. The wording seems to imply that it's safe to use non-exact solvers on 64-bit machines.

[Jeroen Demeyer]

Also, this isn't quite correct:

Tickets :trac:`23658` and :trac:`23798` are fixed::

followed by a test with GLPK.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​910fb83

trac #23798: reviewers comments

[David Coudert]

Is this more appropriate ?

[Jeroen Demeyer]

Well, it depends. Do you consider the code here to be a fix or a workaround? I am asking because you need to decide what to do with

sage: g.fractional_chromatic_index(solver='GLPK') # known bug (#23798)

You cannot say that this ticket is a known bug while at the same time fixing this ticket.

[David Coudert]

The problem is not fixed. That's why I changed the text to Issue reported in :trac:`23658` and :trac:`23798` with non exact solvers::. What else can I write to be more correct/specific?

[Jeroen Demeyer]

Replying to dcoudert:

The problem is not fixed.

Then I'm moving your branch to a new ticket: #24099.

[David Coudert]

OK, thanks.

[David Coudert]

Since #24824, we use GLPK 4.65. Does anyone with access to a 32-bit machine still see the bug ?

[Matthias Köppe]

Setting new milestone based on a cursory review of ticket status, priority, and last modification date.

[Dave Morris]

Replying to dcoudert:

Since #24824, we use GLPK 4.65. Does anyone with access to a 32-bit machine still see the bug ?

I still see the bug (on a 32-bit debian virtual machine). The default solver seems instantaneous, but I let solver='GLPK' run for about 15 minutes and did not get an answer.

[David Coudert]

This is unfortunate.

The only solutions I see are:

• Force to use PPL, but this is not nice for users with a 64 bits machine (most of the users I guess)
• Raise an error when the solver is glpk on a 32 bits machine

and none of them are satisfactory.

[Matthias Köppe]

Replying to dcoudert:

I suspect that we need to change if M.solve(log = verbose) <= 1: to if M.solve(log = verbose) <= 1 + tol:, where tol = 0 if solver=='PPL' else 1e-6. I don't like this solution, but I don't know what else we can do.

Using a tolerance is exactly the right solution. The test for exact <= 1 and == 1 is meaningless with a numerical LP solver. LP solvers use perturbations systematically. It is not a bug if the result is not an exact integer.

[Matthias Köppe]

See also my explanations in ​https://trac.sagemath.org/ticket/30635#comment:20 and following.

[Dima Pasechnik]

there are two LPs involved, one of them for a maximum weight matching, something that can be instead done by a combinatorial algorithm, see e.g. Blossom V in ​http://pub.ist.ac.at/~vnk/software.html

[Dima Pasechnik]

If I force PPL on the inner (matching) LP:

src/sage/graphs/graph_coloring.pyx

@@ -825 +825 @@
     frozen_edges = [frozenset(e) for e in G.edges(labels=False, sort=False)]
 
     # Initialize LP for maximum weight matching
-    M = MixedIntegerLinearProgram(solver=solver, constraint_generation=True)
+    M = MixedIntegerLinearProgram(solver="PPL", constraint_generation=True)
 
     # One variable per edge
     b = M.new_variable(binary=True, nonnegative=True)

then on a 32-bit system it's all fine (GLPK from the system, unpatched, so these extra messages)

sage: G=graphs.PetersenGraph()
sage: G.fractional_chromatic_index(solver="GLPK")
Long-step dual simplex will be used
Long-step dual simplex will be used
Long-step dual simplex will be used
Long-step dual simplex will be used
Long-step dual simplex will be used
Long-step dual simplex will be used
3.0

[David Coudert]

Following above discussion, I added a tolerance gap for numerical LP solvers.

Note that we can use the networkx implementation of the blossom algorithm via the matching method, but it does not solve the issue. Actually, it's slower and worse for the rounding as I observe the issue on a 64 bits machine...

---

New commits:

​ebcde7c

trac #23798: add tolerance gap for numerical LP solvers

[Dima Pasechnik]

I don’t like this approach. Without explicit guarantees that these tolerances are correct, it is replacing correct algorithms with heuristics.

[Matthias Köppe]

         matching = [fe for fe in frozen_edges if M.get_values(b[fe]) == 1]

This line also needs changing because the test "== 1" is not robust.

[Dima Pasechnik]

I don’t see how one can make the oracle (the inner LP) inexact, without potentially returning a very wrong answer.

The oracle checks that there is no maximum weight matching of weight >1. Say, we let it error by epsilon, i.e we terminate with oracle returning 1+epsilon. Potentially, there could be K maximum matchings with this weight, if they are disjoint this means that the final error K times epsilon, oops…

[David Coudert]

I don't like this solution either but I don't know what to do when a solver returns 0.99999... instead of 1 although we have set the variable type to binary. The solvers are aware of the type of the variable and so should return a value with the correct type and not a double. The solution might be in the backends.

[Dima Pasechnik]

Replying to dcoudert:

I don't like this solution either but I don't know what to do when a solver returns 0.99999... instead of 1 although we have set the variable type to binary. The solvers are aware of the type of the variable and so should return a value with the correct type and not a double. The solution might be in the backends.

No, my point is that without a special analysis it's not possible to argue that solving the oracle problem (with non-integer objective function) inexactly provides a correct result, even if you "correctly" round 0.9999... to 1. It's because a small oracle error may get amplified a lot in the main LP. Welcome to floating point hell :-)|

[Dima Pasechnik]

Replying to dcoudert:

Following above discussion, I added a tolerance gap for numerical LP solvers.

Note that we can use the networkx implementation of the blossom algorithm via the matching method, but it does not solve the issue. Actually, it's slower and worse for the rounding as I observe the issue on a 64 bits machine...

The oracle implementation here is naive, and bound to get very slow; it's integer LP without Edmonds' constraints, instead of a "normal" LP over the matching polytope with Edmonds' constraints (aka blossom inequalities). So this would need yet another oracle (as there are too exponentially many inequalities there), but well, it's polynomial time then. The generated constraints can stay, so this should be fast.

---

New commits:

​ebcde7c

trac #23798: add tolerance gap for numerical LP solvers

[Matthias Köppe]

I took a quick look at the function now. I would suggest the following changes:

1. Before adding a new constraint to the master problem, verify that matching is indeed a matching. In this way, the master problem will always be a correct relaxation, even if an inexact oracle is used.

2. When the numerical solver that is used for solving the separation problem does not find a matching of value greater than 1 + epsilon, you can switch to PPL - then, with a bit of luck, it can prove the bound <= 1.

3. It will make sense to have separate parameters for the solver used for the master problem and the one(s) used for the separation problem.

[Dima Pasechnik]

Actually, it seems that even with PPL, the code is just wrong, as PPL does not do MILP, it only does LP, right?

[Matthias Köppe]

The PPL does have a (very limited) MIP solver.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​1926be5	trac #23798: merged with 9.5.beta5
​43e8873	trac #23798: ideas from comment 35

[David Coudert]

I tried the ideas from #comment:35. I have let some code for debugging as the code may loop forever when using GLPK for both master and separation problems. The patchbot will complain...

We should search for another method not relying on LP solvers, if any...