Posts Tagged ‘graph theory’

In previous posts about graph minors, we reviewed the definitions of graph minors and topological minors, then looked at a relationship between the two; and explored whether there was a relationship between the computational hardness of a set of forbidden subgraphs and the graph class defined as all graphs that avoid the set of forbidden subgraphs.  Both of those questions are fundamentally related to the Graph Minor Theorem of Robertson and Seymour.  Today, I’d like to take a step back and look big-picture at the Graph Minor Theorem, and its relationship to computer science, by featuring an answer Timothy Chow gave to a question Ryan Williams asked about the role of the Axiom of Choice in TCS.

Ryan’s question, which he asked in October 2010, was, “What interesting theorems in TCS rely on the Axiom of Choice (or, alternatively the Axiom of Determinacy)?” The Axiom of Choice, of course, is a (sometimes controversial) axiom extending Zermelo-Fraenkel set theory, and at this point is generally accepted as part of the foundations of mathematics.  Intuitively, the Axiom of Choice says that anytime you have a mathematical construction in which there is a set of nonempty bins, then it is legal to produce a set that contains exactly one item from each bin.  While this might seem “obviously true,” once this intuition is applied to infinitely-many bins, there are nonconstructive and perhaps counter intuitive consequences.  So it is natural to wonder, in the world of TCS where the sets of bins we encounter are unlikely to be “pathologically infinite,” whether we really need the Axiom of Choice to prove anything.  (I won’t discuss the Axiom of Determinacy, except to say that it is consistent with the Zermelo-Fraenkel axioms, but inconsistent with the Axiom of Choice, and set theorists sometimes put it forth as a “better” axiom, but it has not been widely accepted by mathematicians outside of logic.)

Janne Korhonen answered Ryan’s question by noting that the proof of the Graph Minor Theorem fundamentally uses Kruskal’s Tree Theorem, which in turn uses the Axiom of Choice.  A fine answer, and one might think that was the end of the story — but one would be wrong, as Timothy Chow demonstrated in a followup answer which I will now quote from.

What about arithmetical statements whose proof requires something like Koenig’s lemma or Kruskal’s tree theorem? Don’t these require a weak form of the axiom of choice? The answer is that it depends on exactly how you state the result in question. For example, if you state the graph minor theorem in the form, “given any infinite set of unlabeled graphs, there must exist two of them such that one is a minor of the other,” then some amount of choice is needed to march through your infinite set of data, picking out vertices, subgraphs, etc. However, if instead you write down a particular encoding by natural numbers of the minor relation on labeled finite graphs, and phrase the graph minor theorem as a statement about this particular partial order, then the statement becomes arithmetical and doesn’t require AC in the proof.

So the question comes down to, “What do we really mean in theoretical computer science when we talk about a mathematical proof of an infinitary statement?”  To quote from Chow’s answer again:

Most people feel that the “combinatorial essence” of the graph minor theorem is already captured by the version that fixes a particular encoding, and that the need to invoke AC to label everything, in the event that you’re presented with the general set-theoretic version of the problem, is sort of an irrelevant artifact of a decision to use set theory rather than arithmetic as one’s logical foundation. If you feel the same way, then the graph minor theorem doesn’t require AC. (See also this post by Ali Enayat to the Foundations of Mathematics mailing list, written in response to a similar question that I once had.)

So, in the final analysis, at least in Chow’s opinion, there appears to be no example of a TCS theorem that requires the Axiom of Choice to prove.  His final suggestion to Ryan is that it may be more fruitful to ask whether there are TCS statements that require large cardinal axioms to prove, instead of the Axiom of Choice.  Harvey Friedman has apparently demonstrated some artificial examples of finitary graph theory statements that require the consistency of large cardinals to prove.  So if artificial examples exist already, perhaps natural examples are around the corner.

The previous graph theory post mentioned the concept of forbidden minors — a class of graphs that contains no minors of a particular type.  We can also, of course, consider the notion of forbidden subgraphs, in which a graph class $latex \mathcal{C}$ is defined by avoidance of another graph class $latex \mathcal{H}$: graph $latex G$ is a member of $latex \mathcal{C}$ iff $latex G$ contains no subgraph that is a member of $latex \mathcal{H}$.  A partial list of graph classes defined in this way can be found here.

Suppose the forbidden subgraph family $latex \mathcal{H}$ is finite.  If we want to check whether $latex G \in \mathcal{C}$ “all we have to do” is to confirm that $latex G$ does not contain finitely many subgraphs of a particular type, which can be done in time polynomial in the size of $latex G$.  While this has tremendous real-world applications, it also lands us in the world of “nonconstructive polynomial time”: there are graph classes that are provably characterized by finite forbidden families, and yet, we don’t know what the families are.  Moreover, there may be no reasonable way to find out, as shown by the following theorem of Fellows and Langston:

Theorem: There is no algorithm to compute, from a finite description of a minor-closed family represented by a Turing machine that accepts precisely the graphs in the family, the set of obstructions for that family.

I will turn now to a question about what happens when the forbidden family is infinite but easy (or hard) to recognize.  Nonconstructive polynomial time, and the use of finite forbidden family characterization in graph algorithms, is dealt with in more detail in many places, including Algorithmic Implications of the Graph Minor Theorem, by Bienstock and Langston.

Consider chordal graphs, which are graphs where every cycle of length four or more contains a chord, that is to say, an edge between two vertices that are part of the cycle.  So chordal graphs have a simple forbidden graph characterization: no subgraphs that are cycles of size four or more with no chords.  This is an infinite forbidden family, but one that is computationally simple to recognize.  Of course, not all graph classes are simple to recognize, which motivated Vincius Santos to ask the question on CSTheory: is there a relationship between the computational complexity of recognizing a forbidden family of subgraphs, and the computational complexity of the graph class defined by the forbidden family?

Hugo Nobrega gave a remarkable answer to this question, but I wanted to blog about this question both to publicize the answer, and also because the original question remains open.  Nobrega’s answer boils down to “probably not,” though it provides a beautiful example that shows any straightforward “intuitions” will not work.  Perhaps someone reading this blog knows the answer, or has further insight.  If you know any work in this direction, please do leave a comment on the post, or on the original question.  Nobrega’s answer is brief enough that I will reproduce it in full (lightly edited).

A folk theorem is a mathematical fact that is generally known (at least by experts), even though its proof may never have appeared in print, or may be difficult to locate.  Recently, cstheory user Eli asked for a citation to a proof of the following folk theorem.

If $latex G$ is a graph with maximum degree 3 and is a minor of $latex H$, then $latex G$ is a topological minor of $latex H$.

Wikipedia mentions this to be true, and refers to Reinhard Diestel’s excellent book on graph theory.  Diestel’s book asserts its truth as well, but does not give a proof.

Graph $latex G$ is a minor of graph $latex H$, if $latex G$ can be obtained from $latex H$ by removing edges in $latex H$ and identifying the vertices that were previously the endpoints of the now-deleted edge.  So, in a sense, we can say that $latex H$ “has local structures that looks like” $latex G$ if $latex G$ is a minor of $latex H$.  It is sometimes possible to prove strong properties about the global structure of $latex H$ if we are guaranteed that $latex H$ has no minors of a certain type, because then no local area of $latex H$ can look a certain way, so it is possible to say something about the entire graph.  The most famous example of this is Wagner’s Theorem, which states that a graph is planar iff neither $latex K_5$ nor $latex K_{3,3}$ is a minor of it.

A topological minor is a related notion.  Informally, $latex G$ is a topological minor of $latex H$ if we can start with $latex G$, and transform it into a subgraph of $latex H$ by only adding vertices into edges that already exist (i.e., subdividing edges, so edge $latex (u,v)$ becomes two edges, $latex (u,x)$ and $latex (x,v)$).  Figure 1 on this web page shows examples of a minor and a topological minor.  (Note: that figure may be behind a paywall.  If anyone knows a good diagram explaining minor vs. topological minor that is freely available, please put it in the comments.)

Fortunately for Eli, Robin Kothari and co-authors had needed this folk theorem for a paper, and they had been unable to find a proof of it either, so they reproved it themselves.  Kothari’s proof (and answer to Eli’s question) appears here.  It is an example of a growing number of original proofs that appear on cstheory.  That proof by Kothari is definitely my “favorite answer of the week.”  As Eli put it, “This is stellar.”