Many months (and only one post) ago, I wrote about an analysis of the Gabber-Galil graphs using a simple combinatorial lemma and the discrete Cheeger inequality. Here is a note I posted recently carrying the study a bit further.
Given any two invertible, integral matrices , one can consider the family of graphs , where contains edges from every to each of
The Gabber-Galil graphs correspond to the choice and .
Consider also the countably infinite graph with vertex set and edges
Using some elementary Fourier analysis and a discrete Cheeger inequality, one can prove the following relationship (we use to represent the transpose of a matrix ).
Theorem 1 For any , if has positive Cheeger constant, then is a family of expander graphs.
We recall that an infinite graph with uniformly bounded degrees has positive Cheeger constant if there is a number such that every finite subset has at least edges with exactly one endpoint in .
While Theorem 1 may not seem particularly powerful, it turns out that in many interesting cases, proving a non-trivial lower bound on the Cheeger constant of is elementary. For the Gabber-Galil graphs, the argument is especially simple. The following argument is, in fact, significantly simpler than the analysis of the previous post.
In the next lemma, let where and and let be the corresponding edge set. We will prove that has positive Cheeger constant.
Lemma 2 For any finite subset , we have .
Proof: Define . This is the first quadrant, without the -axis and the origin. Define similarly by rotating by , , and degrees, respectively, and note that we have a partition .
Let . We will show that . Since our graph is invariant under rotations of the plane by , this will imply our goal:
It is immediate that . Furthermore, we have because maps points in above (or onto) the line and maps points of below the line . Furthermore, and are bijections, thus In particular, this yields , as desired.
One can generalize the Gabber-Galil graphs in a few different ways. As a prototypical example, consider the family for any . An elementary analysis yields the following characterization.
Theorem 3 For any , it holds that is an expander family if and only if .
For instance, the preceding theorem implies that if has order dividing then is not a family of expander graphs, but if has order or infinite order (the other possibilities) and then the graphs are expanders.
Here is a different sort of generalization. Let be the reflection across the line . The Gabber-Galil graphs can also be seen as where and . We can characterize expansion of these graphs as well.
Theorem 4 For any , it holds that is an expander family if and only if .
It is an interesting open problem to find a complete characterization of the pairs such that is a family of expanders.
Finally, in the realm of re-proving theorems, let me mention this absolutely gorgeous proof by Lovett and Meka of Spencer’s six standard deviations suffice theorem in discrepancy theory. The proof is constructive (i.e., there exists a polynomial-time algorithm to construct the 2-coloring) as in recent groundbreaking work of Bansal, but unlike Bansal’s proof is independent of Spencer’s bound. It also happens to be elementary and beautiful; my thanks to Aravind Srinivasan for pointing it out.
I’m currently teaching a course on spectral graph theory, and I decided to lecture on Margulis’ expander construction and the analysis of its spectral gap by Gabber and Galil. I had never realized how intuitive the analysis could be; the lectures notes I had seen didn’t quite reflect this. In particular, we won’t do any unsightly calculations. Everything I’m about to say is probably well-known, but I thought I would write it down anyway.
The idea is to first start with an initial “expanding object,” and then try to construct a family of graphs out of it. First, consider the infinite graph with vertex set . The edges are given by two maps and , where and . So the edges are and . Clearly is not adjacent to anything. Except for the origin, this graph is an expander in the following sense.
Lemma 1 For any subset , we have
where denotes the edges leaving .
The following simple proof is inspired by a paper of Linial and London.
Proof: First consider a subset that does not intersect the coordinate axes. Let represent the four quadrants of , and let . Consider that and . The latter fact follows because if then . Similarly, and and , while all the respective pairs and and are disjoint.
Combining this with the fact that and are bijections on immediately shows that is at least
Dealing with the coordinate axes is not too hard. Suppose now that is arbitrary. Let . The preceding argument shows . We will show that . Averaging these two inequalities with weights and completes the proof.
Let , and . Then we have the bounds:
The first equation follows since each element of is connected to exactly two elements of , and each element of is connected to exactly two elements of . For instance, is connected to and , while is connected to and .
The second follows because every point of is connected to two unique points of , e.g. is connected to and . The final inequality follows from the fact that from our first argument (since does not touch the axes), and because has no edges to . Summing these three inequalities yields .
Of course, is not a finite graph, so for a parameter , we define the graph with vertex set , where . There are four types of edges in : A vertex is connected to the vertices
This yields a graph of degree at most 8.
Theorem 2 There is a constant such that for every ,
where is the second-smallest eigenvalue of the Laplacian on .
Of course, the graphs now have torsion, and thus our expansion result for is not, a priori, very useful. The non-trivial idea of the proof is that the torsion doesn’t matter, making Lemma 1 applicable. This is not difficult to show using some Fourier analysis. It turns out to be better though, if we first move to the continuous torus.
Let be the 2-dimensional torus equipped with the Lebesgue measure, and consider the complex Hilbert space
equipped with the inner product .
We might also define a related value,
Note that this is just defined as a number; the eigenvalue notation is merely suggestive, but we have not introduced an operator on the torus.
Claim 1 There is some such that for any , we have \
Proof: We simply sketch a proof, which is rather intuitive. Suppose we are given some map such that . Define its continuous extension as follows: Under the natural embedding of into , every point sits inside a grid square with four corners . Writing a local convex combination , we define
Now it is elementary to verify that . It is also easy to verify that for some . (Even if , we still get a contribution from to on this square because we are taking a weighted average.)
Finally, for any , there is a path of length in connecting each of the corners of ‘s square to the corners of ‘s square. A similar fact holds for . In fact, this is the only place where we need to use the fact that edges of the form and are present in . Thus any contribution to can be charged against a term in , and similarly for .
Now our goal is to show that . We will use the Fourier transform to “remove the torsion.” The point is that and , being shift operators, will act rather nicely on the Fourier basis.
We recall that if and we define by , then forms an orthonormal Hilbert basis for . In particular, every can be written uniquely as
where . The Fourier transform is defined as a map , where . In particular, is a linear isometry.
Define and . Then for any , we have
In particular, for any , we have and . The final thing to note is that . So now if we simply apply the Fourier transform (a linear isometry) to the expression in (1), we get a reformulation that is precisely
Here we have simply replaced by in (1), and then written .
But now recall our initial infinite graph on . If we denote by the Laplacian on , then we can rewrite this as,
In other words, it is precisely the first Dirichlet eigenvalue on , subject to the boundary condition .
But now the discrete Cheeger inequality tells us that
where is the minimal expansion of any set not containing the origin. Thus we have indeed unwrapped the torsion and returned to our initial question. Lemma 1 shows that , yielding the desired lower bound on .
The Klein-Plotkin-Rao (KPR) Theorem is a powerful statement about the geometry of planar graphs and their generalizations. Here, I’ll present a new, very simple proof of the theorem that was discovered in joint work with Cyrus Rashtchian. (This will appear in a preprint soon, together with some new results.) In the next post, I’ll give some applications in geometry and algorithms.
Recall that a graph is planar if it can be drawn in the plane without any edge crossings. Wagner’s theorem gives an intrinsic characterization of planar graphs in terms of excluded minors. Recall that a graph is a minor of a graph if can be obtained from by a sequence of (i) edge and vertex deletions and (ii) contraction of edges. A graph excludes as a minor if is not a minor of . Kuratowki’s theorem states that planar graphs are precisely those which exclude both and as minors, where we use and to denote the complete graph on vertices and the complete -by- bipartite graph, respectively. In this post, we are particularly concerned with -minor-free graphs, i.e. those which exclude as a minor for some .
I’ll first state and prove a simpler version of the KPR theorem. In the next post, I’ll discuss a stronger statement (in the language of random partitions) that follows directly from the proof. Then using these partitions, we will show that the observable diameter of every -minor-free graph is “large,” and use that fact to prove an upper bound on the uniform multi-commodity flow gap in such graphs.
2. Low-diameter graph partitioning
Consider a finite graph equipped with its shortest-path metric (much of what we say here extends to infinite graphs). For now, all the edges of will have length one, although we will generalize to arbitrary weighted graphs for some applications in the next post.
Given a subset , we write A weak but simpler version of the KPR Theorem can be stated as follows.
Theorem 1 Let be a graph that excludes as a minor. Then for every number , there exists a partition such that for every and at most an -fraction of edges of go between different sets in the partition.
The theorem was originally proved with a dependence of , but this was improved to by Fakcharoenphol and Talwar. Today I will prove the bound. The partitioning will be accomplished via an iterative operation which we will call chopping.
Consider any connected graph and a number . We will describe an operation which we call a -chop of . Let be any node of , which we will call the “root node” of the chop, and let (the “initial offset”).
The chopping operation is as follows: We partition , where , and the rest of the sets are the disjoint annuli,
for . See Figure 1 for an example of these cuts (the red and blue alternating regions) on a grid graph. Of course, since is finite, eventually the annuli are empty.
We define a -chop on a possibly disconnected graph as the partition arising from doing a -chop on each of its connected components. Finally, we define a -chop on a sequence of disjoint sets as the result of doing a -chop on each of the induced graphs for . Thus if we have an initial partition of , then a -chop of produces a refined partition of . See Figure 2 for the result of two iterated -chops applied to the grid graph. The yellow circles represent the root nodes in the second chop.
We can now state the main technical result needed to prove Theorem 1.
Lemma 2 If excludes as a minor, then for any , any sequence of iterated -chops on results in a partition such that for each .
Observe that refers to the diameter in , the shortest-path metric on . Also, note that we do not constrain the root node or the initial offset of the chops. Klein, Plotkin, and Rao prove this lemma with a dependence of on the diameter. FT use a more complicated approach.
To see how Lemma 2 implies Theorem 1, one proceeds as follows. First, let be large enough so that setting in Lemma 2 yields a partition into sets of diameter at most . After fixing the root node for a -chop of , one can consider the initial offsets . An edge can be cut (i.e. and end up in distinct sets of the partition) in at most one of these offsets. Thus there exists an offset that cuts only a -fraction of edges. Since we perform iterated -chops, there exists a choice of initial offsets that cuts at most -fraction of edges. That completes the reduction.
3. A sketch
Before moving onto the formal argument, I’ll present a simple sketch that contains the main ideas. The proof is by contradiction; if we perform a sequence of -chops and the diameter of any remaining piece fails to be , then we will construct a minor in .
First, we give an equivalent characterization of when a graph has a graph as a minor: There exist disjoint connected subsets , one corresponding to each vertex of . We call these supernodes. Furthermore, there should be an edge between supernodes whenever there is an edge between the corresponding vertices in .
Now, the proof is by induction. Note that the base case is trivial since a minor is a single vertex. By induction, we can assume that if a sequence of chops fails, then there must be a minor contained in some offending annulus. See Figure 3. If we could ensure that every supernode of the minor touched the upper boundary of the annulus as in Figure 3, we could easily construct a minor and be done, by simplying choosing the -st supernode to be a ball around .
Thus we need to enforce this extra property of our minor. The (very simple) idea is contained in Figure 4.
After finding a minor that intersects the annuli, we extend the supernodes to touch the upper boundary of the annulus from the preceding chop (which is represented by the purple line in the picture). The point is that we can choose these paths to be contained above the red boundary (and thus disjoint from the supernodes), and also each of length at most since the width of the “purple” annulus is only . The same can be done for . (If we didn’t care that the paths have to be above the red boundary, we could choose them of length only .)
The only issue is that we need these new paths to be disjoint. Since the paths are always short (length at most ), we can enforce this by making sure that each supernode contains a representative and these representatives are pairwise far apart; then we grow the paths from the representatives. Initially, the representatives will be apart, and then as we go up the inductive chain, they will get closer by at most at every step. By choosing the initial separation large enough, they will remain disjoint. That’s the sketch; it should be possible to reproduce the proof from the sketch alone, but we now present a more formal proof.
4. The proof
We need a couple definitions. First, given a subset of vertices and a number , we say that a set is -dense in if every element of can reach an element of by a path of length that is contained completely in (the induced graph on ). Second, we say that an -minor is -represented by if every supernode of contains a representative from and these representatives are pairwise distance more than apart in the metric (the global shortest-path metric on ). We now state a lemma that we can prove by induction and implies Lemma 2.
Lemma 3 Let and be any numbers. Suppose is a graph and there is any sequence of iterated -chops on that leaves a component of diameter more than . Then for any set that is -dense in , one can find a -minor that is -represented by .
Proof: We proceed by induction on . The case is trivial since a minor is simply a single vertex. Thus we may assume that .
The following figure will be a useful reference.
In general, we argue as follows. Let be any set satisfying the assumptions of the lemma. Assume there is a sequence of iterated -chops that leaves a component of diameter more than . Then there must be some annulus of the first -chop such that iterated -chops on leaves a component of diameter more than .
Suppose the first -chop has root node and initial offset . Let be the set of nodes at distance exactly , i.e. the upper boundary of . Observe that is -dense in by construction. Thus by induction, there is a -minor in that is -represented by . Let be the supernodes of this minor, and let be the representatives which are further than apart.
We now extend this to a -minor which is -represented by . First, we may assume that . Otherwise, all the points of lie in a ball of radius at most , and hence has diameter at most . In particular, we know that for every .
Now for each , we choose a point such that is connected to by a path of length at most in . This can be done by first going up a shortest-path from to of length to reach a point , and then choosing any point of within distance of (which can always be done since is -dense in ). We add this path to to get a new supernode . Observe that the sets are all connected and pairwise disjoint since the new paths are outside the annulus and the paths themselves are pairwise disjoint because they are each of length at most , but they emanate from representatives that are more than apart. In fact, this also shows that the representatives are further than apart, as required.
Finally, we construct a new supernode as follows. For each , let be the closest node to , and let be the closest node in to . For each , let be a shortest-path from to without its endpoint , and let be a shortest-path to , including . We now set . We claim that forms a -minor which is -represented by . First, it is clear that since (again, because is -dense in ). For the same reason, the path is disjoint from all the sets .
Thus the only possible obstruction to having a valid -minor is if some path intersects a set for . We now show that this cannot happen. We know that if intersects , then it must have already traveled distance at least away from . But contains a node adjacent to (by construction), which means it continues an additional distance of (the distance between and . This additional distance is also moving away from , implying that intersects , which is impossible. This completes the proof.
5. The dependence on
The best-known lower bound requires the conclusion of Theorem 1 to cut at least an -fraction of edges. (One can take to be an -vertex 3-regular expander graph, which obviously excludes as a minor. Now it is easy to see that for some constant , partitioning into pieces of diameter at most must cut at least an -fraction of edges.) In some special cases, e.g. graphs of genus (which exclude as a minor for some constant ), one can reduce the bound to (see this joint work with Sidiropoulos). This leads to the following open problem.
Open problem: Show that under the assumptions of Theorem 1, one can find a partition that cuts only an -fraction of edges.
A positive resolution would yield an optimal unifom multi-commodity flow/cut gap for -minor-free graphs. (See the next post for details.)
By now, it is known that integrality gaps for the standard Unique Games SDP (see the paper of Khot and Vishnoi or Section 5.2 of this post) can be used to obtain integrality gaps for many other optimization problems, and often for very strong SDPs coming from various methods of SDP tightening; see, for instance, the paper of Raghavendra and Steurer.
Problematically, the Khot-Vishnoi gap is rather inefficient: To achieve the optimal gap for Unique Games with alphabet size , one needs an instance of size . As far as I know, there is no obstacle to achieving a gap instance where the number of variables is only .
The Walsh-Hadamard code
The Khot-Vishnoi construction is based on the Hadamard code.
(See Section 5.2 here for a complete description.) If we use to denote the Hilbert space of real-valued functions , then the Walsh-Hadamard basis of is the set of functions of the form
Of course, for two such sets , we have the orthogonality relations,
In their construction, the variables are essentially all functions of the form , of which there are , while there are only basis elements which act as the alphabet for the underlying Unique Games instance. This is what leads to the exponential relationship between the number of variables and the label size.
A PSD lifting question
In an effort to improve this dependence, one could start with a much larger set of nearly orthogonal vectors, and then somehow lift them to a higher-dimensional space where they would become orthogonal. In order for the value of the SDP not to blow up, it would be necessary that this map has some kind of Lipschitz property. We are thus led to the following (possibly naïve) question.
Let be the smallest number such that the following holds. (Here, denotes the -dimensional unit sphere and denotes the unit-sphere of .)
There exists a map such that and whenever satisfy , we have .
(Recall that .)
One can show that
by randomly partitioning so that all vectors satisfying end up in different sets of the partition, and then mapping all the points in a set to a different orthogonal vector.
My question is simply: Is a better dependence on possible? Can one rule out that could be independent of ? Note that any solution which randomly maps points to orthogonal vectors must incur such a blowup (this is essentially rounding the SDP to an integral solution).
Today it’s raining in both Seattle and Paris. Here are a few things I think are worth reading while the weather clears up.
Stop collecting coupons. Recently, Batson, Spielman, and Srivastava gave a beautiful sparsification result for graphs: Every graph has a linear-sized spectral sparsifier. Here is one statement of their result (taken from Srivastava’s thesis):
Theorem [BSS]: For every and , the following holds. For every , there are non-negative numbers of which at most are non-zero, and such that for all ,
Assaf Naor has given a very nice account of some recent geometric applications of this idea. These involves breaking the “coupon collecting” barrier from a number of results which were previously based on random sampling. There is still an outstanding open problem left open here: Embedding -dimensional subspaces of into .
Lecture notes that are lecture notes. Some people write lecture notes like they are preliminary book drafts. These lecture notes by Ryan O’Donnell for an undergraduate course on “Probability and Computing” are like transcribed lectures. They’re conversational, with philosophical asides–a great example of the style.
An important epsilon. Finally, there is the recent result of Gharan, Saberi, and Singh giving a approximation for the Traveling Salesman Problem in unweighted graphs. I haven’t gotten a chance to digest the paper yet. Here’s hoping someone else will write an overview of the key ideas.