Optimal Lower Bounds for Sketching Graph Cuts

We study the space complexity of sketching cuts and Laplacian quadratic forms of graphs. We show that any data structure which approximately stores the sizes of all cuts in an undirected graph on n vertices up to a 1+ϵ error must use Ω(n n/ϵ^2) bits of space in the worst case, improving the Ω(n/ϵ^2) bound of Andoni et al. and matching the best known upper bound achieved by spectral sparsifiers. Our proof is based on a rigidity phenomenon for cut (and spectral) approximation which may be of independent interest: any two d-regular graphs which approximate each other's cuts significantly better than a random graph approximates the complete graph must overlap in a constant fraction of their edges.

Authors

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1 Introduction

An spectral sparsifier [ST04] of a weighted graph is a weighted graph such that for every we have the multiplicative guarantee:

 (1−ϵ)xTLGx≤xTLHx≤(1+ϵ)xTLGx. (1)

An cut sparsifier [BK96] is a graph such that (1) holds for all ; the former condition clearly implies the latter, and it is not hard to see that the converse is not true.

There have been many papers on efficiently constructing sparsifiers with few edges (e.g. [SS11, FHHP11, AZLO15, LS17]). The best known sparsity-approximation tradeoff is achieved in [BSS12], who showed that every graph on vertices has an -spectral sparsifier with edges (and this is also the best known upper bound for -cut sparsification). Since every weighted edge can be stored using bits555After discretizing the weights to, say, precision, which introduces negligible error., storing such a sparsifier requires bits.

The recent work [ACK16] studied the question of whether it is possible to use substantially fewer bits if one simply wants a data structure (not necessarily a graph) which answers cut and quadratic form queries. We will use the following definition, which is inspired by the one in [JS17].

Definition 1.1.

An spectral sketch of a graph is a function such that for every :

 (1−ϵ)xTLGx≤f(x)≤(1+ϵ)xTLGx. (2)

An cut sketch is a function such that (2) holds for all .

A sketching scheme is a deterministic map from graphs on vertices to (-spectral or -cut) sketches , along with specified procedures for storing the functions as bit strings and for evaluating any given on a query 666

We will not be concerned with the details of these procedures since we are only interested in the space used by the sketches and not computational parameters such as query time / success probability of randomized schemes.

. The number of bits required to store is called its size.

In [ACK16] it was shown that any -cut sketch must use bits in the worst case, leaving a logarithmic gap between the best known upper and lower bounds. In this paper, we close this gap by showing that any -cut sketching scheme must in fact use bits whenever (Theorem 3.2), which means that it is not in general possible to obtain any asymptotic savings by considering sketches which are not graphs. We also give a lowerbound for -spectral sketching with a simpler proof and slightly better constants (Theorem 2.1).

1.1 Related Work

The paper [ACK16] also studied the problem of producing a “for each” sketching algorithm, which has the property that for any particular query , approximates with constant probability. They showed that in this weaker model it is possible to obtain cut sketches of size and -spectral sketches of size . The latter bound was recently improved to and generalized to include queries to the pseudoinverse in [JS17].

In contrast, in this paper we study only the “for all” model, in which the sketch is required to work for all queries simultaneously.

1.2 Techniques

Our proof is based on the following rigidity phenomenon: there is a constant such that if two regular graphs -cut approximate each other then they must overlap in a constant fraction of edges (Lemmas 2.2 and 3.1). This allows us to argue that below this approximation threshold any sketch encodes a constant fraction of the information in the graph, and so any sketch must use bits.

Interestingly, this phenomenon is very sensitive to the value of the constant — in particular, a well-known theorem of Friedman [Fri03]

says that the eigenvalues of the Laplacian of a random

regular graph are contained in the range with high probability, which implies that almost every regular graph has a -spectral sketch of constant size, namely the complete graph.

Note that according to the Alon-Boppana bound [Nil91] it is not possible to approximate by a regular graph with error less than . Thus, our result can be interpreted as saying that a regular graph can only be approximated by nearby graphs when the error is substantially below the Alon-Boppana bound.

We remark that the proof of [ACK16] was based on showing a bit lower bound for a constant sized graph by reducing to the Gap-Hamming problem in communication complexity, and then concatenating instances of this problem to obtain an instance of size . This method is thereby inherently incapable of recovering the logarithmic factor that we obtain (in contrast, we work with random regular graphs).

1.3 Notation and Organization

We will denote -spectral approximation as

 (1−ϵ)LG⪯LH⪯(1+ϵ)LG

and cut approximation as

 (1−ϵ)LG⪯□LH⪯□(1+ϵ)LG.

We will use and to denote binary and natural logarithms.

We prove our lower bound for spectral sketching in Section 2, and the lower bound for cut sketching in Section 3. Although the latter is logically stronger than the former, we have chosen to present the spectral case first because it is extremely simple and introduces the conceptual ideas of the proof.

Acknowledgments

We would like to thank MSRI and the Simons Institute for the Theory of Computing, where this work was carried out during the “Bridging Discrete and Continuous Optimization” program.

2 Lower Bound for Spectral Sketches

In this section, we prove the following theorem.

Theorem 2.1.

For any , any -spectral sketching scheme for graphs with vertices must use at least bits in the worst case.

The main ingredient is the following lemma.

Lemma 2.2 (Rigidity of Spectral Approximation).

Suppose and are simple regular graphs such that

 (1−ϵ)LG⪯LH⪯(1+ϵ)LG.

Then and must have at least edges in common.

Proof.

Since and are regular, we have and . Thus the hypothesis implies that:

 −2dϵI⪯−ϵ⋅LG⪯LH−LG=AG−AH⪯ϵ⋅LG⪯2dϵI,

which means . Passing to the Frobenius norm, we find that

 ∑ij(AG(i,j)−AH(i,j))2=∥AG−AH∥2F≤n∥AG−AH∥2≤4ϵ2d2n.

The matrix has entries in , with exactly two nonzero entries for every edge in , so the left hand side is equal to

 2|E(G)ΔE(H)|=2(|E(G)|+|E(H)|−2|E(G)∩E(H)|)=2(dn−2|E(G)∩E(H)|),

since . Rearranging yields the desired claim. ∎

Note that the above lemma is vacuous for but indicates that and must share a fraction of their edges for below this threshold.

Lemma 2.3.

For any function and :

 lg|{G∈Gn,d:G is ϵ−spectrally approximated by f}|≤dn/2+5ϵ2d2nlgn,

where denotes the set of all regular graphs on vertices.

Proof.

If does not approximate any regular graph then we are done. Otherwise, let be the lexicographically (in some pre-determined ordering on regular graphs) first graph which -approximates. Suppose is another graph that approximates. Notice that by applying (2) twice, we have that for every :

 1−ϵ1+ϵ≤xTLHxxTLGx=f(x)xTLGx⋅xTLHxf(x)≤1+ϵ1−ϵ,

so -spectrally approximates . By Lemma 2.2, and must share

 k:=dn2(1−9ϵ2d/2)

edges. Thus, can be encoded by specifying:

1. Which edges of occur in . This is a subset of the edges of , which requires at most bits.

2. The remaining edges of . Each edge requires at most bits to specify, so the number of bits needed is at most .

Thus, the total number of bits required is at most

 dn/2+18ϵ2d2nlgn/4,

as desired. ∎

Theorem 2.4.

([Wor99]Cor. 2.4) For , the number of regular graphs on vertices is:

 (dn)!(dn/2)!2dn/2(d!)nexp(1−d24−d312n+O(d2n))≥exp(dnln(n/d)/2⋅(1−on(1))).
Proof of Theorem 2.1.

Let be the number of distinct sketches produced by and let . By Lemma 2.3, the binary logarithm of the number of regular graphs -spectrally approximated by any single sketch is at most

 dn/2+dnlgn/5≤dnlgn/5⋅(1+o(1)).

On the other hand by Theorem 2.4, since we have

 lg|Gn,d|≥dnlg(n/d)/2⋅(1−o(1))≥dnlgn/4⋅(1−o(1)). (3)

Since every regular graph receives a sketch, we must have

 (1+o(1))lgN≥dnlgn/4−dnlgn/5=dnlgn/20=nlgn/500ϵ2,

as desired.

Remark 1.

The proof above actually shows that any -spectral sketching scheme for regular graphs on vertices must use at least bits on average, since the same proof goes through if we only insist that the sketches work for most graphs.

Remark 2.

The result of [BSS12] produces -spectral sparsifiers with edges, which yield -spectral sketches with bits by discretizing the edge weights up to error, so the bound above is tight up to a factor of . We have not made any attempt to optimize the constants.

3 Lower Bound for Cut Sketches

In this section we prove Theorem 3.2. The new ingredient is a rigidity lemma for cuts, which may be seen as a discrete analogue of Lemma 2.2. The lemma holds for bipartite graphs and is proven using a Goemans-Williamson [GW95] style rounding argument.

Lemma 3.1 (Rigidity of Cut Approximation).

Suppose and are simple regular bipartite graphs with the same bipartition , such that

 (1−ϵ)LG⪯□LH⪯□(1+ϵ)LG. (4)

Then and must have at least edges in common.

Proof.

We will show the contrapositive. Assume

 |E(G)∖E(H)|=|E(H)∖E(G)|=δdn/2

for some . To show that (4

) does not hold, it is sufficient to exhibit a vector

such that

 xTMx:=xT(LG−LH)x>2ϵdn≥ϵ⋅xTLGx,

where , since both graphs are regular and the latter inequality follows from . To find such an we will first construct vectors such that

 n∑i,j=1Mij⟨yi,yj⟩>πϵdn (5)

and then use hyperplane rounding to find scalars

which satisfy:

 n∑i,j=1Mijxixj>2ϵdn. (6)

Let be the columns of

 I+M√d,

and note that since and every vertex is incident with at most edges in each of and . The relevant inner products of the are easy to understand:

 Mij⟨zi,zj⟩=⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩0if Mij=02√d=1⋅(1⋅1√d+1⋅1√d)if ij∈E(H)∖E(G)2√d=(−1)⋅(1⋅−1√d+1⋅−1√d)if ij∈E(G)∖E(H),

where in the latter two cases we have used the fact that and cannot have any common neighbors because they lie on different sides of the bipartition. Letting , we therefore have

 n∑i,j=1Mij⟨yi,yj⟩≥2⋅δdn⋅23√d>πϵdn (7)

by our choice of . Let be a random unit vector and let

 xi={+1if ⟨yi,w⟩≥0−1if ⟨yi,w⟩<0.

We denote the angle between vectors as . We recall that since and have unit length. It follows that the probability that is equal to and the probability of is equal to . Thus,

 E[n∑i,j=1Mijxixj] =n∑i,j=1MijE[xixj] =n∑i,j=1Mij(θijπ(−1)+(1−θijπ)(1)) =n∑i,j=1Mij(1−2θijπ) =2πn∑i,j=1Mij(π2−θij) =2πn∑i,j=1Mij(π2−cos−1⟨yi,yj⟩) =2πn∑i,j=1Mijsin−1⟨yi,yj⟩ ≥2πn∑i,j=1Mij⟨yi,yj⟩since sin−1(x)≥x >2ϵdn,

by (7), as desired.

Remark 3.

The analysis of the rounding scheme above can be improved from to the Goemans-Williamson constant , but we have chosen not to do so for simplicity.

Theorem 3.2.

For any , any -cut sketching scheme for graphs with vertices must use at least bits in the worst case.

Proof.

Assume is divisible by (add a constant if this is not the case). Let and let be the set of bipartite graphs on vertices with respect to a fixed bipartition. We proceed as in the proof of Theorem 2.1. Let be any function which is an cut sketch for some graph . Arguing as in Lemma 2.3, any other graph which has the same sketch must -cut approximate , so by Lemma 3.1, any such must have at most edges which are not present in . Thus, the encoding length of such is at most

 ℓ:=dn/2+9d3/2nϵlgn=dnlgn/16⋅(1+o(1))

bits, by our choice of , so any particular can only be an -cut sketch for graphs in

On the other hand, is quite large. Recall that the bipartite double cover of a graph on vertices is the graph on

vertices obtained by taking its tensor product with

, and two distinct graphs must have distinct double covers. Thus, by (3) we have

 lg|Bn,d|≥lg|Gn/2,d|≥dnlgn/8⋅(1−o(1)).

Thus, if is the number of distinct sketches produced by , we must have

 (1+o(1))lgN≥dnlgn/8−dnlgn/16=dnlgn/16=nlgn/(16)2⋅9ϵ2,

as desired. ∎

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