Bounds for graph regularity and removal lemmas

We show, for any positive integer k, that there exists a graph in which any equitable partition of its vertices into k parts has at least ck^2/\log^* k pairs of parts which are not \epsilon-regular, where c,\epsilon>0 are absolute constants. This bound is tight up to the constant c and addresses a question of Gowers on the number of irregular pairs in Szemer\'edi's regularity lemma. In order to gain some control over irregular pairs, another regularity lemma, known as the strong regularity lemma, was developed by Alon, Fischer, Krivelevich, and Szegedy. For this lemma, we prove a lower bound of wowzer-type, which is one level higher in the Ackermann hierarchy than the tower function, on the number of parts in the strong regularity lemma, essentially matching the upper bound. On the other hand, for the induced graph removal lemma, the standard application of the strong regularity lemma, we find a different proof which yields a tower-type bound. We also discuss bounds on several related regularity lemmas, including the weak regularity lemma of Frieze and Kannan and the recently established regular approximation theorem. In particular, we show that a weak partition with approximation parameter \epsilon may require as many as 2^{\Omega(\epsilon^{-2})} parts. This is tight up to the implied constant and solves a problem studied by Lov\'asz and Szegedy.Comment: 62 page

Balanced Allocation on Graphs: A Random Walk Approach

In this paper we propose algorithms for allocating $n$ sequential balls into $n$ bins that are interconnected as a $d$-regular $n$-vertex graph $G$, where $d\ge3$ can be any integer.Let $l$ be a given positive integer. In each round $t$, $1\le t\le n$, ball $t$ picks a node of $G$ uniformly at random and performs a non-backtracking random walk of length $l$ from the chosen node.Then it allocates itself on one of the visited nodes with minimum load (ties are broken uniformly at random). Suppose that $G$ has a sufficiently large girth and $d=\omega(\log n)$. Then we establish an upper bound for the maximum number of balls at any bin after allocating $n$ balls by the algorithm, called {\it maximum load}, in terms of $l$ with high probability. We also show that the upper bound is at most an $O(\log\log n)$ factor above the lower bound that is proved for the algorithm. In particular, we show that if we set $l=\lfloor(\log n)^{\frac{1+\epsilon}{2}}\rfloor$, for every constant $\epsilon\in (0, 1)$, and $G$ has girth at least $\omega(l)$, then the maximum load attained by the algorithm is bounded by $O(1/\epsilon)$ with high probability.Finally, we slightly modify the algorithm to have similar results for balanced allocation on $d$-regular graph with $d\in[3, O(\log n)]$ and sufficiently large girth

Span programs and quantum algorithms for st-connectivity and claw detection

We introduce a span program that decides st-connectivity, and generalize the span program to develop quantum algorithms for several graph problems. First, we give an algorithm for st-connectivity that uses O(n d^{1/2}) quantum queries to the n x n adjacency matrix to decide if vertices s and t are connected, under the promise that they either are connected by a path of length at most d, or are disconnected. We also show that if T is a path, a star with two subdivided legs, or a subdivision of a claw, its presence as a subgraph in the input graph G can be detected with O(n) quantum queries to the adjacency matrix. Under the promise that G either contains T as a subgraph or does not contain T as a minor, we give O(n)-query quantum algorithms for detecting T either a triangle or a subdivision of a star. All these algorithms can be implemented time efficiently and, except for the triangle-detection algorithm, in logarithmic space. One of the main techniques is to modify the st-connectivity span program to drop along the way "breadcrumbs," which must be retrieved before the path from s is allowed to enter t.Comment: 18 pages, 4 figure

Clustering, Hamming Embedding, Generalized LSH and the Max Norm

We study the convex relaxation of clustering and hamming embedding, focusing on the asymmetric case (co-clustering and asymmetric hamming embedding), understanding their relationship to LSH as studied by (Charikar 2002) and to the max-norm ball, and the differences between their symmetric and asymmetric versions.Comment: 17 page

Heavy Hitters and the Structure of Local Privacy

We present a new locally differentially private algorithm for the heavy hitters problem which achieves optimal worst-case error as a function of all standardly considered parameters. Prior work obtained error rates which depend optimally on the number of users, the size of the domain, and the privacy parameter, but depend sub-optimally on the failure probability. We strengthen existing lower bounds on the error to incorporate the failure probability, and show that our new upper bound is tight with respect to this parameter as well. Our lower bound is based on a new understanding of the structure of locally private protocols. We further develop these ideas to obtain the following general results beyond heavy hitters. $\bullet$ Advanced Grouposition: In the local model, group privacy for $k$ users degrades proportionally to $\approx \sqrt{k}$, instead of linearly in $k$ as in the central model. Stronger group privacy yields improved max-information guarantees, as well as stronger lower bounds (via "packing arguments"), over the central model. $\bullet$ Building on a transformation of Bassily and Smith (STOC 2015), we give a generic transformation from any non-interactive approximate-private local protocol into a pure-private local protocol. Again in contrast with the central model, this shows that we cannot obtain more accurate algorithms by moving from pure to approximate local privacy

The number of independent sets in a graph with small maximum degree

Let ${\rm ind}(G)$ be the number of independent sets in a graph $G$. We show that if $G$ has maximum degree at most $5$ then $${\rm ind}(G) \leq 2^{{\rm iso}(G)} \prod_{uv \in E(G)} {\rm ind}(K_{d(u),d(v)})^{\frac{1}{d(u)d(v)}}$$ (where $d(\cdot)$ is vertex degree, ${\rm iso}(G)$ is the number of isolated vertices in $G$ and $K_{a,b}$ is the complete bipartite graph with $a$ vertices in one partition class and $b$ in the other), with equality if and only if each connected component of $G$ is either a complete bipartite graph or a single vertex. This bound (for all $G$) was conjectured by Kahn. A corollary of our result is that if $G$ is $d$-regular with $1 \leq d \leq 5$ then $${\rm ind}(G) \leq \left(2^{d+1}-1\right)^\frac{|V(G)|}{2d},$$ with equality if and only if $G$ is a disjoint union of $V(G)/2d$ copies of $K_{d,d}$. This bound (for all $d$) was conjectured by Alon and Kahn and recently proved for all $d$ by the second author, without the characterization of the extreme cases. Our proof involves a reduction to a finite search. For graphs with maximum degree at most $3$ the search could be done by hand, but for the case of maximum degree $4$ or $5$, a computer is needed.Comment: Article will appear in {\em Graphs and Combinatorics