On Approximating the Number of kk-cliques in Sublinear Time

We study the problem of approximating the number of $k$-cliques in a graph when given query access to the graph. We consider the standard query model for general graphs via (1) degree queries, (2) neighbor queries and (3) pair queries. Let $n$ denote the number of vertices in the graph, $m$ the number of edges, and $C_k$ the number of $k$-cliques. We design an algorithm that outputs a $(1+\varepsilon)$-approximation (with high probability) for $C_k$, whose expected query complexity and running time are O\left(\frac{n}{C_k^{1/k}}+\frac{m^{k/2}}{C_k}\right)\poly(\log n,1/\varepsilon,k). Hence, the complexity of the algorithm is sublinear in the size of the graph for $C_k = \omega(m^{k/2-1})$. Furthermore, we prove a lower bound showing that the query complexity of our algorithm is essentially optimal (up to the dependence on $\log n$, $1/\varepsilon$ and $k$). The previous results in this vein are by Feige (SICOMP 06) and by Goldreich and Ron (RSA 08) for edge counting ($k=2$) and by Eden et al. (FOCS 2015) for triangle counting ($k=3$). Our result matches the complexities of these results. The previous result by Eden et al. hinges on a certain amortization technique that works only for triangle counting, and does not generalize for larger cliques. We obtain a general algorithm that works for any $k\geq 3$ by designing a procedure that samples each $k$-clique incident to a given set $S$ of vertices with approximately equal probability. The primary difficulty is in finding cliques incident to purely high-degree vertices, since random sampling within neighbors has a low success probability. This is achieved by an algorithm that samples uniform random high degree vertices and a careful tradeoff between estimating cliques incident purely to high-degree vertices and those that include a low-degree vertex

On the grade of modules over Noetherian rings

Let $\Lambda$ be a left and right noetherian ring and $\mod \Lambda$ the category of finitely generated left $\Lambda$-modules. In this paper we show the following results: (1) For a positive integer $k$, the condition that the subcategory of $\mod \Lambda$ consisting of $i$-torsionfree modules coincides with the subcategory of $\mod \Lambda$ consisting of $i$-syzygy modules for any $1\leq i \leq k$ is left-right symmetric. (2) If $\Lambda$ is an Auslander ring and $N$ is in $\mod \Lambda ^{op}$ with \grade N=k<\infty, then $N$ is pure of grade $k$ if and only if $N$ can be embedded into a finite direct sum of copies of the $(k+1)$st term in a minimal injective resolution of $\Lambda$ as a right $\Lambda$-module. (3) Assume that both the left and right self-injective dimensions of $\Lambda$ are $k$. If \grade {\rm Ext}_{\Lambda}^k(M, \Lambda)\geq k for any $M\in\mod \Lambda$ and \grade {\rm Ext}_{\Lambda}^i(N, \Lambda)\geq i for any $N\in\mod \Lambda ^{op}$ and $1\leq i \leq k-1$, then the socle of the last term in a minimal injective resolution of $\Lambda$ as a right $\Lambda$-module is non-zero.Comment: 17 pages. To appear in Communications in Algebr

On monotone circuits with local oracles and clique lower bounds

We investigate monotone circuits with local oracles [K., 2016], i.e., circuits containing additional inputs $y_i = y_i(\vec{x})$ that can perform unstructured computations on the input string $\vec{x}$. Let $\mu \in [0,1]$ be the locality of the circuit, a parameter that bounds the combined strength of the oracle functions $y_i(\vec{x})$, and $U_{n,k}, V_{n,k} \subseteq \{0,1\}^m$ be the set of $k$-cliques and the set of complete $(k-1)$-partite graphs, respectively (similarly to [Razborov, 1985]). Our results can be informally stated as follows. 1. For an appropriate extension of depth-$2$ monotone circuits with local oracles, we show that the size of the smallest circuits separating $U_{n,3}$ (triangles) and $V_{n,3}$ (complete bipartite graphs) undergoes two phase transitions according to $\mu$. 2. For $5 \leq k(n) \leq n^{1/4}$, arbitrary depth, and $\mu \leq 1/50$, we prove that the monotone circuit size complexity of separating the sets $U_{n,k}$ and $V_{n,k}$ is $n^{\Theta(\sqrt{k})}$, under a certain restrictive assumption on the local oracle gates. The second result, which concerns monotone circuits with restricted oracles, extends and provides a matching upper bound for the exponential lower bounds on the monotone circuit size complexity of $k$-clique obtained by Alon and Boppana (1987).Comment: Updated acknowledgements and funding informatio

On the Ces\`aro average of the "Linnik numbers"

Let $\Lambda$ be the von Mangoldt function and $r_{Q}\left(n\right)=\sum_{m_{1}+m_{2}^{2}+m_{3}^{2}=n}\Lambda\left(m_{1}\right)$ be the counting function for the numbers that can be written as sum of a prime and two squares (that we will call Linnik numbers, for brevity). Let $N$ a sufficiently large integer, let $k>3/2$ and let $M_{i}\left(N,k\right),\, i=1,\dots,k$ suitable parameters depending on $J_{v}\left(u\right)$, where $J_{v}\left(u\right)$ denotes the Bessel function of complex order $v$ and real argument $u$. We prove that $$\sum_{n\leq N}r_{Q}\left(n\right)\frac{\left(N-n\right)^{k}}{\Gamma\left(k+1\right)}=M_{1}\left(N,k\right)+M_{2}\left(N,k\right)+M_{3}\left(N,k\right)+M_{4}\left(N,k\right)+O\left(N^{k+1}\right).$$ We also prove that with this technique the bound $k>3/2$ is optimal.Comment: Accepted on Acta Arithmetic

On higher-order discriminants

For the family of polynomials in one variable $P:=x^n+a_1x^{n-1}+\cdots +a_n$, $n\geq 4$, we consider its higher-order discriminant sets $\{ \tilde{D}_m=0\}$, where $\tilde{D}_m:=$Res$(P,P^{(m)})$, $m=2$, $\ldots$, $n-2$, and their projections in the spaces of the variables $a^k:=(a_1,\ldots ,a_{k-1},a_{k+1},\ldots ,a_n)$. Set $P^{(m)}:=\sum _{j=0}^{n-m}c_ja_jx^{n-m-j}$, $P_{m,k}:=c_kP-x^mP^{(m)}$. We show that Res$(\tilde{D}_m,\partial \tilde{D}_m/\partial a_k,a_k)= A_{m,k}B_{m,k}C_{m,k}^2$, where $A_{m,k}=a_n^{n-m-k}$, $B_{m,k}=$Res$(P_{m,k},P_{m,k}')$ if $1\leq k\leq n-m$ and $A_{m,k}=a_{n-m}^{n-k}$, $B_{m,k}=$Res$(P^{(m)},P^{(m+1)})$ if $n-m+1\leq k\leq n$. The equation $C_{m,k}=0$ defines the projection in the space of the variables $a^k$ of the closure of the set of values of $(a_1,\ldots ,a_n)$ for which $P$ and $P^{(m)}$ have two distinct roots in common. The polynomials $B_{m,k},C_{m,k}\in \mathbb{C}[a^k]$ are irreducible. The result is generalized to the case when $P^{(m)}$ is replaced by a polynomial $P_*:=\sum _{j=0}^{n-m}b_ja_jx^{n-m-j}$, $0\neq b_i\neq b_j\neq 0$ for $i\neq j$