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Bonne and Censor-Hillel (ICALP 2019) initiated the study of distributed subgraph finding in dynamic networks of limited bandwidth. For the case where the target subgraph is a clique, they determined the tight bandwidth complexity bounds in…
Valiant's famous VP vs. VNP conjecture states that the symbolic permanent polynomial does not have polynomial-size algebraic circuits. However, the best upper bound on the size of the circuits computing the permanent is exponential.…
We prove the existence of a function f: N^2 -> N such that for all p,k in N every (k(p-3) + 14p+14) - connected graph either has k disjoint K_p minors or contains a set of at most f(p,k) vertices whose deletion kills all its K_p minors. For…
In machine learning and big data, the optimization objectives based on set-cover, entropy, diversity, influence, feature selection, etc. are commonly modeled as submodular functions. Submodular (function) maximization is generally NP-hard,…
We prove that a random Cayley graph on a group of order $N$ has clique number $O(\log N \log \log N)$ with high probability. This bound is best possible up to the constant factor for certain groups, including~$\mathbb{F}_2^n$, and improves…
A polynomial identity testing algorithm must determine whether an input polynomial (given for instance by an arithmetic circuit) is identically equal to 0. In this paper, we show that a deterministic black-box identity testing algorithm for…
We introduce the following submodular generalization of the Shortest Cycle problem. For a nonnegative monotone submodular cost function $f$ defined on the edges (or the vertices) of an undirected graph $G$, we seek for a cycle $C$ in $G$ of…
Clique-width is a well-known graph parameter. Many NP-hard graph problems admit polynomial-time solutions when restricted to graphs of bounded clique-width. The same holds for NLC-width. In this paper we study the behavior of clique-width…
The paper discusses the gate complexity of reversible circuits consisting of NOT, CNOT and 2-CNOT gates. The Shannon gate complexity function $L(n, q)$ for a reversible circuit, implementing a Boolean transformation $f\colon \mathbb Z_2^n…
We prove that if conditions I-II (below) hold and there is a sequence of Boolean functions $f_n$ hard to approximate by p-size circuits such that p-size circuit lower bounds for $f_n$ do not have p-size proofs in Extended Frege system EF,…
We study the minimum number of constraints needed to formulate random instances of the maximum stable set problem via linear programs (LPs), in two distinct models. In the uniform model, the constraints of the LP are not allowed to depend…
We introduce techniques for proving superlinear conditional lower bounds for polynomial time problems. In particular, we show that CircuitSAT for circuits with m gates and log(m) inputs (denoted by log-CircuitSAT) is not decidable in…
A {+,x}-circuit counts a given multivariate polynomial f, if its values on 0-1 inputs are the same as those of f; on other inputs the circuit may output arbitrary values. Such a circuit counts the number of monomials of f evaluated to 1 by…
Finding the clique of maximum cardinality in an arbitrary graph is an NP-Hard problem that has many applications, which has motivated studies to solve it exactly despite its difficulty. The great majority of algorithms proposed in the…
The problem of minimizing a polynomial over the standard simplex is one of the basic NP-hard nonlinear optimization problems --- it contains the maximum clique problem in graphs as a special case. It is known that the problem allows a…
We develop techniques to prove lower bounds for the BCAST(log n) Broadcast Congested Clique model (a distributed message passing model where in each round, each processor can broadcast an O(log n)-sized message to all other processors). Our…
We extend the work of Narasimhan and Bilmes [30] for minimizing set functions representable as a difference between submodular functions. Similar to [30], our new algorithms are guaranteed to monotonically reduce the objective function at…
We show that every algorithm for testing $n$-variate Boolean functions for monotonicity must have query complexity $\tilde{\Omega}(n^{1/4})$. All previous lower bounds for this problem were designed for non-adaptive algorithms and, as a…
In this note, we consider the minimum number of NOT operators in a Boolean formula representing a Boolean function. In circuit complexity theory, the minimum number of NOT gates in a Boolean circuit computing a Boolean function $f$ is…
One of the major outstanding foundational problems about boolean functions is the sensitivity conjecture, which (in one of its many forms) asserts that the degree of a boolean function (i.e. the minimum degree of a real polynomial that…