We consider strings which are succinctly described. The description
is in terms of straight-line programs in which the constants are
symbols and the only operation is the concatenation. Such
descriptions correspond to the systems of recurrences or to
context-free grammars generating single words. The descriptive ... more >>>

In this paper we approach the problem of computing the characteristic
polynomial of a matrix from the combinatorial viewpoint. We present
several combinatorial characterizations of the coefficients of the
characteristic polynomial, in terms of walks and closed walks of
different kinds in the underlying graph. We develop algorithms based
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We present an improved list decoding algorithm for decoding
Reed-Solomon codes. Given an arbitrary string of length n, the
list decoding problem is that of finding all codewords within a
specified Hamming distance from the input string.

It is well-known that this decoding problem for Reed-Solomon
codes reduces to the ... more >>>

The Chinese Remainder Theorem states that a positive
integer m is uniquely specified by its remainder modulo
k relatively prime integers p_1,...,p_k, provided
m < \prod_{i=1}^k p_i. Thus the residues of m modulo
relatively prime integers p_1 < p_2 < ... < p_n
form a redundant representation of m if ... more >>>

Deciding whether a vertex in a graph is reachable from another
vertex has been studied intensively in complexity theory and is
well understood. For common types of graphs like directed graphs,
undirected graphs, dags or trees it takes a (possibly
nondeterministic) logspace machine to decide the reachability
problem, and ... more >>>

We consider the ``minor'' and ``homeomorphic'' analogues of the maximum clique problem, i.e., the problems of determining the largest $h$ such that the input graph has a minor isomorphic to $K_h$ or a subgraph homeomorphic to $K_h,$ respectively.We show the former to be approximable within $O(\sqrt {n} \log^{1.5} n)$ by ... more >>>

Consider a homogeneous polynomial $p(z_1,...,z_n)$ of degree $n$ in $n$ complex variables .
Assume that this polynomial satisfies the property : \\

$|p(z_1,...,z_n)| \geq \prod_{1 \leq i \leq n} Re(z_i)$ on the domain $\{(z_1,...,z_n) : Re(z_i) \geq 0 , 1 \leq i \leq n \}$ . \\

We prove that ... more >>>

We study the structure of EG[2], the class of Eisenberg-Gale markets
with two agents. We prove that all markets in this class are rational and they
admit strongly polynomial algorithms whenever
the polytope containing the set of feasible utilities of the two agents can be described
via a combinatorial LP. ... more >>>

This paper considers the tradeoff between divisibility and the hardness of approximating
equilibrium prices. Tight bounds are obtained for smooth Fisher markets that obey a relaxed
weak gross substitutes property (WGS). A smooth market is one in which small changes in
prices cause only proportionately small changes ... more >>>

Different techniques have been used to prove several transference theorems of the form "nontrivial algorithms for a circuit class C yield circuit lower bounds against C". In this survey we revisit many of these results. We discuss how circuit lower bounds can be obtained from derandomization, compression, learning, and satisfiability ... more >>>

Leveraging tools of De, Mossel, and Neeman [FOCS, 2019], we show two different results pertaining to the tolerant testing of juntas. Given black-box access to a Boolean function $f:\{\pm1\}^{n} \to \{\pm1\}$ we give a poly$(k, \frac{1}{\varepsilon})$ query algorithm that distinguishes between functions that are $\gamma$-close to $k$-juntas and $(\gamma+\varepsilon)$-far from ... more >>>