Derandomization and Group Testing

The rapid development of derandomization theory, which is a fundamental area in theoretical computer science, has recently led to many surprising applications outside its initial intention. We will review some recent such developments related to combinatorial group testing. In its most basic setting, the aim of group testing is to identify a set of "positive" individuals in a population of items by taking groups of items and asking whether there is a positive in each group. In particular, we will discuss explicit constructions of optimal or nearly-optimal group testing schemes using "randomness-conducting" functions. Among such developments are constructions of error-correcting group testing schemes using randomness extractors and condensers, as well as threshold group testing schemes from lossless condensers.Comment: Invited Paper in Proceedings of 48th Annual Allerton Conference on Communication, Control, and Computing, 201

A deterministic version of Pollard's p-1 algorithm

In this article we present applications of smooth numbers to the unconditional derandomization of some well-known integer factoring algorithms. We begin with Pollard's $p-1$ algorithm, which finds in random polynomial time the prime divisors $p$ of an integer $n$ such that $p-1$ is smooth. We show that these prime factors can be recovered in deterministic polynomial time. We further generalize this result to give a partial derandomization of the $k$-th cyclotomic method of factoring ($k\ge 2$) devised by Bach and Shallit. We also investigate reductions of factoring to computing Euler's totient function $\phi$. We point out some explicit sets of integers $n$ that are completely factorable in deterministic polynomial time given $\phi(n)$. These sets consist, roughly speaking, of products of primes $p$ satisfying, with the exception of at most two, certain conditions somewhat weaker than the smoothness of $p-1$. Finally, we prove that $O(\ln n)$ oracle queries for values of $\phi$ are sufficient to completely factor any integer $n$ in less than $\exp\Bigl((1+o(1))(\ln n)^{{1/3}} (\ln\ln n)^{{2/3}}\Bigr)$ deterministic time.Comment: Expanded and heavily revised version, to appear in Mathematics of Computation, 21 page

Monomial Testing and Applications

In this paper, we devise two algorithms for the problem of testing $q$-monomials of degree $k$ in any multivariate polynomial represented by a circuit, regardless of the primality of $q$. One is an $O^*(2^k)$ time randomized algorithm. The other is an $O^*(12.8^k)$ time deterministic algorithm for the same $q$-monomial testing problem but requiring the polynomials to be represented by tree-like circuits. Several applications of $q$-monomial testing are also given, including a deterministic $O^*(12.8^{mk})$ upper bound for the $m$-set $k$-packing problem.Comment: 17 pages, 4 figures, submitted FAW-AAIM 2013. arXiv admin note: substantial text overlap with arXiv:1302.5898; and text overlap with arXiv:1007.2675, arXiv:1007.2678, arXiv:1007.2673 by other author

Derandomized Graph Product Results using the Low Degree Long Code

In this paper, we address the question of whether the recent derandomization results obtained by the use of the low-degree long code can be extended to other product settings. We consider two settings: (1) the graph product results of Alon, Dinur, Friedgut and Sudakov [GAFA, 2004] and (2) the "majority is stablest" type of result obtained by Dinur, Mossel and Regev [SICOMP, 2009] and Dinur and Shinkar [In Proc. APPROX, 2010] while studying the hardness of approximate graph coloring. In our first result, we show that there exists a considerably smaller subgraph of $K_3^{\otimes R}$ which exhibits the following property (shown for $K_3^{\otimes R}$ by Alon et al.): independent sets close in size to the maximum independent set are well approximated by dictators. The "majority is stablest" type of result of Dinur et al. and Dinur and Shinkar shows that if there exist two sets of vertices $A$ and $B$ in $K_3^{\otimes R}$ with very few edges with one endpoint in $A$ and another in $B$, then it must be the case that the two sets $A$ and $B$ share a single influential coordinate. In our second result, we show that a similar "majority is stablest" statement holds good for a considerably smaller subgraph of $K_3^{\otimes R}$. Furthermore using this result, we give a more efficient reduction from Unique Games to the graph coloring problem, leading to improved hardness of approximation results for coloring

Faster Deterministic Algorithms for Packing, Matching and tt-Dominating Set Problems

In this paper, we devise three deterministic algorithms for solving the $m$-set $k$-packing, $m$-dimensional $k$-matching, and $t$-dominating set problems in time $O^*(5.44^{mk})$, $O^*(5.44^{(m-1)k})$ and $O^*(5.44^{t})$, respectively. Although recently there has been remarkable progress on randomized solutions to those problems, our bounds make good improvements on the best known bounds for deterministic solutions to those problems.Comment: ISAAC13 Submission. arXiv admin note: substantial text overlap with arXiv:1303.047