Evaluating Matrix Circuits

The circuit evaluation problem (also known as the compressed word problem) for finitely generated linear groups is studied. The best upper bound for this problem is $\mathsf{coRP}$, which is shown by a reduction to polynomial identity testing. Conversely, the compressed word problem for the linear group $\mathsf{SL}_3(\mathbb{Z})$ is equivalent to polynomial identity testing. In the paper, it is shown that the compressed word problem for every finitely generated nilpotent group is in $\mathsf{DET} \subseteq \mathsf{NC}^2$. Within the larger class of polycyclic groups we find examples where the compressed word problem is at least as hard as polynomial identity testing for skew arithmetic circuits

Unconstraining Graph-Constrained Group Testing

In network tomography, one goal is to identify a small set of failed links in a network using as little information as possible. One way of setting up this problem is called graph-constrained group testing. Graph-constrained group testing is a variant of the classical combinatorial group testing problem, where the tests that one is allowed are additionally constrained by a graph. In this case, the graph is given by the underlying network topology. The main contribution of this work is to show that for most graphs, the constraints imposed by the graph are no constraint at all. That is, the number of tests required to identify the failed links in graph-constrained group testing is near-optimal even for the corresponding group testing problem with no graph constraints. Our approach is based on a simple randomized construction of tests. To analyze our construction, we prove new results about the size of giant components in randomly sparsified graphs. Finally, we provide empirical results which suggest that our connected-subgraph tests perform better not just in theory but also in practice, and in particular perform better on a real-world network topology

Lower bounds for identifying subset members with subset queries

An instance of a group testing problem is a set of objects \cO and an unknown subset $P$ of \cO. The task is to determine $P$ by using queries of the type ``does $P$ intersect $Q$'', where $Q$ is a subset of \cO. This problem occurs in areas such as fault detection, multiaccess communications, optimal search, blood testing and chromosome mapping. Consider the two stage algorithm for solving a group testing problem. In the first stage a predetermined set of queries are asked in parallel and in the second stage, $P$ is determined by testing individual objects. Let n=\cardof{\cO}. Suppose that $P$ is generated by independently adding each x\in \cO to $P$ with probability $p/n$. Let $q_1$ ($q_2$) be the number of queries asked in the first (second) stage of this algorithm. We show that if $q_1=o(\log(n)\log(n)/\log\log(n))$, then \Exp(q_2) = n^{1-o(1)}, while there exist algorithms with $q_1 = O(\log(n)\log(n)/\log\log(n))$ and \Exp(q_2) = o(1). The proof involves a relaxation technique which can be used with arbitrary distributions. The best previously known bound is q_1+\Exp(q_2) = \Omega(p\log(n)). For general group testing algorithms, our results imply that if the average number of queries over the course of $n^\gamma$ ($\gamma>0$) independent experiments is $O(n^{1-\epsilon})$, then with high probability $\Omega(\log(n)\log(n)/\log\log(n))$ non-singleton subsets are queried. This settles a conjecture of Bill Bruno and David Torney and has important consequences for the use of group testing in screening DNA libraries and other applications where it is more cost effective to use non-adaptive algorithms and/or too expensive to prepare a subset $Q$ for its first test.Comment: 9 page

Boolean Compressed Sensing and Noisy Group Testing

The fundamental task of group testing is to recover a small distinguished subset of items from a large population while efficiently reducing the total number of tests (measurements). The key contribution of this paper is in adopting a new information-theoretic perspective on group testing problems. We formulate the group testing problem as a channel coding/decoding problem and derive a single-letter characterization for the total number of tests used to identify the defective set. Although the focus of this paper is primarily on group testing, our main result is generally applicable to other compressive sensing models. The single letter characterization is shown to be order-wise tight for many interesting noisy group testing scenarios. Specifically, we consider an additive Bernoulli($q$) noise model where we show that, for $N$ items and $K$ defectives, the number of tests $T$ is $O(\frac{K\log N}{1-q})$ for arbitrarily small average error probability and $O(\frac{K^2\log N}{1-q})$ for a worst case error criterion. We also consider dilution effects whereby a defective item in a positive pool might get diluted with probability $u$ and potentially missed. In this case, it is shown that $T$ is $O(\frac{K\log N}{(1-u)^2})$ and $O(\frac{K^2\log N}{(1-u)^2})$ for the average and the worst case error criteria, respectively. Furthermore, our bounds allow us to verify existing known bounds for noiseless group testing including the deterministic noise-free case and approximate reconstruction with bounded distortion. Our proof of achievability is based on random coding and the analysis of a Maximum Likelihood Detector, and our information theoretic lower bound is based on Fano's inequality.Comment: In this revision: reorganized the paper, added citations to related work, and fixed some bug

Adaptive group testing as channel coding with feedback

Group testing is the combinatorial problem of identifying the defective items in a population by grouping items into test pools. Recently, nonadaptive group testing - where all the test pools must be decided on at the start - has been studied from an information theory point of view. Using techniques from channel coding, upper and lower bounds have been given on the number of tests required to accurately recover the defective set, even when the test outcomes can be noisy. In this paper, we give the first information theoretic result on adaptive group testing - where the outcome of previous tests can influence the makeup of future tests. We show that adaptive testing does not help much, as the number of tests required obeys the same lower bound as nonadaptive testing. Our proof uses similar techniques to the proof that feedback does not improve channel capacity.Comment: 4 pages, 1 figur

An Efficient Quantum Algorithm for some Instances of the Group Isomorphism Problem

In this paper we consider the problem of testing whether two finite groups are isomorphic. Whereas the case where both groups are abelian is well understood and can be solved efficiently, very little is known about the complexity of isomorphism testing for nonabelian groups. Le Gall has constructed an efficient classical algorithm for a class of groups corresponding to one of the most natural ways of constructing nonabelian groups from abelian groups: the groups that are extensions of an abelian group $A$ by a cyclic group $Z_m$ with the order of $A$ coprime with $m$. More precisely, the running time of that algorithm is almost linear in the order of the input groups. In this paper we present a quantum algorithm solving the same problem in time polynomial in the logarithm of the order of the input groups. This algorithm works in the black-box setting and is the first quantum algorithm solving instances of the nonabelian group isomorphism problem exponentially faster than the best known classical algorithms.Comment: 20 pages; this is the full version of a paper that will appear in the Proceedings of the 27th International Symposium on Theoretical Aspects of Computer Science (STACS 2010