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Publication# Graph-Constrained Group Testing

Abstract

Non-adaptive group testing involves grouping arbitrary subsets of $n$ items into different pools. Each pool is then tested and defective items are identified. A fundamental question involves minimizing the number of pools required to identify at most $d$ defective items. Motivated by applications in network tomography, sensor networks and infection propagation, a variation of group testing problems on graphs is formulated. Unlike conventional group testing problems, each group here must conform to the constraints imposed by a graph. For instance, items can be associated with vertices and each pool is any set of nodes that must be path connected. In this paper, a test is associated with a random walk. In this context, conventional group testing corresponds to the special case of a complete graph on $n$ vertices. For interesting classes of graphs a rather surprising result is obtained, namely, that the number of tests required to identify $d$ defective items is substantially similar to what required in conventional group testing problems, where no such constraints on pooling is imposed. Specifically, if $T(n)$ corresponds to the mixing time of the graph $G$, it is shown that with $m=O(d^2T^2(n)\log(n/d))$ non-adaptive tests, one can identify the defective items. Consequently, for the Erd\H{o}s-R'enyi random graph $G(n,p)$, as well as expander graphs with constant spectral gap, it follows that $m=O(d^2\log^3n)$ non-adaptive tests are sufficient to identify $d$ defective items. Next, a specific scenario is considered that arises in network tomography, for which it is shown that $m=O(d^3\log^3n)$ non-adaptive tests are sufficient to identify $d$ defective items. Noisy counterparts of the graph constrained group testing problem are considered, for which parallel results are developed.

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Random walk

In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space. An elementary example of a random walk is the random walk on the integer number line which starts at 0, and at each step moves +1 or −1 with equal probability. Other examples include the path traced by a molecule as it travels in a liquid or a gas (see Brownian motion), the search path of a foraging animal, or the price of a fluctuating stock and the financial status of a gambler.

Special case

In logic, especially as applied in mathematics, concept A is a special case or specialization of concept B precisely if every instance of A is also an instance of B but not vice versa, or equivalently, if B is a generalization of A. A limiting case is a type of special case which is arrived at by taking some aspect of the concept to the extreme of what is permitted in the general case. A degenerate case is a special case which is in some way qualitatively different from almost all of the cases allowed.

Graph isomorphism problem

The graph isomorphism problem is the computational problem of determining whether two finite graphs are isomorphic. The problem is not known to be solvable in polynomial time nor to be NP-complete, and therefore may be in the computational complexity class NP-intermediate. It is known that the graph isomorphism problem is in the low hierarchy of class NP, which implies that it is not NP-complete unless the polynomial time hierarchy collapses to its second level.

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