We consider the problem of dualizing a monotone CNF (equivalently, computing all minimal transversals of a hypergraph), whose associated decision problem is a prominent open problem in NP-completeness. We present a number of new polynomial time resp. output-polynomial time results for significant cases, which largely advance the tractability frontier and improve on previous results. Furthermore, we show that duality of two monotone CNFs can be disproved with limited nondeterminism. More precisely, this is feasible in polynomial time with O(log² n/log log n) suitably guessed bits. This result sheds new light on the complexity of this important problem.

Approaches to DNA-based computing by self-assembly require the use of DNA nanostructures, called tiles, that have efficient chemistries, expressive computational power, and convenient input and output (I/O) mechanisms. We have designed two new classes of DNA tiles, TAO and TAE, both of which contain three double-helices linked by strand exchange. Structural analysis of a TAO molecule has shown that the molecule assembles efficiently from its four component strands. Here we demonstrate a novel method for I/O whereby multiple tiles assemble around a single-stranded (input) scaffold strand. Computation by tiling theoretically results in the formation of structures that contain single-stranded (output) reported strands, which can then be isolated for subsequent steps of computation if necessary. We illustrate the advantages of TAO and TAE designs by detailing two examples of massively parallel arithmetic: construction of complete XOR and addition tables by linear assemblies of DNA t...

The potential of DNA as a truly parallel computing device is enormous. Solution-phase DNA chemistry, though not unlimited, provides the only currently-available experimental system. Its practical feasibility, however, is controversial. We have sought to extend the feasibility and generality of DNA computing by a novel application of the theory of counting . The biochemically equivalent operation for DNA counting is well known. We propose a DNA algorithm that employs this new operation. We also present an implementation of this algorithm by a novel DNA-chemical method. Preliminary computer simulations suggest that the algorithm can significantly reduce the DNA space complexity (i.e., the maximum number of DNA molecules that must be present in the test tube during computation) for solving 3SAT to O(2 0:4n ). If the observation is correct, our algorithm can solve 3SAT instances of size up to or exceeding 120 variables. 1 Introduction 1.1 Two major issues in DNA computing Adleman [Ad...

The number of molecular strands used by a molecular algorithm is an important measure of the algorithm's complexity. This measure is also called the volume used by the algorithm. We prove that three important polynomial-time models of molecular computation with bounded volume are equivalent to models of polynomial-time Turing machine computation with bounded nondeterminism. Without any assumption, we show that the Split operation does not increase the power of polynomial-time molecular computation. Assuming a plausible separation between Turing machine complexity classes, the Amplify operation does increase the power of polynomial-time molecular computation. 1. Introduction Molecular computation was first studied in [1, 15], which identified the number of molecular strands used as an important resource. This measure is also Research performed at Yale University and at the University of Maryland. Supported in part by the National Science Foundation under grant CCR-8958528, CCR-94154...

DNA computation investigates the potential of DNA as a massively parallel computing device. Research is focused on designing parallel computation models executable by DNA-based chemical processes and on developing algorithms in the models. In 1994 Leonard Adleman initiated this area of research by presenting a DNA-based method for solving the Hamilton Path Problem. That contribution raised the hope that parallel computation by DNA could be used to tackle NP-complete problems which are thought of as intractable. The current realization, however, is that NP-complete problems may not be best suited for DNA-based (more generally, molecule-based) computing. A better subject for DNA computing could be large-scale evaluation of parallel computation models. Several proposals have been made in this direction. We overview those methods, discuss technical and theoretical issues involved, and present some possible applications of those methods. 1 Introduction Biomolecular computing is the computi...

We develop a general technique for constructing molecular-based approximation algorithms for NP optimization problems. Our algorithms exhibit a useful volume--accuracy tradeoff. In particular we solve the Covering problem of Hochbaum and Maass using polynomial time and O i ` 2 (log `)n 2 \Gamma n\Delta(n\Gamma1) l 2 =2 \Delta j volume with error ratio (1 + 1 ` ) 2 . We also present the first candidate for a problem that can be solved more efficiently with the Amplify operation than without. 1. Introduction Molecular computers were introduced by Adleman [1, 8], but so far the field lacks a "killer application." It is well known that a DNA computer can solve SAT in linear time [8], but using an exponential number of DNA strands. The number of strands used by an algorithm is called the "volume." Although recent papers [5, 4, 9] solve NP problems using smaller exponential volume, we believe that it is essential to find applications of DNA computers that use subexponential vol...

Length of DNA strands is an important resource in DNA computing. We show how to decrease strand lengths in known molecular algorithms for some NP-complete problems, such as like 3-SAT and Independent Set, without substantially increasing their running time or volume. 1. Introduction Since Adleman's pioneering experiment [1], many researchers have explored efficient molecular algorithms for NP-complete problems. The running time for a molecular algorithm is to the number of operations on test tubes. The volume is the maximum number of strings in all test tubes at any time, counting multiplicities. The strand-length complexity of a molecular algorithm is the length of the longest DNA strand used in the computation. Although time and volume complexity have been well studied [13, 6, 2, 14, 5, 9, 10, 8], strand length has received less attention. Yet Roweis et al [16], state that 2500-base sequences decay at a rate of 10% per hour, and Sambrook [17] states that DNA strands longer than 1000...

We survey the theoretical use of DNA computing to solve intractable problems. We also discuss the relationship between problems in DNA computing and questions in complexity theory. 1. Introduction Adleman's pioneering experiment [1] opened the possibility that moderately large instances of NP-complete problems might be solved via techniques from molecular biology. Since then numerous papers have explored more efficient molecular algorithms for particular problems in NP [27, 10, 3, 30, 8, 20, 21, 18], molecular solutions to PSPACE-complete problems [7, 37], and fault tolerant molecular algorithms [12, 25]. Other papers have examined the relationships between molecular complexity classes and classical complexity classes [38, 19]. We will survey some of these advances in this paper. For previous surveys in DNA computing, see [24, 36, 34, 32]. 2. Biological Background DNA is the storage medium for genetic information. It is composed of units called nucleotides, distinguished by the che...

We design volume-efficient molecular algorithms for all problems in #P, using only reasonable biological operations. In particular, we give a polynomial-time O(2 n n 2 log 2 n)-volume algorithm to compute the number of Hamiltonian paths in an n-node graph. This improves Adleman's celebrated n!-volume algorithm for finding a single Hamiltonian path. 1. Introduction Molecular computation was first proposed by Feynman [10], but his idea was not implemented by experiment for a few decades. In 1994 Adleman [1] succeeded to practically solve an instance of the Hamiltonian path problem in a test tube, just by handling DNA strings. DNA is the storage medium for genetic information. It is composed of units called nucleotides, distinguished by the chemical group (base) attached to them. The four bases are adenine, guanine, cytosine, and thymine, abbreviated as A, G, C, and T. Single nucleotides are linked end-to-end to form DNA strands. Each DNA strand has two chemically distinguishable...

Introduction Biomolecular computing is the computing methodology in which biologically important molecules are used as memory. The `aperiodic' nature of these polymeric molecules makes them suitable as memory units [GRY56, WHR + 87]. Instead of monotonous repeating units of most synthetic polymers (such as polyethylene), certain biological polymers (such as DNA, RNA and protein) have sets of repeating (or information encoding) units, and these units can appear in any order. The use of atomic or molecular-scale particles for building machinery was first conceived by Feynman in 1959 [Fey61], which gave rise to the recently emerging field of nanotechnology---the fabrication technology of molecular machines [Cra96, Dre92]. The biomolecular computation we discuss here is somewhat different from what conventional nanotechnology pursues: here the definition of machines is obscure. These computational machines have specific geometrical forms only in terms of