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The Maximum Clique Problem
, 1999
"... Contents 1 Introduction 2 1.1 Notations and Definitions . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Problem Formulations 4 2.1 Integer Programming Formulations . . . . . . . . . . . . . . . . . . . 5 2.2 Continuous Formulations . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Computation ..."
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Cited by 195 (21 self)
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Contents 1 Introduction 2 1.1 Notations and Definitions . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Problem Formulations 4 2.1 Integer Programming Formulations . . . . . . . . . . . . . . . . . . . 5 2.2 Continuous Formulations . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Computational Complexity 12 4 Bounds and Estimates 15 5 Exact Algorithms 19 5.1 Enumerative Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 19 5.2 Exact Algorithms for the Unweighted Case . . . . . . . . . . . . . . 21 5.3 Exact Algorithms for the Weighted Case . . . . . . . . . . . . . . . . 25 6 Heuristics 27 6.1 Sequential Greedy Heuristics . . . . . . . . . . . . . . . . . . . . . . 28 6.2 Local Search Heuristics . . . . . . . . . . . . . . . . . . . . . . . . . 29 6.3 Advanced Search Heuristics . . . . . . . . . . . . . . . . . . . . . . . 30 6.3.1 Simulated annealing . . . . . . . . . . . . . . . . . . . . . . . 30 6.3.2 Neural networks . . . . . . . . . . . . . . . . . . . . . . . .
The program-size complexity of self-assembled squares
- In Proceedings of the thirty-second annual ACM symposium on Theory of computing
, 2000
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Algorithmic Self-Assembly of DNA
, 1998
"... How can molecules compute? In his early studies of reversible computation, Bennett imagined an enzymatic Turing Machine which modified a hetero-polymer (such as DNA) to perform computation with asymptotically low energy expenditures. Adleman's recent experimental demonstration of a DNA computat ..."
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Cited by 166 (6 self)
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How can molecules compute? In his early studies of reversible computation, Bennett imagined an enzymatic Turing Machine which modified a hetero-polymer (such as DNA) to perform computation with asymptotically low energy expenditures. Adleman's recent experimental demonstration of a DNA computation, using an entirely different approach, has led to a wealth of ideas for how to build DNA-based computers in the laboratory, whose energy efficiency, information density, and parallelism may have potential to surpass conventional electronic computers for some purposes. In this thesis, I examine one mechanism used in all designs for DNA-based computer -- the self-assembly of DNA by hybridization and formation of the double helix -- and show that this mechanism alone in theory can perform universal computation. To do so, I borrow an important result in the mathematical theory of tiling: Wang showed how jigsaw-shaped tiles can be designed to simulate the operation of any Turing Machine. I propose...
Logical computation using algorithmic self-assembly of dna triple-crossover molecules
- Nature
, 2000
"... Recent work has demonstrated the self-assembly of designed periodic two-dimensional arrays composed of DNA tiles, in which the intermolecular contacts are directed by 'sticky ' ends. In a mathematical context, aperiodic mosaics may be formed by the self-assembly of 'Wang ' tiles ..."
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Cited by 115 (20 self)
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Recent work has demonstrated the self-assembly of designed periodic two-dimensional arrays composed of DNA tiles, in which the intermolecular contacts are directed by 'sticky ' ends. In a mathematical context, aperiodic mosaics may be formed by the self-assembly of 'Wang ' tiles 4, a process that emulates the operation of a Turing machine. Macroscopic self-assembly has been used to perform computations 5; there is also a logical equivalence between DNA sticky ends and Wang tile edges 6, 7. This suggests that the self-assembly of DNA-based tiles could be used to perform DNA-based computation 8. Algorithmic aperiodic self-assembly requires greater fidelity than periodic self-assembly, because correct tiles must compete with partially correct tiles. Here we report a one-dimensional algorithmic self-assembly of DNA triple-crossover molecules 9 that can be used to execute four steps of a logical (cumulative XOR) operation on a string of binary bits. A variety of different DNA tile types have been used in previous assemblies, including double-crossover molecules 1, triple-crossover molecules 9, and parallelograms produced from Holliday junction analogues 3.
Universal computation via self-assembly of DNA: Some theory and experiments
- In DNA Based Computers II, volume 44 of DIMACS
, 1996
"... In this paper we examine the computational capabilities inherent inthehybridization of DNA molecules. First we consider theoretical models, and show that the self-assembly of oligonucleotides into linear duplex DNA can only generate sets of sequences equivalent to regular languages. If branched DNA ..."
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Cited by 100 (12 self)
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In this paper we examine the computational capabilities inherent inthehybridization of DNA molecules. First we consider theoretical models, and show that the self-assembly of oligonucleotides into linear duplex DNA can only generate sets of sequences equivalent to regular languages. If branched DNA is used for self-assembly of dendrimer structures, only sets of sequences equivalent tocontext-free languages can be achieved. In contrast, the self-assembly of double crossover molecules into two dimensional sheets or three dimensional solids is theoretically capable of universal computation. The proof relies on a very direct simulation of a universal class of cellular automata. In the second part of this paper, we present results from preliminary experiments which investigate the critical computational step in atwo-dimensional self-assembly process. 1
Simulations of Computing by Self-Assembly
, 1998
"... Winfree (1996) proposed a Turing-universal model of DNA self-assembly. In this abstract model, DNA double-crossover molecules self-assemble to form an algorithmically-patterned two-dimensional lattice. Here, we develop a more realistic model based on the thermodynamics and kinetics of oligonucleo ..."
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Cited by 95 (15 self)
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Winfree (1996) proposed a Turing-universal model of DNA self-assembly. In this abstract model, DNA double-crossover molecules self-assemble to form an algorithmically-patterned two-dimensional lattice. Here, we develop a more realistic model based on the thermodynamics and kinetics of oligonucleotide hydridization. Using a computer simulation, we investigate what physical factors influence the error rates, i.e., when the more realistic model deviates from the ideal of the abstract model. We find, in agreement with rules of thumb for crystal growth, that the lowest error rates occur at the melting temperature when crystal growth is slowest, and that error rates can be made arbitrarily low by decreasing concentration and increasing binding strengths. 1 Introduction Early work in DNA computing (Adleman 1994; Lipton 1995; Boneh et al. 1996; Ouyang et al. 1997) showed how computations can be accomplished by first creating a combinatorial library of DNA and then, through successiv...
On the Computational Power of DNA Annealing and Ligation
- DNA Based Computers, volume 27 of DIMACS
, 1995
"... In [Winfree] it was shown that the DNA primitives of Separate, Merge, and Amplify were not sufficiently powerful to invert functions defined by circuits in linear time. Dan Boneh et al [Boneh] show that the addition of a ligation primitive, Append, provides the missing power. The question becomes, & ..."
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Cited by 87 (19 self)
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In [Winfree] it was shown that the DNA primitives of Separate, Merge, and Amplify were not sufficiently powerful to invert functions defined by circuits in linear time. Dan Boneh et al [Boneh] show that the addition of a ligation primitive, Append, provides the missing power. The question becomes, "How powerful is ligation? Are Separate, Merge, and Amplify necessary at all?" This paper proposes to informally explore the power of annealing and ligation for DNA computation. We conclude, in fact, that annealing and ligation alone are theoretically capable of universal computation. 1 Introduction When Len Adleman introduced the paradigm of using DNA to solve combinatorial problems [Adleman], his computational scheme involved two distinct phases. To solve the directed Hamiltonian path problem, he first mixed together in a test tube a carefully designed set of DNA oligonucleotide "building blocks", which anneal to each other and are ligated to create long strands of DNA representing paths t...
A DNA and restriction enzyme implementation of Turing Machines.
- DIMACS SERIES IN DISCRETE MATHEMATICS AND THEORETICAL COMPUTER SCIENCE
"... Bacteria employ restriction enzymes to cut or restrict DNA at or near specific words in a unique way. Many restriction enzymes cut the two strands of double-stranded DNA at different positions leaving overhangs of single-stranded DNA. Two pieces of DNA may be rejoined or ligated if their terminal ov ..."
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Cited by 85 (1 self)
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Bacteria employ restriction enzymes to cut or restrict DNA at or near specific words in a unique way. Many restriction enzymes cut the two strands of double-stranded DNA at different positions leaving overhangs of single-stranded DNA. Two pieces of DNA may be rejoined or ligated if their terminal overhangs are complementary. Using these operations fragments of DNA, or oligonucleotides, may be inserted and deleted from a circular piece of plasmid DNA. We propose an encoding for the transition table of a Turing machine in DNA oligonucleotides and a corresponding series of restrictions and ligations of those oligonucleotides that, when performed on circular DNA encoding an instantaneous description of a Turing machine, simulate the operation of the Turing machine encoded in those oligonucleotides. DNA based Turing machines have been proposed by Charles Bennett but they invoke imaginary enzymes to perform the state-symbol transitions. Our approach differs in that every operation can be pe...
