Hamiltonian path

About: Hamiltonian path is a research topic. Over the lifetime, 3350 publications have been published within this topic receiving 57423 citations.

Molecular computation of solutions to combinatorial problems

Abstract: The tools of molecular biology were used to solve an instance of the directed Hamiltonian path problem. A small graph was encoded in molecules of DNA, and the "operations" of the computation were performed with standard protocols and enzymes. This experiment demonstrates the feasibility of carrying out computations at the molecular level.

How to generate and exchange secrets

Abstract: In this paper we introduce a new tool for controlling the knowledge transfer process in cryptographic protocol design. It is applied to solve a general class of problems which include most of the two-party cryptographic problems in the literature. Specifically, we show how two parties A and B can interactively generate a random integer N = p?q such that its secret, i.e., the prime factors (p, q), is hidden from either party individually but is recoverable jointly if desired. This can be utilized to give a protocol for two parties with private values i and j to compute any polynomially computable functions f(i,j) and g(i,j) with minimal knowledge transfer and a strong fairness property. As a special case, A and B can exchange a pair of secrets sA, sB, e.g. the factorization of an integer and a Hamiltonian circuit in a graph, in such a way that sA becomes computable by B when and only when sB becomes computable by A. All these results are proved assuming only that the problem of factoring large intergers is computationally intractable.

On the complexity of the parity argument and other inefficient proofs of existence

Abstract: We define several new complexity classes of search problems, ''between'' the classes FP and FNP. These new classes are contained, along with factoring, and the class PLS, in the class TFNP of search problems in FNP that always have a witness. A problem in each of these new classes is defined in terms of an implicitly given, exponentially large graph. The existence of the solution sought is established via a simple graph-theoretic argument with an inefficiently constructive proof; for example, PLS can be thought of as corresponding to the lemma ''every dag has a sink.'' The new classes, are based on lemmata such as ''every graph has an even number of odd-degree nodes.'' They contain several important problems for which no polynomial time algorithm is presently known, including the computational versions of Sperner's lemma, Brouwer's fixpoint theorem, Chevalley's theorem, and the Borsuk-Ulam theorem, the linear complementarity problem for P-matrices, finding a mixed equilibrium in a non-zero sum game, finding a second Hamilton circuit in a Hamiltonian cubic graph, a second Hamiltonian decomposition in a quartic graph, and others. Some of these problems are shown to be complete.

Planar Formulae and Their Uses

Abstract: We define the set of planar boolean formulae, and then show that the set of true quantified planar formulae is polynomial space complete and that the set of satisfiable planar formulae is NP-complete. Using these results, we are able to provide simple and nearly uniform proofs of NP-completeness for planar node cover, planar Hamiltonian circuit and line, geometric connected dominating set, and of polynomial space completeness for planar generalized geography.The NP-completeness of planar node cover and planar Hamiltonian circuit and line were first proved elsewhere [M. R. Garey and D. S. Johnson, The rectilinear Steiner tree is NP-complete, SIAM J. Appl. Math., 32 (1977), pp. 826–834] and [M. R. Garey, D. S. Johnson and R. E. Tarjan, The planar Hamilton circuit problem is NP-complete, SIAM J. Comp., 5 (1976), pp. 704–714].

How to Generate and Exchange Secrets (extended abstract)

Abstract: In this paper we introduce a new tool for controlling the knowl­ edge transfer process in cryptographic protocol design. It is applied to solve a general class of problems which include most of the two-party cryptographic problems in the literature. -- -_.Specifically, we show how two parties A and B can interactively generate a random integer N =p' q such that its secret, i.e., the prime factors (p, q), is hidden from either party individually but is recoverab~~ jointly if desired. This can be utilized to give a protocol for two parties with private values i and j to compute any polynomially computable functions f(i,j) and g(i,j) with minimal knowledge transfer and a strong fairness property. As a special case, A and B can exchange a pair of secrets SA, SB, e.g. the factorizatio~ of an integer and a Hamiltonian circuit in a graph, in such a way that SA becomes computable by Bwhen and only when SB becomes computable by A. All these results are proved assuming only that the problem of factoring large intergers is computationally intractable.