A DNA Computing Algorithm for Directed Hamiltonian Paths

An algorithm for an NP-complete problem is presented, namely the existence of a Directed Hamiltonian Path in a directed graph. The main advantage of this algorithm is that it uses exponentially less DNA than original approach of Adleman. It is an implementation of a classic result in a 1964 paper of R. Held and R.M. Karp. For a graph withn vertices,O(n) time steps are used, each requiring up ton simultaneous laboratory operations. Morimoto, Arita, and Suyama have proposed another algorithm which would also conserve DNA (it can be shown to implement the Held-Karp algorithm). It also uses O(n) time steps. However, each time step involves a sequence of laboratory operations, rather than n simpler simultaneous operations. 1 Molecular Computing Becomes Feasible: Directed Hamiltonian Path In November 1994, Leonard Adleman reported (Adleman 1994) a laboratory experiment in which he used fabricated strands of DNA to solve the directed Hamiltonian path (DHP) problem for a very small instance. For DHP, we are given a set of cities along with available one-way flights connecting them. The objective of DHP is to find whether or not a travel itinerary exists connecting two given cities. The itinerary must also satisfy one other important criterion: it must visit all cites exactly once. DHP is a particularly interesting problem because it is known to be in the problem class NP-complete. The worst instances of such problems are considered to be very hard to solve when they are large. If it were ever discovered how to solve even one NP-complete problem easily, all of the 2,000 or so (considered very hard) NP-complete problems would automatically also have easy solutions. This is because transforming any other NP-complete problem into DHP (or any other NP-complete problem, for that matter) is in principle an “easy” problem (having polynomial complexity). However, using this equivalence is not always natural or practical. 2 The Held-Karp Algorithm The objective of DHP is to find a travel itinerary, if it exists, from an initial city to a final city that includes exactly one visit to each intermediate city in a preassigned set. Not all pairs of cities necessarily have connecting flights. This section presents a implementation of an algorithm that avoids incorporating duplicate cities as it constructs travel itineraries of exactly the right length, if there are any. Its greatest virtue is that is more suited to scaling up than are the present methods. This DHP algorithm uses dynamic programming. It is essentially “Algorithm B” from The Traveling Salesman Problem (Lawler et al. 1985, page 40) which in turn was taken from a 1962 article by Held and Karp (Held and Karp 1962). Actually, the underlying idea is quite simple. The candidate travel itineraries are repeatedly increased in length by adding one city at at time avoiding duplicating any prior cities. At any time step each city (let’s focus on, say, Chicago) corresponds to one parallel “laboratory task” (see Figure 1). Each task is given a copy of the available partial itineraries. To prevent duplicate Chicagos from being included in an itinerary, the first part of the task is to remove all partial itineraries that already include Chicago. To the remaining partial itineraries, all flights to Chicago are appended whenever possible, along with Chicago itself. After removing any itineraries that were not extended, all tasks combine their outputs. The cycle is repeated exactly enough times for all partial itineraries to have a chance to be extended to full length. Any resulting full length itineraries necessarily solve the DHP. 2.1 DNA Version of Held-Karp Algorithm