Using DNA to Solv e NP-Com pl e te Ric hard J. Lipton y Princeton Princeton, NJ 08540 rjl@princeton.edu Abstract: shoentional w how to use DNA exp erimen to solv e the of Computer Science. This is a sp ecial case of ats more general metho that NP-completeWe problems, rst in tro duced in [3]. The adv an tage the h uge parallelism inheren t in DNA based It has the oten tial yield ast sp eedups overthe con v electronic basedUniversity computers for suc searc 1. In tro In a recen t breakthrough Adleman sho w ed ho w to use biological solv e instances of famous Hamiltonian P ath Problem (HPP). Recalltheseunderstandingeisbavthat.solvisey:thatatoparallelproblemproblemwcantourytsT".toathisTheresultsNPproblems./SAwdbiologicalerimenNPhofylarge:expaProblemsofpbasedfamouscomputationalhv that is: Giv en a set of /cities" directed paths b et w een them Find directed starts at a giv en cit y , ends at a en cit ycomputing. , and visits ev ery other cit exactly onc This problem (HPP) is wn to b e NP-completeproblem.esFurther,, [2]. A problem is in NP provided itsense. can e form ulated as a /searc h" problem NP-complete pro vided, has an ecien solution, then do al l One of the ma jor ac hiev emen ts Computer Science the last w o decades is the that man y imp ortan t searctin h problems are not only in but are NP- complete. Another ma jor ac hiev emen t is the wing evidence that no general ecien t solution exists for an NP-complete problem.groHPP Th us, Adleman's result that HPP can b e solv ed b a DNA exp er- imen t is exciting. Ho w ev er, it es not mean that all instances of NP problems can e solv ed in aductionyofhoifwbitandknogiv[1] fe asible Adleman solv es the in a brute force he designs a biological system /tries" all p ossible tours the giv cities. The sp eed of an y computer, biological or not, is determined b y tty w o factors: ho man pro cesses it has (ii) man y steps eac h can p erform p er unit time. exciting p oin t ab out biology iscomputational thatthatdo the rst of these factors canofsobtotally(i)en e ery recall y Supp orted in part b y NSF CCR-9304718. Draft of Jan. 19, 1995.

Thus, thet antage biological computers their evenhalldiculttodproblems. this adv anadvpractical tage do es not allo10 w y instance of an NP problem solv ed. The y is that ev enof with 22 parallel computers oneugebhange try tours for problem 100 cities. The brute force algorithm simply The o news is that biological computers cisis an solv eh an HPP sa y 70 or less edges. Ho w ev er, a issue is that there do es not seem need solv e suc HPP's. It app ears p ossible to routinely solv m uc h larger HPP's on con en mac hineswith One migh b e tempted to conclude that this means that biological computations are only a curious fo otnote to the history of computing. This is incorrect: v sho wn that itecomputations isgoIn p ossible to usejust biologicalane computationse to v astly sp eed man y imp ortan t computations [3]. In particular, w e can extend the metho d of Adleman wa y that allo ws biological computers to p oten tially radically c the w a that w do not HPP's. W will sho w ho w to solv e another famous NP-complete problem, the so called SA T problem. [3] w e sho w ho w to solv essen tially an problem from NP The goal here is present the full details of the results rst sketched 2.
NP-complete SA T In this w e dene SApresent T. It is a simple searchy: problem as the Let us it y giving an example:that wing form ula: F = ( x _b/true"._Then, y ) ^ ( x y ) The v ariables x ywill are bo ole an : they are allo w ed to range onlyConsiderwdirectly.tional.ea1er,tialrst0evwerationtoeessenfolloHovopyofvhaant.theeallinonefeasiblyWtofeb[3].greatwupvatoinoinecienparallelism.eoytocannotto er o alues Usuallysection , one thinks of 0 as /false" and 1 as _ is the /logical or" and ^ is the /logical and" op eration. 2