Critique of Feinsteins Proof that P is not Equal to NP

Critique of F einstein’s Pro of that P 6 = N P Kyle Sab o Ry an Sc hmitt Mic hael Silv erman June 14, 2007 Abstract W e ex a mine a pro of b y Cra ig A lan F einstein tha t P 6 = N P . W e present counterexamples to claims ma de in his paper and exp ose a flaw in the metho do logy he uses to make his assertions. The fault in his argument is the incorr ect use of reduction. F einstein makes incorr ect assumptions a bout the complexity of a pr oblem based on the fact that there is a mo re complex problem that ca n be used to solve it. His pap er in tro duces the terminology “imag inary pr ocessor ” to describ e how it is p ossible to b eat the brute for c e reduction he o ff ers to so lv e the Subset-Sum pro blem . The cla ims made in the pap er would not be v alidly established even were imaginar y pr ocessor s to exist. 1 In tro duction In this pap er we analyze the argument set forth by Craig Alan F einstein in his pap er, ‘A New and Elegan t Argument that P 6 = N P ’ [F e i07 ].W e present his argumen t and a counterargumen t b y using his reasoning to “pro ve ” a clearly trivial problem to b e n on-polynomially difficu l t. 2 F einstein’s Argumen t F einstein argues that P 6 = N P through the u se of “imaginary pro cessors.” The author b egins by in tro ducing the problem of searching for a record out of n unsorted records. W e state this as the follo wing f o rmal language. Definition 2.1 Find-Re cord = {h R, r , i i | r = r i ∈ { r 1 , r 2 , ..., r n } = R } F einstein then claims that Find-Record is Θ( n ). Ho we ve r, he con tin ues b y arguing that usin g multiple pro cessors to searc h subsets of R it is p ossible to ac hieve a b etter runtime. The au th o r’s definition of “m ultiple pro cessors” 1 is one su c h that if there w ere a Find-Record algorithm using n p rocessors, it w ould b e p ossible to find the record in Θ(1) time. In this w a y , F einstein’s use of multiple pro cessors is in tuitiv ely similar to a non-deterministic T uring Mac hine. E a c h branc h of the compu ta tion would use another p r ocessor. Ho w ev er, the author never make s this comparison and it is p ossible, therefore, that his mo del of compu ta tion has s ome extra prop erties th at a traditional nondeterministic T uring Mac hine do es not. W e use F einstein’s notation of m ultiple pro cessors in order to not b ia s the reader to disagree with his argu m e nt durin g its presen tation. W e will see that under ev ery p ossible reasonable definition of m ultiple pro cessor computation F einstein’s arguments fail to p ro v e his claim. F einstein notes th at usin g n pro cessors to solv e Find-Record is exp en- siv e, and that the b est solution, he argues, is to instead optimize the s um of pro cessors used and computations p er pr o cessor. This, under his nota- tion, im p lie s that the b est solution is to use Θ( √ n ) pro cessors, eac h doing Θ( √ n ) compu t ations. F urtherm o re, he implies that the only w a y to b eat the efficiency of br u te force is to u se these m ultiple pr ocessors. F einstein then describ es the Subse t-Sum problem. In his pap er, the Subset -Sum pr oblem is th e task of deciding if for a giv en set of in tegers, S ( || S || = n ), and another in teger, x , it is p ossible to find set S ′ ( || S ′ || = m ≤ n ) ⊆ S suc h that the sum of all the elemen ts in S ′ is equal to x . W e can state this alternativ ely as: Definition 2.2 Subset -Sum =    h S, x, y i | y =    1 if ther e is a set S ′ ⊆ S | x = X w ∈ S ′ w 0 otherwise    T o solve the Sub set-Sum problem, one can simply c hec k all p ossible su b- sets, whic h w ould tak e Θ(2 n ) time, h o w ev er, F einstein notes there are faster solutions, suc h as the Meet-In -The-Middle algorithm whic h is O ( √ 2 n ). F einstein argues th at it is p ossible to b eat the bru te force approac h of Find- Record b ecause there is an inherent mathematical structure to Subset- Sum . That is, there is some prop erty th e Su bset-Sum problem has that the Find-Recor d pr oblem d o es not h a v e. Essen tially , F einstein argues that Su bset-Sum is a searc h for a record where instead of a sp ecial num b er or lab el, the record is a subset with the sp ecial prop ert y of h a ving a sum equal to the d esired v alue x . This 2 sp ecial r ecord is a sp ecific su bset or set of sub sets of S out of all p ossible subsets. There are clearly 2 n subsets, where again n is the size of S , and therefore th e Find-Reco rd searc h through eac h sub set will ta ke Θ(2 n ). More f ormally , the p ap er offers a non-p olynomial time reduction to Find- Record from Subset-Sum , though the pap er nev er formally defines the reduction. Giv en an instance of S ubset-Su m F einstein describ es h o w to en umerate all sub sets and their sums, Θ(2 n ), and then searc hes th at list for a record that has a sum of x . How ev er, as F einstein notes, it is clear the algorithm Meet -In-The-Middle h as a r untime of O ( √ 2 n ). T o reconcile this, F einstein states there must b e some inh eren t math- ematical structure to Subset-S um which allo ws for a faster s olution than using the reduction to Find-Record . Using the reduction ab o v e, in essence F einstein implies that th e since the Find-Reco rd searc h is Θ( e ), where e is the num b er of elemen ts to search thr ough, and that sin ce S has 2 n subsets, where n is the size of S , it is imp ossible to b eat the ru ntime O (2 n ) withou t someho w searc hing the 2 n subsets of S in faster than linear time. Since there is in fact a wa y to solv e Subset -Sum faster than su c h a linear searc h thr ough the 2 n subsets of S , and since, according to F eins tein, the only w a y to go faster is to searc h through the 2 n subsets in less than 2 n op erations, there must b e some asp ect of the Me et-In-The-Middle algorithm that recreates m ultiple pr o cessors which F einstein calls imaginary pro cessors. That is, since chec king ev ery p ossib le subset wo uld tak e longer than Meet-In-The-Middle there must b e some wa y that Meet -In-The- Middle ac h iev es this sp eedup. Since there is an inheren t mathematical structure to Subse t-Sum , acco rdin g to F einstein, Meet-In-The-Middle can create these “imaginary pro cessors” that giv e the same b o ost in com- putation that using extra real p ro cessors do es for Find-Reco rd . F einstein then states th at there is a computational p enalt y to creating these Θ ( √ r ) imaginary p ro cessors, where r is the n umber of records to searc h through an d w e ha v e ea c h of the √ r imaginary pro cessors searc hing √ r elemen ts. He argues that b ecause eac h creation requires a sorting op eration, the amount of time it take s to create eac h “imaginary p ro cessor,” is Θ( p ) where p is the n umb er of pro cessors. Therefore, h e argues, w e are b oun d again by minimizing the p ro du ct of the n um b er of ‘imaginary pro cessors,’ times the num b er of elements pro cessed by eac h, giving us a lo we r b ound of Θ( √ 2 n ). F einstein states that s ince Subset-Sum is n o w b ound ed to Θ( √ 2 n ), the b est s olution for an N P complete pr oblem requires more than p olynomially m an y computations and therefore P 6 = N P . This b ou n d is based entirely on F einstein’s idea that it is to o exp ensiv e to create more imaginary pro cessors and that these imaginary p ro cessors are the only w a y 3 to get a sp eedup. 3 Brief Summary In the earlier section w e p resen t F einstein’s argum ent in as rigorous and formal a w a y as p ossible without c hanging the spirit or meaning of h is ar- gumen t. W e no w rewo rd his argument to the equiv alen t and more simp le logica lly v alid statemen ts. 1. A br ute force searc h for Sub set-Sum tak es Θ (2 n ) u n less w e u se m ul- tiple pr o cessors. This is b ecause we h a v e a non-p olynomial time re- duction f rom S ubset-S um to Find-Reco rd and records cannot b e solv ed faster than linear time w ithout multiple p ro cessors. 2. M eet-In-The-Middle is Θ( √ 2 n ) 3. M eet-In-The-Middle m ust s im ulate multiple p ro cessors b y creating imaginary pr o cessors to ac hieve b etter th an Θ(2 n ) b ecause of 1 and 2. 4. T he only w a y to get a faster runtime for S ubset-Su m without multiple pro cessors is to use imaginary pr o cessors. 5. T he b est run time p ossible for Sub set-Sum with im aginary pro cessors is Θ ( √ 2 n ) due to p enalties of creating new imaginary pro cessors. 6. S ince 4 and 5, P 6 = N P . 4 Coun terargumen t The fault in F einstein’s pap er lies in neglec ting pro of of some nont rivial statemen ts th at are implied b y his argument. Most simply , the pap er fails to p ro v e that the reduction given in argumen t 1 ab o v e is the most efficient reduction. The author argues that since the reduction sho wn to exist in argumen t 1 ab o v e is Θ(2 n ), an y such algorithm that h as faster than Θ(2 n ) runtime for Su bset-Sum must b e imitating some more p ow erful form of computation. This is simply n ot true. T h ere could in concept b e a more efficien t redu ction that could ev en b e p olynomial time. Without pro of that the reduction in 3 is the most efficien t suc h reduction, it is imp ossible to mak e the author’s claim. Ev en so, let’s assume the reduction in 3 is optimal. The author’s argu- men t then relies on the fact that th e non-p olynomial solution of brute force 4 has a non-p olynomial runtime to imp ly that the p roblem the non-p olynomial time algorithm b eing used to s olve must someho w b e using imaginary pro- cessors. If F einstein w ere correct, then m any things use imaginary p ro cessors. F or instance, consider c hec king if the fi rst num b er in a list of n num b ers is 3, where n v a ries on input, called First-Three . W e offer a non -p olynomial reduction f rom F irst-Three to records. W e can tak e the first num b er of our input list, and pu t it in a list of n rand om but non-3 elemen ts, and then we can rep eat p utting the ele ment into new random lists n ! times. This algorithm is a non-p olynomial reduction from F irst-Three to Find- Record . Therefore, according to F einstein’s reasoning, F irst-Three not only cannot b e done in p olynomial time, b ut to do b etter than the sp eed of the redu ction it m ust u se imaginary pro cessors. Ho w ev er, it seems clear that the co de that outpu ts tr ue if the fir s t elemen t is 3, do es not use imaginary pro cessors, and is constant time. If F einstein w ere correct, First-Three could not b e solv ed this wa y . References [F ei07] C .A. F einstein. A New and Elegan t Argument that P is not equal to NP. 2007. arXiv:cs/060 7093v2 5