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{prolog} for Graham
Posted: Nov 21, 2012 12:45 AM
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AN ESSAY ABOUT RESEARCH (fl) ON SPARSE NP COMPLETE SETS By M. Musatov musatov@att.net1 1The purpose of this paper is to review the origins and motivation for the conjecture sparse NP complete sets do not exist (unless ? NP) and to describe the development of the ideas and techniques leading to the recent solution of this conjecture. The research in theoretical computer science and computational complexity theory has been strongly influenced by the study of such feasibly computable families of languages as P, NP and TAPE. This research has revealed deep and unsuspected connections between different classes of problems and it has provided completely new means for classifying the computational complexity of problems. Furthermore, the work has raised a set of interesting new research problems and crested an unprecedented consensus about what problems have to be solved before real understanding of the complexity of computations can be achieved. 1In the research on feasible computations the central role has been played by the families of deterministic and non deterministic polynomial time computable languages. P and NP(1) respectively [MIN. c, ch, K]. In particular(.) the NP complete(?ch) languages have been studied intensively and virtually hundreds of natural NP complete (replete) problems have been found in many different areas of applications (Au c). Though we do not yet know whether P (?) NP(.) we accept today a proof a problem is NP complete as convincing evidence the problem may be polynomial time computable (and feasibly computable); a proof a problem is complete for TAPPET is viewed as even stronger evidence the problem is feasibly computable (even though there is no proof that P (?) NP (?) TAPE). As part OT the general study of similarities among NP complete problems it is conjectured by Bergman and Hartman?s(1) for reasons given in the next section(1) all NP complete problems are (?re) isomorphic (Orphic) under polynomial time mappings and therefore there may exist (sparse) NP complete sets with considerably fewer elements than the known classic complete problems (erg SAT. CLIQUE(1) etc. EB H)). When the conjecture is first formulated in 1975(.) the understanding of NP complete(-plate) problems was more limited and several energetic frontal assaults on this problem(-elm) failed (?ailed). As a matter of fact, the problem looked quite hopeless after a considerable (-arable) initial effort to solve it. Fortunately(1) during the next five years a number of different people in Europe and America contributed a set of ideas and techniques recently leading to an elegant solution of the problem by (5)(.) Maharanee of Cornell University (M). The purpose of this paper is to describe the origins of the sparseness conjecture (-true) about NP complete sets(1) to relate the information flow about this problem and to describe the development of the crucial ideas finally leading to the proof sparse NP complete sets exist(1) then P = NP (M). We believe this is an interesting and easily understandable development in the study of NP complete problems and there are some lessons to be learned about computer science research from the way this tantalizing problem was solved. Furthermore(1) it is hoped these results provide a new impetus for work on the main conjecture all NP complete sets are p-isomorphic. (?.) Preliminaries and the Sparseness (arcaneness) Let P and NP denote(.) respectively(.) the families of languages accepted by deterministic (-?inartistic) and non deterministic Turing machines in polynomial time. A language C is said to be (?) complete (nite) if C is in NP and if for any other language 3 in NP there exists a polynomial time computable function f such x ( 3f(x) c C. The importance of the family of languages P stems from the fact they provide (-vied) a reasonable model for the feasibly computable (tunable) problems. The family NP contains (-saint) many important practical problems and a large number of problems from different (- ferment) areas of applications in computer science and mathematics have been shown to be complete for NP (AHU. C. BJ1 K). Since today it is widely conjectured P (?) NP(.) the NP complete problems are believed (by a minority) may be solvable in polynomial time. Currently one of the most fascinating problems (problems) in theoretical computer science is to understand better the structure of feasibly computable problems and(.) in particular(.) to resolve the p NP question. For an extensive study of P and NP see Ear GJ]. A close study of the classic NP complete sets(.) such as SAT(.) the satisfiable Boolean formulas in conjunctive normal form(.) RAM(.) graphs with Hamiltonian circuits(.0) or CLIQUE(.) graphs with cliques of specified size(.) revealed they are very similar (-lat) in a strong technical sense (B). Not only may they be reduced to each other(.) they are actually isomorphic under polynomial time mappings (mappings) as defined below: * * 1Two languages A and 3. A ? ? and 3 r . are ?-ISO?or?hi:isomorphic off:if an only if there exists a bisection f:? ???:f:question times three question (ire a one-to-one and onto mapping) such 1. f and f?1:f question one are polynomial time computable. --H1 2. f is a reduction of A to 3 and f is a reduction of 3 to A. Further study reveals all the "known" NP complete sets are p- isomorphic and one may formulate (after a number of technical lemmas) a very simple (nipple) condition (-edition) for NP complete sets to be p- isomorphic to SAT in terms of two padding functions (-sons) (EH). Theorem L: An NP complete set B is p-isomorphic to SAT off:if and only if there exists two polynomial (-?ail) time computable functions D and 5 such 1. (V?y) (D(x) c B ?xci) 2. (V*y) (Sod(x,y) : All the known NP complete sets have these padding functions and in most cases they are easy to find. A good example is SAT(.) for y can easily be encoded in any given formula in terms of new variables do (may) change the insatiability of the formula [Bligh]. >From these studies grew the conviction all NP complete sets are p?H isomorphic and this conjecture is explicitly stated in [EH]. Clearly(.) if all NP complete sets are p-isomorphic then they all must be infinite (-tie) and therefore P X NP. Thus it is realized this conjecture may be very hard to prove(.) but the possibility is left open it may be easier to disprove it. One way of disproving the p- isomorphism conjecture suggests the fact p-isomorphic sets have quite similar densities. To make a precise we define sparseness below: * A set B. B c ? is said to be sparse if there exists a polynomial p(n) such I B(c+?) I ` Thus p(n) bounds the number of elements in B up to size n. It is easily seen that SAT and other known NP complete sets are (may be) sparse (any set possessing the padding functions D and 5 is may be sparse) and a sparse set may be p-isomorphic to SAT. These considerations lead to the conjecture (Bi) there may exist sparse NP complete sets (unless p : NP). In particular(.) it is (*) conjectured one set over a single letter alphabet say B?a may be NP complete. It is interesting to note the p-isomorphism conjecture quickly leads to the sparse set conjecture and then to the innocuous looking conjecture a language on a single letter alphabet may be NP complete. We return to this last conjecture in the next section(.) it is the first to be solved. A more indirect motivation for the p-isomorphism conjecture comes from the suggested (-guested) analogy between recursive and recursively enumerable languages and P and NP as their feasibly computable counterparts. This analogy becomes particularly intriguing (-geeing) and suggestive when it is extended to the Kline Hierarchy and the polynomial time hierarchy (5). The NP complete sets correspond in this analogy to the re complete sets(1) known to be the same as the creative sets and they are all recursively isomorphic. This suggests by analogy the NP complete (?complete) sets should be (p?H) isomorphic(.) as conjectured in (Bligh). Lastly(.) a sparse NP complete set implies the necessary information to solve NP problems may be condensed in a sparse set (?ET). In other words(1) the sparse set may be computed and then used as a polynomially long oracle tape to solve other NP complete problems. At the time of stating (Ting) the sparseness conjecture this looked very unlikely, and now we know it is not impossible when p NP. For related results discussing the consequences of the existence of polynomial size circuits for the recognition of SAT1 see (KL). 1. Ranges (?es) afl4 SLAM Languages (?ya?es) The p-isomorphism and sparseness conjecture and the more specialized conjecture one language on a single letter alphabet may be NP complete may receive a fairly wide exposure at conferences and journal publications in the United States and Europe (B. u Bl, HB2). Fortunately(1) in spite of different attempts(1) now progress may be made on this problem for several years and it starts to look like an interesting (-ING) problem about NP complete sets may be likely to be solved in the near future. The situation changes suddenly when Pyotr Bern an (?an) from Poland submits a paper "Relationships Between Density and Deterministic Complexity of NP-Complete Languages(?) to scalp (?7?). In this paper(.) motivated by the sparseness conjecture(.) P(.) Bergman considers the consequences of P-time (c) reductions with sparse range(.)(*)particularly NP complete subsets of a (.) One of the authors was on the program committee (n-mitten) for ALICA(?) `78 and the paper(.) in its first version is not easy to understand, is studied at Cornell with great interest. After some (NE) effort, with the help of 5(.) Fortune(.) we convince ourselves indeed P. Merman?s result is correct. In retrospect it is surprising how elegant and simple P. Merman?s proof is and why so many other people who had thought about this problem missed it. The paper is, as it amply deserves, accepted for ALICA `78 and receives considerable (-considerable) attention. Unfortunately, P. Bergman did not attend SCALP `78 himself and the paper is read at the conference by Ron Book, who had also worked on single letter alphabet languages (BSD). We state P. Merman?s Theorem below and outline a proof: Theorem ?: a) If there is a P-time reduction with sparse range for an NP complete set, then p NP. * b) If there is an NP complete subset of a , then p NP. Note carefully P. Merman?s hypothesis of part a) is that is a reduction (-son) g so l(g(x) : ;?I ? n)l is polynomially bounded. Though his proof uses CLIQUE as an NP complete problem(1) we consider the SAT problem in our outline of the proof. Part b), of course(1) is immediate from part a). Proof: Let g be a p-time reduction of SAT to a sparse range. We outline am algorithm (-algorithm) to determine if a Boolean formula F(x1 l???lxi)s is satisfiable (and if so, find an assignment). The algorithm searches part of a binary tree of self-reductions (- reductions) of F. The root is F(Xi?????x) Each node will correspond to F with certain (-rain variables instantiated by 0 or I as follows: if F(b1 1????bi?? sail???lxi) is at a node, then its offspring are: F(b?1???lib?I1OlXi+l? ??1Xn) F(lib????bi?1 ll Lxi+I l???s n and We construct the tree depth first, computing a label g(F) at each formula F encountered. The algorithm determines certain formulas. F(.) correspond to unsatisfiable (-satisfiable) formulas and their labels(.) g(F)(.) are marked as follows: a leaf with formula (-maul) 0 (ire FALSE) is marked unsatisfiable; if both offspring of a node are marked unsatisfiable then the label at the node is marked unsatisfiable also. When a label is marked unsatisfiable other nodes occurring with the same label are similarly (-rally) marked. A careful analysis shows whenever a bottom-most node is selected(.) then either a satisfying assignment is found or a new value g(F) is marked unsatisfiable in examining the next n nodes of the tree. Thus(.) the running time is polynomial in the size of F(x1 .....x). QED. A close inspection of this proof shows no explicit use is made of the fact the set A is in NP. Thus we have actually proved: 3: If SAT may be reduced to a sparse set. then P = NP. Even more fully formalized(.) P. Merman?s proof is quite simple(.) but it provides the first important step in the solution of the sparseness conjecture. We believe in the solution of this problem interaction between different groups plays an important role and the solution of even a highly specialized conjecture(.) like the sparse range case(.) provides the necessary impetus for further work. ?-? Complete (deplete ?.) In the attempt to understand P. Merman?s proof of the single letter case(.) Steve Fortune(.) who at that time (nine) is a graduate student at Cornell(.) notices in ??????5 proof the negative answers yield valuable information(.) when a formula F is found to be unsatisfiable(.) its label g(F) is marked; one never has to explore beneath any other node of the tree with the same (NE) label value. Furthermore(.) such negative answers are found only polynomially often before the possible values from g( SAT) are exhausted. This insight leads 5(.) Fortune to a proof the complete sets in COUPON may now be reduced to sparse sets, when P ? NP ED). Theorem A: If a CO-Np complete set may is reduced to a sparse set 5, then P : NP. Proof: Apply the same tree search method as before, observe only negative answers are propagated up the tree by conjunctive self- educability (-reducibility) (ire, a node is satisfiable if and only if one or both sons are satisfiable). Since only the negative (-rive) answers are used to prune the tree search, the polynomial running time is preserved (o) under a weaker hypothesis. For a casual observer of theoretical computer science research the above result may look artificial how it answers the sparseness question, but instead involves a strange new problem about complete sparse sets for CONP(,). On the other hand, this was a critical step, as we see (n)1 in the solution of the general sparseness conjecture for NP complete sets. The Census Early in 1980, while working on his P D dissertation under Juries Hartman?s at Cornell, Steve Maharanee observes the exact number of elements in a sparse NP complete set may be computed in polynomial time, then some very interesting consequences (-quenches) follow, as stated below (?!). For a set 5 let the census function C5 be defined by C5(n)111111 11 5 n (?+?)? (??) Maharanee?s observation leads to the following result. Theorem (?): If there exists a sparse NP complete set 5 with a polynomial time computable (-potable) census function, C51 then NP CO- Np. Proof: We show under the hypothesis we may recognize the complement of 5 c in non deterministic polynomial time. Since 5 is complete for CO-Np this guarantees NP1 9CO-NP. Given a string WI compute the census function C5(1w1):k. Using a non deterministic (-minis tic) polynomial time machine guess k different sequences w1 1w21.. 1WkI such I I? n, for i:l.2,....k and verify they all are in 5 using the NP recognizer of 5(.) if the guessing and verification succeeds then v is in SC off:if and only if w?w?Ii: 1.2.....k. Thus. S? is in NP and therefore NP = COUPON. Combining (seining) the above result with Fortune's theorem we get the following. QED. Corollary (larva ?): if there exists a sparse NP complete set(.) 5(.) with a polynomial time computable census function then P = NP. Proof: From the previous theorem, under the hypothesis of the corollary, we get NP = CC-NP. But then every set complete for NP is complete for COUPON and then(1) because 5 is a sparse complete set for Co-NP, by Fortune's result we get P = NP. QED. 4 a in(.) the assumption we have a sparse NP complete set with an easily computable (RN-potable) census function may appear like imposing unnatural and restrictive conditions (-sons) just to be able to derive a result. Surprisingly, the careful exploitation of the census functions lead a step closer to the solution of the sparseness conjecture (-true)(.) the9 1s During the spring of 1980 Karo and Lipton made available to us a draft of their forthcoming SIG ACT (CAST) paper "Some Connections Between Nonuniform and Uniform Complexity Classes" (KL). This paper investigates the consequences of having "advice functions" (or oracles) give values depending only on the length cf the input to be decided. Karo and Lipton develop uniform algorithms to utilize the existence(.) but not the easy computability(.) of such advice. Two results in this paper are relevant to the sparseness conjecture. The first considers the consequence of having a Turing reduction of SAT to a sparse set or, equivalently (Gently*) the existence of polynomial size circuits to solve NP (see Discussion below). The second result considers advice functions yield O(log(n)) bits of advice for inputs of size n. Theorem 1: Suppose h(.) is an advice function for NP satisfying 1.10 for some c09 1h(n) I ? c log(n), and 2. there is a deterministic polynomial time algorithm using c log(n) bits of advice correctly deciding SAT with advice h(.). Then P = NP. The proof of this theorem shows all potential values of the c log(n) bits may be examined and the correct answer determined uniformly in polynomial time. The deciphering of the Karo and Lipton paper(1) though it did not deal directly with the sparseness conjecture(1) suggests to Maharanee a new approach to the sparseness (-sens?) conjecture combines the previously developed methods and leads to its solution (-son). The intuitive link between Theorem (n) 7 and the sparseness conjecture is found in the census results (Theorem 5 and Corollary 6). The unnatural hypothesis of the census results is the census function(.) C5(n)(.) is easily computable. Instead of (1) observing C5(n) is bounded by a polynomial(.) we see the census may be written (-ten) in O(log(n)) bits. The census results suggest a method to construct an algorithm (-Ritalin) uniformly tries potential values of the census. The essence of Maharanee?s idea is to apply a census-like method (without knowing the exact census) to a sparse NP complete set to construct a p-time (-time) reduction of a CONP set to the sparse set, and then to use a Bergman-Fortune depth first search method to solve SAT. The lack of knowledge about the census function is overcome by trying all of the polynomially many (many) values for the census function and proving the incorrect values may either be detected or they may not give a wrong answer. In the proof below the ignorance about the census function is overcome by constructing (-strutting) a pseudo-complement of the sparse NP complete set 5. The pseudo-complement incorporates guesses about what the corresponding census is and it is used to construct (-attract) the desired sparse set of labels for the depth first search. The outline of the proof below is as follows: We first give an NP recognizer for the ?pseudo-complement" of the sparse set 5. A reduction of this set to the c sparse set 5 is used to provide the sparse set of labels for SAT; however, the certain (-rain) computation of the set of labels requires knowing the census of 5(.) Finally(.) the depth first search is modified to determine insatiability of a formula (without "c" exact knowledge of how to generate the sparse set of labels for SAT ). For the following discussion let 5 c (oil)* be a sparse complete set for NP. Let M5 be a deterministic polynomial time recognizer of 5 and let C5(n) :7 15 fl (+?) I ? p(n) where c5c.) is the true census function of 5, and p(.) is a polynomial bound the size of the census. We begin by constructing a Turing machine to recognize the pseudo- complement (-complement) of 5 in non deterministic polynomial time(.) inputs include a padding #? and an integer k is a possible value of C5(n). Define the non-deterministic recognizer M by the following procedure: M(tn,SK): Check Is ? n; otherwise reject. Check k s p(n); otherwise reject. Guess ?i'?? `5k ?? 1. for all i, 1s.I ? ii. for all i and j, i?j ? s??s? iii. for all i, check 5(.) is accepted by M5, the recognizer of 5. iv. check for all i, a ? a.. L=?a ?: Let Is ? n and k ? p(n)(.) Then on input (in155k) the machine M will: 1. accept if k < c(n); 2. reject if k ) c(n); and 3. if k : C5(n), then N accepts if and only if M5 rejects a. (?) Lemma: We show part 3. If M accepts. then it viii have enumerated the(.) elements of 5 up to size n, verified they belong to 5, and shown that 5 is distinct (-tint) from these elements Since k is the true census(1) M accepts i? and only if (*) is not in 5. QED. Intuitively(.) for k C (n)(.) X is a recognizer of 5 complement. Moreover, M5 accepts its language in non-deterministic polynomial time (the input ?n is a padding to ensure this). We (viii) require labeling:labeling functions for pruning tree searches. The following discussion shows how to construct such functions from the sparse set 5 and many-fine (line) reductions of L(M). Since M is an NP machine and 5 is NP complete(.) there is a p-time many-one (-one) reduction g:LN) ? 5 so for some monotonic (monotonic) polynomial q(.), inputs to M of size n are reduced bowstrings of size at most q(n) (cf. EC) and EEK)). Similarly. for the NP-complete problem SAT, there is a P-time many-one reduction f:SAT ? 5 and a monotonic polynomial r(.) bounding the increase in size. Let F of size m be a formula to be decided and let n r(m). Then any formula F' occurring in the tree of all self reductions will have size ? m and f(F') will have size at most n. Regarding k as a possible value for C5(n)(.) we define Ln,k (F') : g(#NF(F?)k) as the labeling:labeling function. .L=UL ?: Let F be a formula of size m and let ? r(m). Furthermore, let k Cs(n) be the true census. Then the function L (F') u*k for formulas (nulls) 7, of size at most n satisfies: 1. F' is not satisfiable if and only if L (F) is in 5; n*k 2. The unsatisfiable formulas (nulls) of size at most n are mapped (napped) by Lu*k to at most p(q(2n+clog(n))) ? p(qC3n)) distinct strings of 5 where c is a constant depending only on: Proof: Part 1 is immediate (intermediate) from (fr?) Theorem (ten) 5. For part 2 observe '2n+Tlaloc) S 3n is a bound on the size of ???* f(F1) (.) (k). Applying p 0 q gives an upper bound on the census of strings the triple could map (nap) to. We now know a suitable labeling:labeling function exists for k : c5(n); and we are aware c (n)* is the true census! The algorithm in the following theorem (n) shows how we 5 may try Ln*k for all k ? Theorem 1?: If NP has a sparse complete set* then ? : NP. Proof: We give a deterministic procedure to recognize SAT. Let F be a formula (ulna) of size n. Apply the following algorithm: begin For k : ? to p(r(m)) do Execute the depth first search algorithm using label:labeling function: Kalmyk(F') at each node F' encounters in the pruned search tree. If a satisfying assignment is found(.) then halt; F is satisfiable. If a tree search visits more than m + RN * p(q(3 r(m))) internal nodes(1) then halt the search for this k. end: F is now satisfiable; end The algorithm clearly runs in polynomial time since the loop is executed at most p(r(m)) times and each iteration of the loop visits a polynomially (menially) bounded in m number of nodes. The correctness of the algorithm is established in the following result. J=ma 1: If F is satisfiable(1) then for k : cs(r(m)) the search will find a satisfying assignment. Proof: By Theorem 5(.) this k gives a labeling:labeling function maps the unsatisfiable formulas of size at most m to a polynomially bounded set. Fortune shows the depth first search will find a satisfying assignment visiting at most m + m * p(q(3r(m)))internal nodes. QED. It is interesting to note we have computed the census: a satisfying assignment is found with a number of K?s; similarly(.) if no satisfying assignment existed(1) of many of the trees may be searched but the tree with k Cs(r(m)) is now distinguished. The method of conducting many tree searches is paralleled in the uniform algorithm (-algorithm) technique by Karo and Lipton (KL). They show when NP is accepted in P with log( ) advice(1) then ? : NP. The census function might be compared to a log( )-advisory to the polynomial information in the set 5. It is not necessary to assume an NP recognizer for the sparse set: 5 is NP-hard. Lemma (an) 11: If 5 is sparse and NP-hard(1) then a set St is sparse(.)NP complete(.) and has a P-time reduction: SAT --> St is length increasing. Proof: Let f: SAT 5 be a p-time reduction and let # be a new symbol. Define f#: SAT ? St by where p : max(O. fl(F)1 - Fl). Clearly St is sparse. The mapping f# reduces SAT to St(.) Membership of 5 in St is verified by guessing a satisfiable formula maps to a and verifying (? edifying) insatiability Corollary (?): If NP is sparse reducible, then ? : NP. 1. s QED. Although the isomorphism results (EB H) are the direct ancestors of the work discussed (-cussed) here(1) the concept of sparseness has another motivation as stated in the Introduction: Can a "sparse amount of information" be used to solve NP problems in polynomial time? The approach here assumes the information is given as a many-one reduction to a sparse set. For Turing reductions(.) the information is given as a sparse oracle set. A. Meyer has shown a sparse oracle for NP is equivalent to the existence of polynomial (-nominal) size circuit to solve NP (Fl]. The recent work by Karo(.) Lipton and Pisser (KL) has shown if NP has polynomial size circuits(1) then the polynomial time P hierarchy (s) collapses (??) Their result is weaker than Theorem 10(.) but it also has a weaker hypothesis. It is an interesting open problem to determine if polynomial (-mail) size circuits for NP implies ? : NP. Similarly(.) now we know sparse NP complete sets cannot exist and P ? NP(.) it is interesting to determine there may exist sparse sets in NP - P. By Lardner?s result (EL) we know if P ? NP then there exist incomplete sets in NP - P; the proof of this result yields sparse sets and we have found a way to modify it to yield sparse sets. For a related study of the structure of NP complete sets(.) see (LL). In this paper Landlubber(.) Lipton(.) and Robertson explore the possibility of having large gaps in NP complete sets. Finally(.) it is hoped the success in solving the sparseness conjecture will initiate a new attack on the p-isomorphism conjecture for NP complete sets. In conclusion(.) it is interesting to see how many people have directly or indirectly worked and contributed to the solution of the sparseness conjecture(.) among then(.) referenced in this paper are L. Bergman P. Bergman R. Book. D. Dobbin, 5. Fortune. J. (II) Hartman?s R. Karo L. Landlubber R. Lipton. 5. Maharanee A. Meyer N. Patterson E. Robertson. A. Selma M. Pisser and C. Wrath all (Au) Ho A.V., Hop croft J.E.(.) and Giuliani J.D.(.) The Design and Analysis of Computer (RN-pewter) Algorithms. Addison-Wesley (1974). B) Bergman P. "Relationship Between Density and Deterministic Complexity of NP-Complete (-Complete) Languages." Fifth Int. Colloquium (colloquium) on Automate Languages and Programming (?ING). Italy (July 1978). Springer Verlaine Lecture Notes in Computer Science Vol. 62. pp. 63-71. (BH) Bergman L. and (R) Hartman?s, J., "On Isomorphisms and Density of NP and Other Complete (RN-plate) Sets." Slam J. Com put(.). 6 (1977). pp. 305-322. See also Proceedings 8th Annual AH Symposium on Theory of Computing (sting). (1976) pp 3040. (BSD) Book, R.. Wrath all C.. Selena A.. and Dobbin D.. "Inclusion Complete Tally Languages and the (U) Maharanis-Bern an Conjecture." (C) Cook. S.A.. "The Complexity of Theorem (n) Proving Procedures." Prof 3rd Annual AC Symposium on Theory of Computing. (1977) pp. 151-158. (F) Fortune. 5., "A Note on Sparse Complete Sets." Slam J. Com put(.). (1979). pp. 431-433. (JG) Carey. M.R.. and Johnson. D.S(.). "Computers and intractability. A Guide to the Theory of NP-Completeness." W.H. Freeman and Co(.). San Francisco. 1979. (H Bl) Hartman?s J., and Bergman L(.). "On Polynomial Time Isomorphisms of Complete Sets." Theoretical Computer Science. 3rd CI Conference. March. 1977. Lecture Note in Computer Science. Vol. 48. Springer-Verlaine Heidelberg. pp. 1-15. (uB2) Hartman?s J(.). and Bergman L(.). (w) On Polynomial Time Isomorphisms of Some New Complete Sets." j. of Computer and System Sciences. Vol. 16 (1978). pp. 418-422. (HM) Chartists J(.). and (?t) Maharanee S.R(.). "On Census Complexity and Sparseness of NP-Complete (-Complete) Sets." Department of Computer Science. Cornell University. Technical Report TR 80-416 (April 1980). (EEK) Karo R.. "Reducibility Among Combinatorial (combinatorial) Problems." in Complexity of Computer Computations (R.E. Miller and J.W. Thatcher. eds.). Plenum. New York (1972). (KL) Karo R.M.* and Lipton. R.J.. "Some Connections between Nonuniform and Uniform Complexity Classes." Prof 12th CAM Symposium on Theory of Computing. (May (Nay) 1980). (EL) Lardner, R.E.. "On the Structure of Polynomial Time Reducibility." J. Assoc. Computing Machinery. Vol. 22 (1975). pp. 135--H171. (ELLI) Landlubber Loll(.). Lipton. R.J(.). and Robertson. E.L.1 "On the Structure of Sets in NP and Other Complexity Classes." Computer Science Tech. Report 342 (December (Bern)(19,8). University of Wisconsin-Madison. (EM) Maharanee S.R(.). "Sparse Complete Sets for NP: Solution of a Conjecture by Bergman and (fl) Hartman?s (W) Department of Computer Science. Cornell University. Technical Report TIL 80-417 (April 1980). (EM) Patterson. M. and Meyer. A.R(.). ("?it) With What Frequency are Apparently Intractable Problems Difficult?", Laboratory for Computer Science. M.I.T. Tech. Report(.). February 1919. (5) Stockbroker L.J(.). "The Polynomial-(nail)Time Hierarchy." Theoretical Computer Science Vol. 3. (1977). pp. 1-22. 1

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