## The NP-completeness column: an ongoing guide (1985)

### Cached

### Download Links

- [www.research.att.com]
- [www2.research.att.com]
- [public.research.att.com]
- [www.research.att.com]
- [www2.research.att.com]
- [www.research.att.com]
- [www2.research.att.com]
- [www.research.att.com]
- [www2.research.att.com]
- [public.research.att.com]
- [www.research.att.com]
- [www2.research.att.com]
- [www.research.att.com]
- [www2.research.att.com]
- [www.research.att.com]
- [www2.research.att.com]
- [www.research.att.com]
- [www2.research.att.com]
- [public.research.att.com]
- [www.research.att.com]
- [www2.research.att.com]
- [www.research.att.com]
- [www2.research.att.com]
- [www.research.att.com]
- [www2.research.att.com]
- [www.research.att.com]
- [www2.research.att.com]
- [www.research.att.com]
- [www2.research.att.com]
- [www.research.att.com]
- [www2.research.att.com]
- [www.research.att.com]
- [www2.research.att.com]
- [www.research.att.com]
- [www2.research.att.com]
- [www.research.att.com]
- [www2.research.att.com]
- DBLP

### Other Repositories/Bibliography

Venue: | Journal of Algorithms |

Citations: | 206 - 0 self |

### BibTeX

@ARTICLE{Johnson85thenp-completeness,

author = {David S. Johnson},

title = {The NP-completeness column: an ongoing guide},

journal = {Journal of Algorithms},

year = {1985},

volume = {6},

pages = {434--451}

}

### Years of Citing Articles

### OpenURL

### Abstract

This is the seventeenth edition of a (usually) quarterly column that covers new developments in the theory of NP-completeness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NP-Completeness,’ ’ W. H. Freeman & Co., New York, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, cross-references will be given to that book and the list of problems (NP-complete and harder) presented there. Readers who have results they would like mentioned (NP-hardness, PSPACE-hardness, polynomial-time-solvability, etc.) or open problems they would like publicized, should

### Citations

4253 | Introduction to Automata Theory, Languages, and Computation - Hopcroft, Motwani, et al. - 2001 |

4080 | Optimization by simulated annealing - Kirkpatrick, Jr, et al. - 1983 |

3316 | A method for obtaining digital signatures and public-key cryptosystems
- Rivest, Shamir, et al.
- 1978
(Show Context)
Citation Context ...of UP − P that has cryptographic significance, it seems only fair now to exhibit a potential member of NP − UP with similar significance (in this case a relation to the RSA public key cryptosystem of =-=[24]-=-). The problem is FACTORIZATION: ‘‘Given positive integers k < n, does n have a factor less than k?’’ This problem is clearly in NP, and one might at first think that it would be in UP. This is becaus... |

3089 | New directions in cryptography
- Diffie, Hellman
- 1976
(Show Context)
Citation Context ...olynomial-time algorithm for computing discrete logs (assuming the inputs are written in binary). Indeed, public key cryptosystems have been proposed based on the assumption that this problem is hard =-=[6]-=-. Nevertheless, suppose we view a UTM as computing a partial function (defined for the inputs it accepts, and, when defined, equal to the contents of its work tape at the end of the single accepting c... |

2620 |
The Design and Analysis of Computer Algorithms
- AHO, HOPCROFT, et al.
- 1974
(Show Context)
Citation Context ...ent rash of complexity results concerning VLSI design. The reader of this column will be assumed to be familiar with the basic theory of NP-completeness, as described in [G&J] or other works, such as =-=[3,33,37,38]-=-. Specific familiarity with [G&J] will not be assumed, although cross-referencing to that book will be included when appropriate. 3. SUBMISSIONS TO THE COLUMN Anyone having results or open problems he... |

1561 |
Reducibility among combinatorial problems
- Karp
- 1972
(Show Context)
Citation Context ...ent rash of complexity results concerning VLSI design. The reader of this column will be assumed to be familiar with the basic theory of NP-completeness, as described in [G&J] or other works, such as =-=[3,33,37,38]-=-. Specific familiarity with [G&J] will not be assumed, although cross-referencing to that book will be included when appropriate. 3. SUBMISSIONS TO THE COLUMN Anyone having results or open problems he... |

1320 |
Graph Theory
- Harary
- 1994
(Show Context)
Citation Context ...P [44] P £ [44] £ P [44] £ £ £ Circular Arc P [78] £ P [44] P [50] P [44] N [36] Circle P [71] £ £ P [GJ] P [50] O? N [36] £ Proper Circ. £ Arc P [77] P [44] P [50] P [44] P [66] £ Edge £ (or Line) P =-=[47]-=- P [GJ] P [T] N [GJ] N [49] £ £s¡ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢sClaw-Free P [T] P [63] O? N [GJ] N [49] TABL... |

1269 |
Probabilistic encryption
- Goldwasser, Micali
- 1984
(Show Context)
Citation Context ...romised [11,47].) QUADRATIC RESIDUOSITY: Given positive integers a and n with gcd(a,n) = 1, is there an integer x such that x 2 ≡ a (mod n)? (This is related to cryptographic schemes such as those in =-=[19]-=-. If n is prime or its prime factorization is given, the problem is in P [44]. If the ERH holds, the problem of finding such an x is in P if either n is prime or its complete factorization is given [4... |

1222 |
Geometric Algorithms and Combinatorial Optimization. Second Edition
- Grötschel, Lovasz, et al.
- 1993
(Show Context)
Citation Context ...rational number is the sum of the lengths of its numerator and denominator, under the assumption that RAM operations do not automatically remove common divisors.) Gr .. otschel, Lovász, and Schrijver =-=[23]-=- retain the bound on number size, but require only that the algorithm work for rational input, with the numerator and denominator of each rational entered into separate registers. (In their unit-cost ... |

1207 |
Algorithmic Graph Theory and Perfect Graphs
- Golumbic
- 1980
(Show Context)
Citation Context ...by V ′ has a chromatic number which is larger than its maximum clique size? Comment. Perfect graphs are of wide interest in combinatorial theory - entire books have been written about them (e.g., see =-=[27]-=-). That this problem is in NP would follow immediately from the famous ‘‘strong perfect graph conjecture,’’ which says that a graph is imperfect if and only if its complement contains an induced subgr... |

788 |
A machine program for theorem proving
- Davis, Logemann, et al.
- 1962
(Show Context)
Citation Context ...tions are (possibly) less potent. SATISFIABILITY [LO1]. Perhaps the most famous algorithm for telling whether a conjunctive normal form formula F is satisfiable is the ‘‘DavisPutnam Procedure’’ (DPP) =-=[14]-=-, and there has recently been a flurry ofs- 6 - seemingly conflicting reports on the average-case behavior of this algorithm. The conflicts arise because the results are not only based on different as... |

782 | Factoring polynomials with rational coefficients
- Lenstra, Lenstra, et al.
- 1982
(Show Context)
Citation Context ...ring it up to date. (For more thorough and historical surveys, see [24,28].) The major inspiration for all these new results was a now-famous 1982 papers- 9 - by A. Lenstra, H. Lenstra, and L. Lovász =-=[41]-=- (commonly referred to as ‘‘L3 ’’), which presented the first known polynomial-time algorithm for factoring univariate polynomials with rational (or, equivalently, integer) coefficients. Assuming that... |

711 | A new polynomial-time algorithm for linear programming
- Karmarkar
- 1984
(Show Context)
Citation Context ...he nature of the challenge can be illustrated by considering the recently developed polynomial-time algorithms for linear programming. The ellipsoid method [28] and its more recent successors such as =-=[25]-=- have left many in the field of mathematical programming wanting more: These algorithms run in time bounded by a polynomial in the size of the input (as measured in bits). However, even if all arithme... |

634 |
How to generate cryptographically strong sequences of pseudorandom bits
- Blum, Micali
- 1984
(Show Context)
Citation Context ...t cryptographic significance.s- 6 - DISCRETE LOG: Given positive integers g, b, and n, find an integer x such that g x ≡ b (mod n), if such an x exists. (This is related to cryptographic schemes from =-=[8,13]-=-. Factoring is randomly reducible to it, and the current best algorithms for it have running times like those for factoring. A once-popular choice for cryptograpic schemes was to consider the analogou... |

553 |
Graphs and Hypergraphs
- Berge
- 1979
(Show Context)
Citation Context ... we consider.s- 10 - Perfect Graphs. These are graphs G such that for every induced subgraph G′ of G, the chromatic number of G′ equals its maximum clique size. For alternative characterizations, see =-=[9,40,56]-=-. As was pointed out in the [Dec 1981] column, it is not known whether perfect graphs can be recognized in polynomial time and it was only recently shown that the class of imperfect graphs is in NP (d... |

527 |
The notions of consistency and predicate locks in a database system
- Eswaran, Gray, et al.
- 1976
(Show Context)
Citation Context ...he analog for deadlock problems would simply be testing whether one’s current state is deadlocked). To guarantee serializability, practitioners have developed such strategies as ‘‘two-phase locking’’ =-=[6]-=-, ‘‘tree locking’’ [8,27], and others [13,32]. According to specified rules, these add lock(x) and unlock(x) actions to the transactions, thus allowing a transaction to prevent other transactions from... |

507 |
Combinatorial Optimization: Networks and Matroids
- Lawler
- 1976
(Show Context)
Citation Context ... can tell (in one step) whether any given set is an independent set of the matroid [53]. (Recall that the 2-matroid intersection problem can be solved in polynomial time under such restrictions - see =-=[45]-=-). [7] PARTIAL ORDER DIMENSION INSTANCE: Directed acyclic graph G = (V,A) that is transitive, i.e., whenever (u,v) ∈A and (v,w) ∈A, then (u,w) ∈A, and a positive integer K≤| V| 2 . QUESTION: Does ther... |

495 |
Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms
- Booth, Lueker
- 1976
(Show Context)
Citation Context ...¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢sP [40] £ £ Undirected Path P [39] P £ [40] £ P [40] P [40] P [40] Directed Path P [38] P [40] P £ [40] £ P [40] P [40] Interval P =-=[17]-=- P [44] P [44] P £ [44] £ P [44] £ £ £ Circular Arc P [78] £ P [44] P [50] P [44] N [36] Circle P [71] £ £ P [GJ] P [50] O? N [36] £ Proper Circ. £ Arc P [77] P [44] P [50] P [44] P [66] £ Edge £ (or ... |

459 |
The complexity of enumeration and reliability problems
- Valiant
- 1979
(Show Context)
Citation Context ...]). Here we make the more realistic assumption that paths cannot be traversed instantaneously, and that your presence in the network can affect its behavior. The static problem is ‘‘only’’ #Pcomplete =-=[43]-=-, and hence possibly easier than the dynamic one. I conclude by returning once more to questions about familiar games. The above results raise the possibility that, with similar techniques, we might a... |

433 | Fast probabilistic algorithms for verification of polynomial identities
- Schwartz
- 1980
(Show Context)
Citation Context ...o collections P = {P 1 ,P 2 ,...,P n} and Q = {Q 1 ,Q 2 ,...,Q m} of multivariate polynomials over the n m rationals yield the same product polynomial, i.e., whether Πi = 1 P i ≡ Πi = 1 Q i. Schwartz =-=[38]-=- has shown that a simple randomized test for this problem has error probability less than 1/2, and can be iterated to reduce this probability to ans- 4 - arbitrarily small ε. The test involves randoml... |

426 | Theoretical improvements in algorithmic efficiency for network flow problems
- Edmonds, Karp
- 1972
(Show Context)
Citation Context ...ime in our sense do not take a polynomially-bounded number of steps on a real number RAM. For instance, consider algorithms based on ‘‘scaling’’ techniques. Scaling was introduced by Edmonds and Karp =-=[10]-=- as a technique for solving the ‘‘minimum cost flow’’ problem. In this problem, we are given a directed graph G = (V,A), a capacity c(a) and a cost p(a) for each arc, and two specified vertices s and ... |

415 | Nondeterministic exponential time has twoprover interactive protocols
- Babai, Fortnow, et al.
- 1991
(Show Context)
Citation Context ...T instance of length n. Let r be the actual number of random bits that the verifier uses on input I, and let b be the actual number of proof bits she queries. We may assume without loss of generality =-=[2,3,7,8,25]-=- that the verifier uses her r random bits simply to determine the addresses of the b proof bits, and does this before receiving any of the answers, i.e., in a nonadaptive fashion. Construct a graph G ... |

409 |
Some complexity questions related to distributive computing (preliminary report
- Yao
- 1979
(Show Context)
Citation Context ...ext) by Abelson [1], who was interested in the amount of communication needed when two processes compute a realvalued function of their combined (real-valued) inputs by interchanging real values. Yao =-=[34]-=- adapted this idea to the computation of finite functions by the exchange of bits. In both models, the input partition is fixed. In the discrete case this means that f can be viewed as a function of t... |

408 |
A polynomial algorithm in linear programming
- Khachiyan
- 1979
(Show Context)
Citation Context ...hat, for 1≤i≤m, V i . X ≤ d i and such that C . X ≥ B? Comment. This problem has been shown to be solvable in polynomial time by the ‘‘ellipsoid method’’ in the by now famous paper of L. G. Khachiyan =-=[40]-=- (See [4,24] for alternative presentations of the algorithm and [46] for an entertaining account of how the paper became ‘‘famous’’). Still open is the question of whether there is an algorithm for th... |

398 |
The Art of Computer Programming Vol. 3: Sorting and Searching
- Knuth
- 1998
(Show Context)
Citation Context ...statements of the form ‘‘if x i ≥ x j, then interchange their values,’’ where i < j. (Note that there is a one-to-one correspondence between such programs and ‘‘comparator networks’’ (as described in =-=[24]-=-, Section 5.3.4), so long as no comparators are allowed to operate in parallel). QUESTION: Is there an input to program P which P fails to sort into nondecreasing order, i.e., are there integer input ... |

392 |
On the complexity of fixed-priority scheduling of periodic, real-time tasks
- Leung, Whitehead
- 1982
(Show Context)
Citation Context ...42]. Transformation from CLIQUE. Comment. Not known to be in NP, but is in PSPACE. The transformation iss- 8 - an indirect one via a variant on SIMULTANEOUS INCONGRUENCES [AN2], proved NP-complete in =-=[43]-=- and to be described in a later column. The current problem is NP-hard for any fixed number m ≥ 1 of processors, although there is a very simple scheduling rule (always work on the available task with... |

387 |
Graph minors II: Algorithmic aspects of treewidth
- Robertson, Seymour
- 1986
(Show Context)
Citation Context ...n Column 16, is already enough to show that MINOR CONTAINMENT(H) is in P for all planar graphs H, as I shall now sketch. (Membership in P was proved by slightly different means in ‘‘Graph Minors II’’ =-=[42]-=-.) The needed auxiliary results are as follows. In [2], Arnborg, Corneil, and Proskurowski describe an O(| V| k + 1 ) algorithm for testing whether a graph G is a partial k-tree and, if so, deducing i... |

370 |
Optimization and approximation in deterministic sequencing and scheduling: A survey
- Graham, Lawler, et al.
- 1979
(Show Context)
Citation Context ... sense more complexity results known for scheduling problems than for all other problems combined! One of their tools is an elaborate shorthand scheme for specifying scheduling problems, described in =-=[7,28,32,33,38]-=-. Using this scheme, the specific results mentioned in the update for PREEMPTIVE SCHEDULING in Section 3 become simply: R | pmtn, r j | L max, Q | pmtn, r j | L max, Q2 | pmtn, prec, r j | L max, and ... |

358 |
On the Shannon capacity of a graph
- Lovász
- 1979
(Show Context)
Citation Context ...P, at least to a limited degree. That hope has now been dashed by Eva Tardos. In [40], she invents a monotone problem related to computing the ‘‘Shannon capacity’’ of a graph (as defined by Lovász in =-=[27]-=-), observes that the Alon-Boppana techniques apply to this problem to yield a true exponential lower bound, and then shows that the problem can be solved in polynomial time using ellipsoid method tech... |

357 | Self-testing/correcting with applications to numerical problems
- Blum, Luby, et al.
- 1993
(Show Context)
Citation Context ...orrect. Shortly after Toda’s result was published, Lund, Fortnow, Karloff, and Nisan [52] did just this. Building on earlier work of Beaver and Feigenbaum [10], Lipton [49], Blum, Luby, and Rubinfeld =-=[19]-=-, and others, they constructed an interactive proof system for the famous #P-complete problem of computing the permanent of a square 0-1 matrix [67]. I would like to say a bit about how this sort of c... |

348 |
A public key cryptosystem and a signature scheme based on discrete logarithms
- Gamal
(Show Context)
Citation Context ...t cryptographic significance.s- 6 - DISCRETE LOG: Given positive integers g, b, and n, find an integer x such that g x ≡ b (mod n), if such an x exists. (This is related to cryptographic schemes from =-=[8,13]-=-. Factoring is randomly reducible to it, and the current best algorithms for it have running times like those for factoring. A once-popular choice for cryptograpic schemes was to consider the analogou... |

346 |
Complexity of finding embeddings in a k-tree
- Arnborg, Corneil, et al.
- 1987
(Show Context)
Citation Context ...¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢sTrees/Forests P [T] P [GJ] £ £ P [T] P [GJ] P [T] £ £ £ Almost Trees (k) P £ P [24] P [T] P? P? Partial £ £ k-Trees P =-=[2]-=- P [1] P [T] P? P [1] Bandwidth-k P [68] P [64] P £ [T] P? £ P [64] Degree-k P [T] N [GJ] P £ [T] N [GJ] N £ [GJ] £ £s¡ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ... |

326 |
Relationships between nondeterministic and deterministic tape complexities
- Savitch
- 1970
(Show Context)
Citation Context ...can be replaced by equality [17].) The exponent in (b) has recently been improved to 2 [12], so that (a) and (b) together provide an alternative (and much more complicated) proof of Savitch’s theorem =-=[37]-=- that NSPACE(S(n)) ⊆ DSPACE(S(n) 2 ). A final nontrivial result for such space bounds is that probabilistic SPACE(S(n)) is closed under complement [36,39]. When the space bound S(n) = log n, there is ... |

323 |
Arthur-Merlin games: A randomized proof system, and a hierarchy of complexity classes
- Babai, Moran
- 1988
(Show Context)
Citation Context ...trarily low levels (k repetitions reduces it to 1/4 k .) The class IP and the concept of interactive proof system were independently proposed in 1985 by Goldwasser, Micali, and Rackoff [37] and Babai =-=[4,9]-=-. (The original definition only required that the verifier be correct 3/4 of the time when x ∈ L, but it was shown in [30] that requiring that she always be correct in this situation did not affect th... |

322 |
Approximation algorithms for NP-complete problems on planar graphs
- Baker
- 1994
(Show Context)
Citation Context ... ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢sPlanar £ P £ [GJ] N [GJ] P [T] N [10] N [GJ] Series Parallel P £ [79] £ P [75] P [T] P? P [74] Outerplanar £ P £ P =-=[6]-=- P [T] P [6] P [67] Halin P P £ [6] £ P [T] P [6] P [74] £ £ k-Outerplanar P P [6] P [T] P £ [6] £ P [6] Grid P P [GJ] P [T] P [GJ] P £ [T] £ K3,3-Free P [4] N £ [GJ] £ P [T] N [10] N [GJ] Thickness-k... |

317 | Digitalized Signatures and Public-Key Functions as Intractable as Factorization - Rabin - 1979 |

314 | Trading group theory for randomness
- Babai
- 1985
(Show Context)
Citation Context ...n 1 covers the two main formalizations of the notion of interactive proof (the original ‘‘interactive proof system’’ of Goldwasser, Micali, and Rackoff [23,24] and the ‘‘Arthur-Merlin game’’ of Babai =-=[4]-=-), illustrating them and discussing the complexity classes they determine. We then turn to the concept of a ‘‘zero-knowledge’’ proof. This is an interactive proof that convinces the verifier of the st... |

311 |
Formal languages and their relation to automata
- Hopcroft, Ullman
- 1969
(Show Context)
Citation Context ...haustive search of the game tree − there are only O(n 2k + 1 ) possible positions). Although problems in P with such nontrivial lower bounds are known to exist by diagonalization arguments (e.g., see =-=[23]-=-), this is to my knowledge the first example of a ‘‘natural’’ problem with this property. Other examples, also from [1], include a variant on the ‘‘cat and mouse’’ game to be discussed in the next sec... |

296 | Algebraic methods in the theory of lower bounds for Boolean circuit complexity
- Smolensky
- 1987
(Show Context)
Citation Context ...positive integer r that is not a power of p, the computation of the mod r function by constant depth, unbounded fan-in circuits requires an exponential number of gates even if mod p gates are allowed =-=[36]-=-. (Moreover, such gates are no more helpful in computing majority than was the parity gate.) In addition, Smolensky’s techniques simplify the proofs of the old results for parity and majority when onl... |

296 |
Parallelism in random access machines
- Fortune, Wyllie
- 1978
(Show Context)
Citation Context ...SP’S, and Turing Machines with various bells and whistles) is in for a real treat when it comes to parallel computation. Here we have vector machines [52], alternating Turing machines [11,12], PRAM’s =-=[22,55]-=-, SIMDAG’s [26], uniform circuits [9,53], conglomerates [26], aggregates [21], and hardware modification machines [21], to name just a few, and to restrict ourselves to ‘‘synchronous’’ models (in whic... |

273 |
Graph minors – XIII: The disjoint paths problem
- Robertson, Seymour
- 1995
(Show Context)
Citation Context ...the latter result.) For only slightly more complicated H, however, such as K 6, the status of the problems is not at all clear. Thus the new results of Robertson and Seymour (in ‘‘Graph Minors XIII’’ =-=[47]-=-), showing that both SUBGRAPH HOMEOMORPHISM(H) and MINOR CONTAINMENT(H) are in P for all graphs H, represent a major breakthrough. Moreover, for good measure, Robertson and Seymour also show membershi... |

272 |
On Numbers and Games
- Conway
- 2001
(Show Context)
Citation Context ... nature of computation, and this aspect of their character will not go totally unrepresented.s- 3 - 2. PUZZLING DEVELOPMENTS Given the many books on the subject of recreational mathematics (e.g., see =-=[2,3,4,12,18,19,20]-=-) and the wide variety of puzzles they describe, researchers have shown admirable restraint in applying the techniques of computational complexity to this area. Although such popular variable-sized re... |

272 |
Fast Probabilistic Algorithms for Hamiltonian Paths and Matchings
- Angluin, Valiant
- 1979
(Show Context)
Citation Context ... a polynomial time algorithm that would construct a tour with probability 1 − O(n − c ) for some c > 0 depending on α [25]. These results were tightened and generalized in 1977 by Angluin and Valiant =-=[2]-=-, who devised an improved algorithm with running time O(n 2 log n). They also extended the results to DIRECTED HAMILTONIAN CIRCUIT and to the related distribution in which for each n we consider only ... |

266 | Complexity of the Movers' Problem and Generalizations - Reif - 1979 |

265 |
Probabilistic algorithms for sparse polynomials
- Zippel
- 1979
(Show Context)
Citation Context ...traight-line programs that compute them. We then apply the appropriate algorithm from [27] to compute straight line programs for the factors (or GCD). By an algorithm in [26] based on ideas of Zippel =-=[61]-=-, these can then be turned back into concise representations in time polynomial in the sizes of those representations and their degrees. This of course re-introduces the potential blow-up in running t... |

263 | Checking computations in polylogarithmic time
- Babai, Fortnow, et al.
- 1991
(Show Context)
Citation Context ...computational resources to restrict. The first is the number of proof bits the verifier can examine. The significance of this was first pointed out in early 1991 by Babai, Fortnow, Levin, and Szegedy =-=[7]-=-. They observed that the MIP = NEXPTIME result meant that proofs of membership in NEXPTIME-complete languages existed that could be probabilistically checked by looking at only a very few randomly cho... |

256 |
M.: Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs
- Tarjan, Yannakakis
- 1984
(Show Context)
Citation Context ... N [10] £ Ns¡ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢s[GJ] £ £ Perfect O! P [42] P [42] P £ [42] £ P [42] Chordal £ P =-=[76]-=- P [40] P [40] P [40] P [40] £ Split £ P £ [40] P [40] P [40] P [40] P [40] £ £ Strongly Chordal P [31] P [40] P £ [40] £ P [40] P [40] Comparability P [40] P [40] P [40] £ P £ [40] P [40] Bipartite P... |

253 | Concurrency control in distributed database systems
- Bernstein, Goodman
- 1981
(Show Context)
Citation Context ...ewed as a tutorial on deadlock prevention and concurrency control, biased of course toward those questions that lend themselves to complexity analysis; tutorials with different biases can be found in =-=[3,15]-=-. Readers interested in further information on this active area of research are directed to these and to the other papers cited in this column’s references. (Readers who decide that the column already... |

250 |
On the inherent intractability of certain coding problems
- Berlekamp, McEliece, et al.
- 1978
(Show Context)
Citation Context ...ER asks whether a given graph G can have its vertices partitioned into K or fewer independent sets (and hence is in a sense dual to PARTITION INTO TRIANGLES, which asks for a partition into cliques). =-=[2]-=- DISTANCE-d CHROMATIC NUMBER INSTANCE: Graph G = (V,E), positive integer K ≤ | V|. QUESTION: Is there a partition V = V 1 ∪ . . . ∪ V K such that if u and v are distinct elements of some set V i in th... |

249 | Integer programming with a fixed number of variables
- Lenstra
- 1983
(Show Context)
Citation Context ..., and the special case of K = 2 when all coefficients are non-negative was recently shown to be solvable in polynomial time [36]. However, the general case for K ≥ 2 remained open until this year. In =-=[48]-=-, H. W. Lenstra uses ideas from the ‘‘geometry of numbers’’ to develop a polynomial time algorithm for each fixed value of K. (Unfortunately, as with all such sequences of algorithms mentioned here, t... |