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## The NP-completeness column: an ongoing guide (1987)

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Venue: | JOURNAL OF ALGORITHMS |

Citations: | 239 - 0 self |

### Citations

2898 |
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... |

1998 |
Reducibility among combinatorial problems
- Karp
- 1962
(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... |

1421 |
Algorithmic Graph Theory and Perfect Graphs
- Golumbic
- 2004
(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... |

617 |
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... |

487 |
A polynomial algorithm for 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... |

346 |
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... |

318 | 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... |

311 |
On Numbers and Games,
- Conway
- 1976
(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... |

222 |
Decomposition of regular matroids,
- Seymour
- 1980
(Show Context)
Citation Context ...x of M whose determinant is ¬ in the set { − 1,0,1}? Comment. This problem turns out to be solvable in polynomial time, as a consequence of an elegant characterization theorem proved by P. D. Seymour =-=[63]-=-. Continuing work has lowered the (high) order of the polynomial, both for the general problem [18] and for special cases [70], and used these results to derive polynomial time algorithms for integer ... |

219 | Computing the minimum fill-in is np-complete.
- Yannakakis
- 1981
(Show Context)
Citation Context ... more than 3 vertices in G ′ , there is some edge in E ′ that is not involved in the cycle but that joins two vertices in the cycle? Comment. This problem has been proved NP-complete by M. Yannakakis =-=[68]-=- via a transformation from OPTIMAL LINEAR ARRANGEMENT [GT42]. [5] CHROMATIC INDEX INSTANCE: Graph G = (V,E) and a positive integer K . QUESTION: Does G have chromatic index K or less, i.e., can E be p... |

218 |
Partially ordered sets,”
- Dushnik, Miller
- 1941
(Show Context)
Citation Context ...contains no chain of length greater than one, so long as K is fixed at a value of 4 or more [69]. The problem for general partial orders was already known to be solvable in polynomial time when K = 2 =-=[17,62]-=-. [8] PRECEDENCE CONSTRAINED 3-PROCESSOR SCHEDULING INSTANCE: Set T of unit length tasks, partial order < . on T , and a deadline D∈Z + . QUESTION: Can T be scheduled on 3 processors so as to satisfy ... |

214 |
The directed subgraph homeomorphism problem.
- Fortune, Hopcroft, et al.
- 1980
(Show Context)
Citation Context ...n, but the ‘‘fixed vertex’’ version of the problem (the input specifies exactly which vertex of G is to correspond to each vertex of H) has been completely classified by Fortune, Hopcroft, and Wyllie =-=[22]-=- for directed graphs. It is polynomial-time solvable if H is a fixed graph all of whose arcs share a common tail, or all of whose arcs share a common head. For all other fixed graphs H the fixed-verte... |

210 |
Parallel sequencing and assembly line problems.
- Hu
- 1961
(Show Context)
Citation Context ... K is arbitrary the problem becomes NP-complete even for such restricted precedence constraints [25]. (Recall that the problem can be solved in polynomial time if < . is an in-forest or an out-forest =-=[34]-=-). The problem can also be solved in polynomial time for every fixed K if < . is a level order (any two tasks with a commons- 7 - immediate predecessor or successor have identical sets of predecessors... |

142 |
On distinguishing prime numbers from composite numbers,
- Adleman, Pomerance, et al.
- 1983
(Show Context)
Citation Context ...ree have been proved NP-complete. In addition, there have been new developments related to all the other six. In order to maintain some suspense, I shall discuss the problems in their original order. =-=[1]-=- GRAPH ISOMORPHISM INSTANCE: Two graphs G 1 = (V 1 ,E 1 ) and G 2 = (V 2 ,E 2 ) . QUESTION: Are G 1 and G 2 isomorphic, i.e., is there a one-to-one onto function f :V 1 →V 2 such that {u,v}∈E 1 if and... |

136 |
The complexity of the partial order dimension problem,
- Yannakakis
- 1982
(Show Context)
Citation Context ... orderings of V such that (u,v) ∈A if and only if u is less than v in each of the orderings? Comment. This has been proved NP-complete independently by Lawler and Vornberger [47] and by M. Yannakakis =-=[69]-=-, although the Yannakakis result is stronger. It shows that NP-completeness holds for every fixed K ≥ 3, whereas the other result was only for arbitrary K. The proof is via a transformation from GRAPH... |

117 |
Games against nature.
- Papadimitriou
- 1985
(Show Context)
Citation Context ...t are especially interesting, as they can be used to model many practical problems involving attempts to optimize performance from a probabilistic point of view. In his paper ‘‘Games against nature’’ =-=[29]-=-, Papadimitriou provides techniques for analyzing how the random element affects the complexity of such problems. His key observation is that, although technically these are one-person games, they can... |

112 |
On the computational complexity of combinatorial problems,”
- Karp
- 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... |

109 |
On the complexity of integer programming,
- Papadimitriou
- 1981
(Show Context)
Citation Context ...s. (If the solution vector is required to be non-negative, INTEGER PROGRAMMING is NP-complete even for two constraints, but can be solved in pseudo-polynomial time for any fixed number of constraints =-=[60]-=-). The second problem concerns perfect graphs, and I am happy to say that it is still open (as of this writing) and hence can be offered as a partial replacement for the six problems already eliminate... |

103 | Complete register allocation problems. In:
- Sethi
- 1973
(Show Context)
Citation Context ...d this may be necessary if k pebbles are to suffice. Ifs- 6 - no re-pebbling is allowed, the problem becomes the REGISTER SUFFICIENCY [PO1] problem of [G&J] and is only NP-complete. This was shown in =-=[37]-=-, which also proved the current problem to be at least NP-hard. In the register allocation interpretation, the graph represents the dependency structure of intermediate results in a computation, and t... |

69 |
Polynomial-time algorithms for permutation groups
- Furst, Hopcroft, et al.
- 1980
(Show Context)
Citation Context ... A subexponential O(n c log n ) algorithm has been found for tournaments [55]. Many of these latter results follow from polynomial-time algorithms for dealing with permutation groups, as described in =-=[23]-=-. They also were foreshadowed by various randomized algorithms for the same problems, such as in [5]. In the realm of probability, [7] presents a method for producing canonical labellings of graphs (a... |

60 |
The complexity of completing partial latin squares
- Colbourn
- 1984
(Show Context)
Citation Context ...STION: Can M be extended to a (full) Latin square, i.e., can we replace each 0 entry in M by a member of {1,2,...,n} in such a way that no column or row contains a repeated entry? Reference. Colbourn =-=[10]-=-. Transformation from EDGE PARTITION INTO TRIANGLES (see Column 3 [June 1982]). Comment. NP-complete, as is the question of whether M has more than one completion to a Latin square (even when given th... |

58 |
Polynomial Solvability of Convex Quadratic Programming.
- Kozlov, Tarasov, et al.
- 1979
(Show Context)
Citation Context ...r the ellipsoid method. A survey of recent research into the ellipsoid method can be found in [8]. In particular, it has been used to solve the convex quadratic programming problem in polynomial time =-=[42]-=- as well as the linear complementarity problem for positive definite symmetric matrices [11] (The general LINEAR COMPLEMENTARITY problem, mentioned in the comments to this problem in [G&J], has been p... |

55 |
The NP-completeness of edge coloring,
- Holyer
- 1981
(Show Context)
Citation Context ...rtitioned into disjoint sets E 1 ,E 2 , . . . , E k , with k≤K , such that, for 1≤i≤k, no two edges in E i share a common endpoint in G? Comment. This problem has been proved NP-complete by I. Holyer =-=[32]-=-, via a transformation from 3SAT, even for K = 3 and cubic graphs. Leven and Galil [50] have shown that NP-completeness also holds for every fixed K > 3, even if the graphs are required to be K-regula... |

55 |
Disjoint paths in graphs”,
- Seymour
- 1980
(Show Context)
Citation Context ...cted to reducible flow graphs [30]. With respect to the undirected version of the problem, new polynomial time algorithms for the case where H consists of two independent edges have been presented in =-=[59,64]-=-, with the latter giving a particularly concise characterization. The original two edge and triangle results mentioned in [G&J] have appeared formally in [65,44]. With respect to the ‘‘non-fixed verte... |

53 |
On linear characterization of combinatorial optimization problems
- Karp, Papadimitriou
- 1982
(Show Context)
Citation Context ...umber of researchers [10,35]). The ellipsoid method has also been used to prove new NP-hardness results, as an equivalence between ‘‘feasability’’ and ‘‘separability’’ problems can be derived from it =-=[28,29,39]-=-. Finally, the furor over the ellipsoid method has revived interest in the theoretical aspects of the simplex method, which works so well in practice even though it has exponential worst case running ... |

52 |
A polynomial solution to the undirected two paths problem
- Shiloach
- 1980
(Show Context)
Citation Context ...dependent edges have been presented in [59,64], with the latter giving a particularly concise characterization. The original two edge and triangle results mentioned in [G&J] have appeared formally in =-=[65,44]-=-. With respect to the ‘‘non-fixed vertex’’ versions of the problem, [52] presents a polynomial time algorithm for H = K 4 in the undirected case and [22] classifies a number of H (both polynomially so... |

52 |
Multiple-Person Alternation,
- Peterson, Reif
- 1979
(Show Context)
Citation Context ...erson teams, with the two team members playing alternately and having their own personal sets of plates, only some of which are visible to their teammates. This new problem (TEAM PEEK) is undecidable =-=[31]-=-. So much for teamwork. For more on the precise nature of the complexity jumps due to incomplete information, including methods of modelling the phenomena using Turing machines with ‘‘private’’ tapes,... |

50 |
Matroid matching and some applications,
- Lovász
- 1980
(Show Context)
Citation Context ... Is there a spanning tree T = (V,E′) for G such that for each E i, 1≤i≤m, either E i ⊆ E′ or E i ∩E′ = ∅? Comment. This problem can be solved in polynomial time by an algorithms- 6 - due to L. Lovász =-=[53,54]-=-, which can be used to solve the parity problem for arbitrary ‘‘representable’’ matroids, of which this is a special case (a matroid is representable if its independent sets correspond to the independ... |

50 |
Isomorphism testing for graphs of bounded genus
- Miller
- 1980
(Show Context)
Citation Context ... although some significant new subcases have been shown to be solvable in polynomial time: graphs embeddable in the projective plane [51], graphs embeddable in a surface of genus K, for every fixed K =-=[20,58]-=-, graphs with maximum vertex degree K, for every fixed K [55], and ‘‘cone graphs’’ of bounded degree (as defined in [31] - note that the ‘‘degree’’ of a cone graph is not its vertex degree), using the... |

50 |
Transitive orientation of graphs and identification of permutation graphs
- Pnueli, Lempel, et al.
- 1971
(Show Context)
Citation Context ...contains no chain of length greater than one, so long as K is fixed at a value of 4 or more [69]. The problem for general partial orders was already known to be solvable in polynomial time when K = 2 =-=[17,62]-=-. [8] PRECEDENCE CONSTRAINED 3-PROCESSOR SCHEDULING INSTANCE: Set T of unit length tasks, partial order < . on T , and a deadline D∈Z + . QUESTION: Can T be scheduled on 3 processors so as to satisfy ... |

48 |
Scheduling intervalordered tasks,
- Papadimitriou, Yannakakis
- 1979
(Show Context)
Citation Context ...of predecessors and successors) [67], and for arbitrary K if < . is an interval order (each task T i corresponds to an interval [a i ,b i ] on the real line, and T i < . T j if and only if b i < a j) =-=[61]-=-. Remains NP-complete for arbitrary K if < . is the disjoint union of an outforest (or in-forest) with a layered order, or a union of layered orders, or an intersection of two layered orders [25] (a l... |

45 |
A polynomial-time algorithm for determining the isomorphism of graphs of fixed genus (working paper
- Filotti, Mayer
- 1980
(Show Context)
Citation Context ... although some significant new subcases have been shown to be solvable in polynomial time: graphs embeddable in the projective plane [51], graphs embeddable in a surface of genus K, for every fixed K =-=[20,58]-=-, graphs with maximum vertex degree K, for every fixed K [55], and ‘‘cone graphs’’ of bounded degree (as defined in [31] - note that the ‘‘degree’’ of a cone graph is not its vertex degree), using the... |

41 | Computing a perfect strategy for n x n Chess requires time exponential in n
- Fraenkel, Liechtenstein
- 1981
(Show Context)
Citation Context ...without a pawn move or a capture yields a draw’’ is generalized to one in which the ‘‘50’’ is replaced by a fixed polynomial in N [42]; if the rule is omitted the problem becomes complete for EXPTIME =-=[16]-=-, as does N × N CHECKERS [36]. (Without such a drawing rule, N × N GO is also a candidate for EXPTIMEcompleteness.) (3). N × N GOBANG (GO-MOKU) is PSPACE-complete, even if the goal of getting 5 stones... |

40 |
Moderately exponential bound for graph isomorphism
- Babai
- 1981
(Show Context)
Citation Context ...ere have been significant improvements over the naive O(n!) bound, where n is the number of vertices: M. K. Goldberg gives an O(c n ) algorithm in [26], and V. N. Zemlyachenko improves on this to O(c =-=[6,71]-=-, using techniques from [55]. Finally, the set of problems which have been proved polynomially equivalent to general graph isomorphism has been considerably broadened [9,12,13,14], although the transf... |

38 |
Universal games of incomplete information
- REIF
- 1979
(Show Context)
Citation Context ...ng this increase in difficulty has until recently been beyond the reach of complexity theory. This section discusses two approaches to extending that reach, both of which use games as their model. In =-=[32]-=-, Reif considers the complexity of games of ‘‘incomplete information,’’ i.e., games in which the players may not know all the details of the current position, as in Poker, where players may not know a... |

36 | Storage requirements for deterministic polynomial time recognizable languages.
- Cook, Sethi
- 1976
(Show Context)
Citation Context ...ate predecessors. This variant is also PSPACE-complete [21]. An interesting variant that rates mention as an ‘‘Open Problem of the Month’’ is THE BLACK-WHITE PEBBLE GAME, introduced by Cook and Sethi =-=[13]-=-. In addition to ‘‘black’’ pebbles that follow the rules (2) and (3), this game also includes white pebbles that follow dual rules: (2′) a white pebble can be placed on any vertex at any time, and (3′... |

31 |
Winning Ways for Your Mathematical Plays, Volume 2
- Berlekamp, Conway, et al.
- 1982
(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... |

28 |
Neither the greedy nor the Delaunay triangulation f a planar point set approximates the optimal triangulation
- Manacher, Zorbrist
- 1979
(Show Context)
Citation Context ... be nonoptimal when [G&J] was prepared, have now been shown to do so poorly on some instances that they may construct a triangulation whose total length is an arbitrary multiple of the optimal length =-=[41,56]-=-. I shall conclude this first column by mentioning two more ‘‘open’’ problems (which should have been on our original list, but weren’t), and by pointing out as- 9 - number of minor errors that have b... |

26 |
Problems polynomially equivalent to graph isomorphism.
- Booth, Colbourn
- 1979
(Show Context)
Citation Context ...ko improves on this to O(c [6,71], using techniques from [55]. Finally, the set of problems which have been proved polynomially equivalent to general graph isomorphism has been considerably broadened =-=[9,12,13,14]-=-, although the transformations involved may not always be able to preserve the exponents in the running times of the above algorithms. [2] SUBGRAPH HOMEOMORPHISM (FOR A FIXED GRAPH H) INSTANCE: Graph ... |

24 |
Isomorphism of bounded valence can be tested in polynomial time
- Luks
- 1980
(Show Context)
Citation Context ...lvable in polynomial time: graphs embeddable in the projective plane [51], graphs embeddable in a surface of genus K, for every fixed K [20,58], graphs with maximum vertex degree K, for every fixed K =-=[55]-=-, and ‘‘cone graphs’’ of bounded degree (as defined in [31] - note that the ‘‘degree’’ of a cone graph is not its vertex degree), using the techniques from [55]. In addition, ‘‘colored’’ graphs, in wh... |

22 |
The pebbling problem is complete in polynomial space
- Gilbert, Lengauer, et al.
- 1980
(Show Context)
Citation Context ... and only if all vertices u such that (u,v) ∈ A currently contain pebbles. (Note that this implies that vertices with in-degree 0 can be pebbled at any time.) Reference. Gilbert, Lengauer, and Tarjan =-=[21]-=-. Transformation from QUANTIFIED BOOLEAN FORMULAS [LO11]. Comment. PSPACE-complete, which is perhaps surprising given that this is not a two-person game and hence there is no obvious form of alternati... |

21 | Monte-carlo algorithms in graph isomorphism testing,”
- Babai
- 1979
(Show Context)
Citation Context ...esults follow from polynomial-time algorithms for dealing with permutation groups, as described in [23]. They also were foreshadowed by various randomized algorithms for the same problems, such as in =-=[5]-=-. In the realm of probability, [7] presents a method for producing canonical labellings of graphs (and hence for testings- 4 - isomorphism) which runs in expected linear time under the standard unifor... |

20 | The subgraph homeomorphism problem
- LaPaugh, Rivest
- 1980
(Show Context)
Citation Context ...dependent edges have been presented in [59,64], with the latter giving a particularly concise characterization. The original two edge and triangle results mentioned in [G&J] have appeared formally in =-=[65,44]-=-. With respect to the ‘‘non-fixed vertex’’ versions of the problem, [52] presents a polynomial time algorithm for H = K 4 in the undirected case and [22] classifies a number of H (both polynomially so... |

20 |
The Scientific American Book of
- Gardner
- 1959
(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... |

19 |
Embedding partial Steiner triple systems is NP-complete
- Colbourn
- 1983
(Show Context)
Citation Context ...s, i.e., assuming the given matrix M is symmetric, can it be extended to a Latin square that is also symmetric, is also NP-complete, even if M is known to be extendable to a nonsymmetric Latin square =-=[9]-=-. The final puzzle in this section will probably not be found in the recreational mathematics literature; its real source is a problem in computer register allocation. [3] THE PEBBLE GAME INSTANCE: Ac... |

19 |
Martin Gardner's New Mathematical Diversions from Scientific American. Simon and
- GARDNER
- 1966
(Show Context)
Citation Context |

17 |
A polynomial algorithm for the two-variable integer programming problem
- Kannan
- 1980
(Show Context)
Citation Context ...estricted to a fixed number K of variables. The problem is trivial for K = 1, 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... |

17 |
Classes of pebble games and complete problems
- Kasai, Adachi, et al.
(Show Context)
Citation Context ...alternate choosing applicable rules and applying them. A player loses if there are no applicable moves on his turn, or if his opponent ever places a pebble on v 0. Reference. Kasai, Adachi, and Iwata =-=[26]-=-. Transformation from LINEAR BOUNDED AUTOMATON ACCEPTANCE [AL3]. Comment. PSPACE-complete. The corresponding one-person game (can a single player pebble v 0 by starting from the intial position and re... |

15 |
The Sixth Book of Mathematical Games from Scientific American
- Gardner
- 1984
(Show Context)
Citation Context |

12 |
Khachiyan's linear programming algorithm
- Aspvall, Stone
- 1980
(Show Context)
Citation Context ...e for every fixed K [21]. A detailed algorithm for the case of K = 1 and cubic graphs is given in [19]. This work on genus led to the isomorphism algorithms for graphs of fixed genus mentioned above. =-=[4]-=- CHORDAL GRAPH COMPLETION INSTANCE: Graph G = (V,E) and a positive integer K . QUESTION: Is there a superset E ′ containing E of unordered pairs of vertices from V that satisfies | E ′ − E| ≤ K and su... |

12 |
Khachian's Algorithm for Linear Programming
- Gacs, Lovasz
- 1979
(Show Context)
Citation Context ...≤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 the problem wh... |

12 |
The matroid matching problem. In:
- Lovasz
- 1978
(Show Context)
Citation Context ... Is there a spanning tree T = (V,E′) for G such that for each E i, 1≤i≤m, either E i ⊆ E′ or E i ∩E′ = ∅? Comment. This problem can be solved in polynomial time by an algorithms- 6 - due to L. Lovász =-=[53,54]-=-, which can be used to solve the parity problem for arbitrary ‘‘representable’’ matroids, of which this is a special case (a matroid is representable if its independent sets correspond to the independ... |

12 |
The two disjoint path problem and wire routing design
- OHTSUKI
- 1981
(Show Context)
Citation Context ...cted to reducible flow graphs [30]. With respect to the undirected version of the problem, new polynomial time algorithms for the case where H consists of two independent edges have been presented in =-=[59,64]-=-, with the latter giving a particularly concise characterization. The original two edge and triangle results mentioned in [G&J] have appeared formally in [65,44]. With respect to the ‘‘non-fixed verte... |

12 |
The computational complexity of recognizing critical sets
- Colbourn, Colbourn, et al.
- 1984
(Show Context)
Citation Context ...TITION INTO TRIANGLES (see Column 3 [June 1982]). Comment. NP-complete, as is the question of whether M has more than one completion to a Latin square (even when given the first as part of the input) =-=[11]-=-. The corresponding question for symmetric Latin squares, i.e., assuming the given matrix M is symmetric, can it be extended to a Latin square that is also symmetric, is also NP-complete, even if M is... |

11 |
On determining the genus of a graph
- Filotti, Miller, et al.
- 1979
(Show Context)
Citation Context ...e integer K . QUESTION: Can G be embedded on a surface of genus K such that no two edges cross one another? Comment. Remains open for general K, but can be solved in polynomial time for every fixed K =-=[21]-=-. A detailed algorithm for the case of K = 1 and cubic graphs is given in [19]. This work on genus led to the isomorphism algorithms for graphs of fixed genus mentioned above. [4] CHORDAL GRAPH COMPLE... |

11 |
The complexity of problems in systems of communicating sequential processes
- Ladner
- 1980
(Show Context)
Citation Context ...1983]; the game involves the progressive conversion of an undirected graph to a directed one, with the goal of Player 1 being to postpone the creation of a directed cycle for as long as possible.) In =-=[27]-=- another game is used to model the ‘‘lockout’’ problem for systems of communicating processes. Such games are usually of little interest outside theirs- 7 - immediate application, however, so I shall ... |

9 |
The Ellipsoid Method: A
- Bland, Goldfarb, et al.
- 1981
(Show Context)
Citation Context ...no chain of length greater than one, so long as K is fixed at a value of 4 or more [69]. The problem for general partial orders was already known to be solvable in polynomial time when K = 2 [17,62]. =-=[8]-=- PRECEDENCE CONSTRAINED 3-PROCESSOR SCHEDULING INSTANCE: Set T of unit length tasks, partial order < . on T , and a deadline D∈Z + . QUESTION: Can T be scheduled on 3 processors so as to satisfy the p... |

9 |
On a class of totally unimodular matrices
- Yannakakis
- 1985
(Show Context)
Citation Context ...onsequence of an elegant characterization theorem proved by P. D. Seymour [63]. Continuing work has lowered the (high) order of the polynomial, both for the general problem [18] and for special cases =-=[70]-=-, and used these results to derive polynomial time algorithms for integer programming when the matrix is totally unimodular [18,57,70]. [11] COMPOSITE NUMBER INSTANCE: Positive integer N . QUESTION: A... |

8 |
On deciding switching equivalence of graphs
- Colbourn, Corneil
- 1980
(Show Context)
Citation Context ...ko improves on this to O(c [6,71], using techniques from [55]. Finally, the set of problems which have been proved polynomially equivalent to general graph isomorphism has been considerably broadened =-=[9,12,13,14]-=-, although the transformations involved may not always be able to preserve the exponents in the running times of the above algorithms. [2] SUBGRAPH HOMEOMORPHISM (FOR A FIXED GRAPH H) INSTANCE: Graph ... |

8 |
An O(max(m, n)) algorithm for finding a subgraph homeomorphic to K4
- Liu, Geldmacher
- 1980
(Show Context)
Citation Context ...rticularly concise characterization. The original two edge and triangle results mentioned in [G&J] have appeared formally in [65,44]. With respect to the ‘‘non-fixed vertex’’ versions of the problem, =-=[52]-=- presents a polynomial time algorithm for H = K 4 in the undirected case and [22] classifies a number of H (both polynomially solvable and NP-complete) in the directed case, but much remains open (a p... |

8 |
K.Winning Ways for your mathematical plays, Volume 1: Games in General
- Berlekamp, Conway, et al.
- 1982
(Show Context)
Citation Context |

8 |
Pr ivate communication
- Berman
(Show Context)
Citation Context ...hnical terms as ‘‘loony endgame’’ and ‘‘double-dealing move,’’ and in general proceeding as one might expect from authors capable of inventing ‘‘Left-Right Hackenbush for Redwood Furniture’’ [GP12].) =-=[5]-=- GENERALIZED DOTS AND BOXES INSTANCE: Planar graph G = (V,E), positive integer handicap K. QUESTION: Can Player 1 guarantee himself a score no worse than K points behind that of his opponent in the fo... |

8 |
On the diameter of permutation groups
- Driscoll, Furst
- 1977
(Show Context)
Citation Context ... (a Parker Brothers puzzle) [GP15]. Perhaps one reason why so few NP-hardness results have been proved is that many of the popular puzzles are not all that difficult. For instance, Driscoll and Furst =-=[15]-=- have shown that many puzzles based on the unraveling of permutations, for example the recent ‘‘Alexander’s Star’’ and ‘‘Hungarian Rings’’ puzzles, have group-theoretic structures that are easy to dea... |

7 |
Isomorphism for graphs embeddable on the projective plane
- Lichtenstein
- 1980
(Show Context)
Citation Context ...only if { f (u) , f (v) }∈E 2 ? Comment. This problem remains open, although some significant new subcases have been shown to be solvable in polynomial time: graphs embeddable in the projective plane =-=[51]-=-, graphs embeddable in a surface of genus K, for every fixed K [20,58], graphs with maximum vertex degree K, for every fixed K [55], and ‘‘cone graphs’’ of bounded degree (as defined in [31] - note th... |

7 |
The Complexity of Distributed Concurrency Control
- Kanellakis, Papadimitriou
- 1985
(Show Context)
Citation Context ...hough conjectured to be PSPACE-complete, is not even known to be NP-hard. 3. GAMES PEOPLE DON’T PLAY Two person games can also serve as models for practical problems about computers. For instance, in =-=[25]-=- such a game is used to model a problem in distributed concurrency control and prove that the problem is PSPACE-complete. (The problem is the DISTRIBUTED SERIALIZABILITY ASSURANCE problem mentioned in... |

5 |
Polynomial algorithms for a class of linear programs
- Maurras, Truemper, et al.
- 1981
(Show Context)
Citation Context ...the polynomial, both for the general problem [18] and for special cases [70], and used these results to derive polynomial time algorithms for integer programming when the matrix is totally unimodular =-=[18,57,70]-=-. [11] COMPOSITE NUMBER INSTANCE: Positive integer N . QUESTION: Are there positive integers p,q > 1 such that N = p . q? Comment. This problem remains open. However, further evidence that the problem... |

4 |
A nonfoctorial algorithm for testing isomorphism of two graphs
- Goldberg
- 1983
(Show Context)
Citation Context ...e for the general problem remains exponential, but there have been significant improvements over the naive O(n!) bound, where n is the number of vertices: M. K. Goldberg gives an O(c n ) algorithm in =-=[26]-=-, and V. N. Zemlyachenko improves on this to O(c [6,71], using techniques from [55]. Finally, the set of problems which have been proved polynomially equivalent to general graph isomorphism has been c... |

4 |
A note on Delaunay and optimal
- Kirkpatrick
- 1980
(Show Context)
Citation Context ... be nonoptimal when [G&J] was prepared, have now been shown to do so poorly on some instances that they may construct a triangulation whose total length is an arbitrary multiple of the optimal length =-=[41,56]-=-. I shall conclude this first column by mentioning two more ‘‘open’’ problems (which should have been on our original list, but weren’t), and by pointing out as- 9 - number of minor errors that have b... |

4 |
Blindfold Games are Harder than Games with Perfect Information
- Jones
- 1978
(Show Context)
Citation Context ... For more on the precise nature of the complexity jumps due to incomplete information, including methods of modelling the phenomena using Turing machines with ‘‘private’’ tapes, see [31,32]. Also see =-=[24]-=- for an earlier, more restricted result along these lines. For entertainment, here is one more problem from [32]. [7] BLIND PURSUIT INSTANCE: Directed graph G = (V,A), specified vertices c, m, and h, ... |

4 |
N by N checkers is EXPTIME-complete
- Robson
- 1984
(Show Context)
Citation Context ...ure yields a draw’’ is generalized to one in which the ‘‘50’’ is replaced by a fixed polynomial in N [42]; if the rule is omitted the problem becomes complete for EXPTIME [16], as does N × N CHECKERS =-=[36]-=-. (Without such a drawing rule, N × N GO is also a candidate for EXPTIMEcompleteness.) (3). N × N GOBANG (GO-MOKU) is PSPACE-complete, even if the goal of getting 5 stones in a row is retained without... |

3 |
The ellipsoid method and its consequences in combinatorial optimization, Combinatorica 1
- TSCHEL, SCHRIJVER
- 1981
(Show Context)
Citation Context ...umber of researchers [10,35]). The ellipsoid method has also been used to prove new NP-hardness results, as an equivalence between ‘‘feasability’’ and ‘‘separability’’ problems can be derived from it =-=[28,29,39]-=-. Finally, the furor over the ellipsoid method has revived interest in the theoretical aspects of the simplex method, which works so well in practice even though it has exponential worst case running ... |

3 |
Computer aided complexity classification of deterministic scheduling problems
- LAGEWEG, LENSTRA, et al.
- 1982
(Show Context)
Citation Context ...ts may qualify for explicit mention in a ‘‘general interest’’ column, but much more is known. A noteworthy example of this type of paper is the survey of scheduling results prepared by Lageweg et al. =-=[43]-=-. 3. Open problems submitted by readers. 4. Updates to the references in [G&J] and this column, reporting on the formal publication of papers originally cited as ‘‘personal communication,’’ ‘‘unpublis... |

2 |
Graph canonization in linear average time
- Babai, Kučera
- 1979
(Show Context)
Citation Context ... algorithms for dealing with permutation groups, as described in [23]. They also were foreshadowed by various randomized algorithms for the same problems, such as in [5]. In the realm of probability, =-=[7]-=- presents a method for producing canonical labellings of graphs (and hence for testings- 4 - isomorphism) which runs in expected linear time under the standard uniform distribution of graphs. The best... |

2 |
A Polynomially Bounded Algorithm for Positive Definite Symmetric LCPs
- CHUNG, MURTY
- 1979
(Show Context)
Citation Context ...n [8]. In particular, it has been used to solve the convex quadratic programming problem in polynomial time [42] as well as the linear complementarity problem for positive definite symmetric matrices =-=[11]-=- (The general LINEAR COMPLEMENTARITY problem, mentioned in the comments to this problem in [G&J], has been proved NP-complete by a number of researchers [10,35]). The ellipsoid method has also been us... |

2 |
Concerning the complexity of deciding isomorphism of block designs
- Colbourn, Colbourn
- 1981
(Show Context)
Citation Context ...ko improves on this to O(c [6,71], using techniques from [55]. Finally, the set of problems which have been proved polynomially equivalent to general graph isomorphism has been considerably broadened =-=[9,12,13,14]-=-, although the transformations involved may not always be able to preserve the exponents in the running times of the above algorithms. [2] SUBGRAPH HOMEOMORPHISM (FOR A FIXED GRAPH H) INSTANCE: Graph ... |

2 |
Expected number of steps of the simplex method for a linear program with a convexity constraint
- Dantzig
- 1980
(Show Context)
Citation Context ...hod has revived interest in the theoretical aspects of the simplex method, which works so well in practice even though it has exponential worst case running time under the standard pivoting rules. In =-=[15]-=-, G. B. Dantzig analyzes the expected running time of the simplex algorithm applied to the special case of LINEAR PROGRAMMING in which there is a convexity constraint, and shows that under reasonable ... |

2 |
An algorithm for embedding cubic graphs in the torus
- Filotti
- 1980
(Show Context)
Citation Context ... two edges cross one another? Comment. Remains open for general K, but can be solved in polynomial time for every fixed K [21]. A detailed algorithm for the case of K = 1 and cubic graphs is given in =-=[19]-=-. This work on genus led to the isomorphism algorithms for graphs of fixed genus mentioned above. [4] CHORDAL GRAPH COMPLETION INSTANCE: Graph G = (V,E) and a positive integer K . QUESTION: Is there a... |

2 |
The great mathematical Sputnik of 1979, The Sciences
- LAWLER
- 1980
(Show Context)
Citation Context ... 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 the problem which runs in polynomial time in the model of computation ... |

2 |
Primality testing algorithms (after
- Lenstra
(Show Context)
Citation Context ...ger N . QUESTION: Are there positive integers p,q > 1 such that N = p . q? Comment. This problem remains open. However, further evidence that the problem is not NP-complete comes from a new algorithm =-=[1,2,49]-=- whose running time is O(n c log(log(n)) ), where n = log(N) is the number of bits needed to represent N. If COMPOSITE NUMBER were NP-complete, all NP-complete problems could be solved in similar runn... |

1 |
A note on the complexity of LCP: the LCP is NP-complete
- CHUNG
(Show Context)
Citation Context ...m for positive definite symmetric matrices [11] (The general LINEAR COMPLEMENTARITY problem, mentioned in the comments to this problem in [G&J], has been proved NP-complete by a number of researchers =-=[10,35]-=-). The ellipsoid method has also been used to prove new NP-hardness results, as an equivalence between ‘‘feasability’’ and ‘‘separability’’ problems can be derived from it [28,29,39]. Finally, the fur... |

1 |
Isomorphism complete problems on matrices
- COLBOURN
- 1980
(Show Context)
Citation Context |

1 |
Scheduling wide graphs
- DOLEV
(Show Context)
Citation Context ...t for each integer i , 0≤i≤D − 1, there are at most 3 tasks t∈T for which σ(t) = i ? Comment. This problem can be solved in linear time if < . is the disjoint union of an in-forest with an out-forest =-=[16,25]-=-, and analogous polynomial time algorithms exist for every fixed number K of processors [25]. However, if K is arbitrary the problem becomes NP-complete even for such restricted precedence constraints... |

1 |
Seymour’s theorem and good algorithms for totally unimodular matrices, in preparation
- EDMONDS
(Show Context)
Citation Context ... in polynomial time, as a consequence of an elegant characterization theorem proved by P. D. Seymour [63]. Continuing work has lowered the (high) order of the polynomial, both for the general problem =-=[18]-=- and for special cases [70], and used these results to derive polynomial time algorithms for integer programming when the matrix is totally unimodular [18,57,70]. [11] COMPOSITE NUMBER INSTANCE: Posit... |

1 |
Scheduling opposing forests, manuscript
- GAREY, JOHNSON, et al.
- 1980
(Show Context)
Citation Context ...t for each integer i , 0≤i≤D − 1, there are at most 3 tasks t∈T for which σ(t) = i ? Comment. This problem can be solved in linear time if < . is the disjoint union of an in-forest with an out-forest =-=[16,25]-=-, and analogous polynomial time algorithms exist for every fixed number K of processors [25]. However, if K is arbitrary the problem becomes NP-complete even for such restricted precedence constraints... |

1 |
Polynomial algorithms for perfect graphs, manuscript
- TSCHEL, SCHRIJVER
- 1981
(Show Context)
Citation Context ...umber of researchers [10,35]). The ellipsoid method has also been used to prove new NP-hardness results, as an equivalence between ‘‘feasability’’ and ‘‘separability’’ problems can be derived from it =-=[28,29,39]-=-. Finally, the furor over the ellipsoid method has revived interest in the theoretical aspects of the simplex method, which works so well in practice even though it has exponential worst case running ... |

1 |
The subgraph homeomorphism problem on reducible flowgraphs, ‘‘Graph Theory and Algorithms
- HIRATU, KIMURA
- 1980
(Show Context)
Citation Context ... if H is allowed to be specified as part of the instance, as follows from [ND38] in [G&J]. The problem is also solvable in polynomial time for each fixed H if G is restricted to reducible flow graphs =-=[30]-=-. With respect to the undirected version of the problem, new polynomial time algorithms for the case where H consists of two independent edges have been presented in [59,64], with the latter giving a ... |

1 |
Testing isomorphisms of cone graphs
- HOFFMANN
(Show Context)
Citation Context ...ive plane [51], graphs embeddable in a surface of genus K, for every fixed K [20,58], graphs with maximum vertex degree K, for every fixed K [55], and ‘‘cone graphs’’ of bounded degree (as defined in =-=[31]-=- - note that the ‘‘degree’’ of a cone graph is not its vertex degree), using the techniques from [55]. In addition, ‘‘colored’’ graphs, in which each vertex is assigned a color, can be tested for isom... |

1 |
Algorithms: Design and Analysis
- HOROWITZ, SAHNI
- 1978
(Show Context)
Citation Context |

1 |
private communication
- HUBERMAN
- 1979
(Show Context)
Citation Context ...m for positive definite symmetric matrices [11] (The general LINEAR COMPLEMENTARITY problem, mentioned in the comments to this problem in [G&J], has been proved NP-complete by a number of researchers =-=[10,35]-=-). The ellipsoid method has also been used to prove new NP-hardness results, as an equivalence between ‘‘feasability’’ and ‘‘separability’’ problems can be derived from it [28,29,39]. Finally, the fur... |

1 |
The partial order dimension problem is NP-complete, manuscript
- LAWLER, VORNBERGER
- 1981
(Show Context)
Citation Context ...a collection of k≤K linear orderings of V such that (u,v) ∈A if and only if u is less than v in each of the orderings? Comment. This has been proved NP-complete independently by Lawler and Vornberger =-=[47]-=- and by M. Yannakakis [69], although the Yannakakis result is stronger. It shows that NP-completeness holds for every fixed K ≥ 3, whereas the other result was only for arbitrary K. The proof is via a... |

1 |
NP-complete problem number 798016, manuscript
- LEVEN, GALIL
- 1981
(Show Context)
Citation Context ...no two edges in E i share a common endpoint in G? Comment. This problem has been proved NP-complete by I. Holyer [32], via a transformation from 3SAT, even for K = 3 and cubic graphs. Leven and Galil =-=[50]-=- have shown that NP-completeness also holds for every fixed K > 3, even if the graphs are required to be K-regular (a surprisingly non-trivial extension). [6] SPANNING TREE PARITY PROBLEM INSTANCE: Gr... |

1 |
Subtree homeomorphism and tree contractability, manuscript
- VALDES
- 1980
(Show Context)
Citation Context ... implies a polynomial time non-fixed vertex algorithm for H, but not vice versa). The special case where G is a tree can be solved in polynomial time even when H is not fixed but is part of the input =-=[66]-=-. n 2/3 )s[3] GRAPH GENUS - 5 - INSTANCE: Graph G = (V,E) and a non-negative integer K . QUESTION: Can G be embedded on a surface of genus K such that no two edges cross one another? Comment. Remains ... |

1 |
private communication
- WARMUTH
- 1981
(Show Context)
Citation Context ...n also be solved in polynomial time for every fixed K if < . is a level order (any two tasks with a commons- 7 - immediate predecessor or successor have identical sets of predecessors and successors) =-=[67]-=-, and for arbitrary K if < . is an interval order (each task T i corresponds to an interval [a i ,b i ] on the real line, and T i < . T j if and only if b i < a j) [61]. Remains NP-complete for arbitr... |

1 |
Low level complexity for combinatorial games
- ADACHI, IWATA, et al.
- 1981
(Show Context)
Citation Context ...nes. Turning from puzzles of the 80’s to puzzles of the 70’s (1870’s, that is − see [18])*, we come to the famous ‘‘15’’ puzzle of Sam Loyd [4,18], and a generalization that does lead to NP-hardness. =-=[1]-=- SLIDING BLOCK REARRANGEABILITY INSTANCE: Polygonal region R, collection C = {b 1 , . . . , b n } of rectangular blocks, each block b i having an associated integer height h i and width w i, along wit... |

1 |
The Way to Play,’’ Bantam
- GROUP
- 1977
(Show Context)
Citation Context ...t, at least in their ungeneralized form, have actually been played. I shall begin with board games (readers unfamiliar with the rules of the following games should consult a general reference such as =-=[14]-=-). As evidenced by the results presented in [G&J], there are three progressively more general (ands- 8 - progressively less natural) ways of taking a standard board game to asymptopia: (1) Allow the d... |

1 |
private communication
- FURST
- 1983
(Show Context)
Citation Context ...ucts of generators. However, their results do not yet extend to Rubik’s Cube®, whose group generators (the slice-wise rotations of the cube) are products of cycles involving the faces of the subcubes =-=[17]-=-. Thus, although the 3 × 3 × 3, 4 × 4 × 4, and 5 × 5 × 5 versions are all well-understood by the cognoscenti, it is not yet known whether the techniques generalize to efficient algorithms. Fortunately... |

1 |
PSPACE-hardness of the motion planning problem for rectangular objects, manuscript
- HOPCROFT, SCHWARTZ, et al.
- 1983
(Show Context)
Citation Context ...the coordinated motion of ensembles of robots, and the current results are extended from rectangular robots to circular ones in [39]. If non-convex robots are allowed, the problem becomes PSPACE-hard =-=[22]-=-. (The problem for one robot is solvable in polynomial time, as was mentioned in the Column 6 [March 1983] discussion of ROBOT ARM REACHABILITY.) A more tractable generalization of the 15-puzzle is th... |

1 |
Black-white pebbles and graph separation, manuscript
- LENGAUER
- 1980
(Show Context)
Citation Context ...red by ordinary machines. However, the complexity of determining just how many registers (pebbles) suffice remains open, except in the case where re-pebbling is forbidden. Here NP-completeness reigns =-=[28]-=-, just as it did in the monochromatic problem. With re-pebbling allowed, the problem, although conjectured to be PSPACE-complete, is not even known to be NP-hard. 3. GAMES PEOPLE DON’T PLAY Two person... |

1 | The complexity of two player games of incomplete information - REIF |

1 |
Gobang ist PSPACE-vollst
- REISCH
- 1980
(Show Context)
Citation Context ...a drawing rule, N × N GO is also a candidate for EXPTIMEcompleteness.) (3). N × N GOBANG (GO-MOKU) is PSPACE-complete, even if the goal of getting 5 stones in a row is retained without generalization =-=[34]-=-. (4). GENERALIZED CHINESE CHECKERS (played on arbitrary graphs) is EXPTIME-complete, via a transformation from YET ANOTHER PEBBLE GAME [26], although the planar and N × N versions (if a star-shaped b... |