Classifying Problems into Complexity Classes

Abstract A fundamental problem in computer science is stated informally as: Given a problem, how hard is it? We measure hardness by looking at the following question: Given a set A what is the fastest algorithm to determine if “x ∈ A?” We measure the speed of an algorithm by how long it takes to run on inputs of length n, as a function of n. For example, sorting a list of length n can be done in roughly n log n steps. Obtaining a fast algorithm is only half of the problem. Can you prove that there is no better algorithm? This is notoriously difficult; however, we can classify problems into complexity classes where those in the same class are roughly equally hard. In this chapter, we define many complexity classes and describing natural problems that are in them. Our classes go all the way from regular languages to various shades of undecidable. We then summarize all that is known about these classes.

[1]  Meena Mahajan,et al.  The complexity of planarity testing , 2000, Inf. Comput..

[2]  Richard Ryan Williams,et al.  Time-Space Tradeoffs for Counting NP Solutions Modulo Integers , 2007, Twenty-Second Annual IEEE Conference on Computational Complexity (CCC'07).

[3]  Ge Xia,et al.  Improved upper bounds for vertex cover , 2010, Theor. Comput. Sci..

[4]  László Babai,et al.  Graph isomorphism in quasipolynomial time [extended abstract] , 2015, STOC.

[5]  David J. Rosenbaum.  Bidirectional Collision Detection and Faster Deterministic Isomorphism Testing , 2013, ArXiv.

[6]  Stephen R. Mahaney.  Sparse Complete Sets of NP: Solution of a Conjecture of Berman and Hartmanis , 1982, J. Comput. Syst. Sci..

[7]  Nutan Limaye,et al.  A Log-space Algorithm for Canonization of Planar Graphs , 2008, ArXiv.

[8]  Scott Aaronson,et al.  The Complexity Zoo , 2008 .

[9]  J. Ferrante,et al.  The computational complexity of logical theories , 1979 .

[10]  John Gill,et al.  Relativizations of the P =? NP Question , 1975, SIAM J. Comput..

[11]  Dana S. Scott,et al.  Finite Automata and Their Decision Problems , 1959, IBM J. Res. Dev..

[12]  Eugene M. Luks,et al.  Isomorphism of graphs of bounded valence can be tested in polynomial time , 1980, 21st Annual Symposium on Foundations of Computer Science (sfcs 1980).

[13]  Nellie Clarke Brown.  Trees , 1896, Savage Dreams.

[14]  J. Büchi.  Weak Second‐Order Arithmetic and Finite Automata , 1960 .

[15]  Johan Håstad,et al.  Almost optimal lower bounds for small depth circuits , 1986, STOC '86.

[16]  W. Browder,et al.  Annals of Mathematics , 1889 .

[17]  Bernard Chazelle,et al.  A minimum spanning tree algorithm with inverse-Ackermann type complexity , 2000, JACM.

[18]  Elvira Mayordomo,et al.  P versus NP , 2004 .

[19]  Michael Sipser,et al.  Expanders, Randomness, or Time versus Space , 1988, J. Comput. Syst. Sci..

[20]  Edsger W. Dijkstra,et al.  A note on two problems in connexion with graphs , 1959, Numerische Mathematik.

[21]  Stephen R. Mahaney.  Sparse complete sets for NP: Solution of a conjecture of Berman and Hartmanis , 1980, 21st Annual Symposium on Foundations of Computer Science (sfcs 1980).

[22]  Noam Nisan,et al.  Hardness vs Randomness , 1994, J. Comput. Syst. Sci..

[23]  A. Yao.  Separating the polynomial-time hierarchy by oracles , 1985 .

[24]  Lane A. Hemaspaandra.  SIGACT news complexity theory column 36 , 2002, SIGA.

[25]  M. Fischer,et al.  STRING-MATCHING AND OTHER PRODUCTS , 1974 .

[26]  G. I. Kustova,et al.  From the author , 2019, Automatic Documentation and Mathematical Linguistics.

[27]  Omer Reingold,et al.  Undirected connectivity in log-space , 2008, JACM.

[28]  Samuel R. Buss,et al.  Limits on Alternation Trading Proofs for Time–Space Lower Bounds , 2012, computational complexity.

[29]  Vasek Chvátal,et al.  A Greedy Heuristic for the Set-Covering Problem , 1979, Math. Oper. Res..

[30]  Peter W. Shor,et al.  Algorithms for quantum computation: discrete logarithms and factoring , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[31]  Stephen Warshall,et al.  A Theorem on Boolean Matrices , 1962, JACM.

[32]  M. Rabin.  Decidability of second-order theories and automata on infinite trees. , 1969 .

[33]  Dana Moshkovitz,et al.  The Projection Games Conjecture and the NP-Hardness of ln n-Approximating Set-Cover , 2012, Theory Comput..

[34]  Alexander A. Razborov,et al.  Natural Proofs , 2007 .

[35]  Albert R. Meyer,et al.  WEAK MONADIC SECOND ORDER THEORY OF SUCCESSOR IS NOT ELEMENTARY-RECURSIVE , 1973 .

[36]  Leslie G. Valiant,et al.  The Complexity of Computing the Permanent , 1979, Theor. Comput. Sci..

[37]  Ronald de Sousa.  Why Do We Think , 2007 .

[38]  Chris Arney,et al.  The Honors Class: Hilbert's Problems and Their Solvers , 2005 .

[39]  William I. Gasarch,et al.  An NP-Complete Problem in Grid Coloring , 2012, ArXiv.

[40]  Yuri Gurevich,et al.  Trees, automata, and games , 1982, STOC '82.

[41]  Michael Sipser,et al.  A complexity theoretic approach to randomness , 1983, STOC.

[42]  Dennis Hinkamp,et al.  Don't Get Fooled Again! , 2003 .

[43]  Walter J. Savitch,et al.  Relationships Between Nondeterministic and Deterministic Tape Complexities , 1970, J. Comput. Syst. Sci..

[44]  Carl Pomerance,et al.  A Tale of Two Sieves , 1998 .

[45]  J. Hartmanis,et al.  On the Computational Complexity of Algorithms , 1965 .

[46]  K. Popper,et al.  The Logic of Scientific Discovery , 1960 .

[47]  L. G. H. Cijan.  A polynomial algorithm in linear programming , 1979 .

[48]  H. Putnam,et al.  The Decision Problem for Exponential Diophantine Equations , 1961 .

[49]  M. Kendall,et al.  The Logic of Scientific Discovery. , 1959 .

[50]  Narendra Karmarkar,et al.  A new polynomial-time algorithm for linear programming , 1984, Comb..

[51]  Volker Strassen,et al.  A Fast Monte-Carlo Test for Primality , 1977, SIAM J. Comput..

[53]  Yuri Gurevich,et al.  The Classical Decision Problem , 1997, Perspectives in Mathematical Logic.

[54]  David Buchfuhrer,et al.  The complexity of Boolean formula minimization , 2008, J. Comput. Syst. Sci..

[55]  Roman Smolensky,et al.  Algebraic methods in the theory of lower bounds for Boolean circuit complexity , 1987, STOC.

[56]  J. Kruskal.  On the shortest spanning subtree of a graph and the traveling salesman problem , 1956 .

[57]  Avi Wigderson,et al.  Algebrization: A New Barrier in Complexity Theory , 2009, TOCT.

[58]  Albert R. Meyer,et al.  The Equivalence Problem for Regular Expressions with Squaring Requires Exponential Space , 1972, SWAT.

[59]  Ryan Williams.  Alternation-Trading Proofs, Linear Programming, and Lower Bounds , 2010, STACS.

[60]  Kenneth W. Regan,et al.  People, Problems, and Proofs: Essays from Gödel's Lost Letter: 2010 , 2013 .

[61]  William I. Gasarch,et al.  Guest Column: the second P =?NP poll , 2012, SIGA.

[62]  Kazuya Kato,et al.  Number Theory 1 , 1999 .

[63]  Michael Sipser,et al.  Expanders, Randomness, or Time versus Space , 1988, Journal of computer and system sciences (Print).

[64]  Seinosuke Toda,et al.  PP is as Hard as the Polynomial-Time Hierarchy , 1991, SIAM J. Comput..

[65]  Stephen A. Cook,et al.  Problems Complete for Deterministic Logarithmic Space , 1987, J. Algorithms.

[66]  Jan van Leeuwen,et al.  Worst-case Analysis of Set Union Algorithms , 1984, JACM.

[67]  S. Sieber.  On a decision method in restricted second-order arithmetic , 1960 .

[68]  Richard E. Ladner,et al.  On the Structure of Polynomial Time Reducibility , 1975, JACM.

[69]  Manindra Agrawal,et al.  PRIMES is in P , 2004 .

[70]  Ketan Mulmuley,et al.  On P vs. NP and geometric complexity theory: Dedicated to Sri Ramakrishna , 2011, JACM.

[71]  R. Prim.  Shortest connection networks and some generalizations , 1957 .

[72]  Clemens Lautemann,et al.  BPP and the Polynomial Hierarchy , 1983, Inf. Process. Lett..

[73]  Ryan Williams.  Nonuniform ACC Circuit Lower Bounds , 2014, JACM.

[74]  M. Rabin.  Probabilistic algorithm for testing primality , 1980 .

[75]  Leslie G. Valiant,et al.  The Complexity of Enumeration and Reliability Problems , 1979, SIAM J. Comput..

[76]  R. Solovay,et al.  Relativizations of the $\mathcal{P} = ?\mathcal{NP}$ Question , 1975 .

[77]  J. Paris,et al.  Accessible Independence Results for Peano Arithmetic , 1982 .