Matrix Transformation Is Complete for the Average Case

In the theory of worst case complexity, NP completeness is used to establish that, for all practical purposes, the given NP problem is not decidable in polynomial time. In the theory of average case complexity, average case completeness is supposed to play the role of NP completeness. However, the average case reduction theory is still at an early stage, and only a few average case complete problems are known. We present the first algebraic problem complete for the average case under a natural probability distribution. The problem is this: Given a unimodular matrix $X$ of integers, a set $S$ of linear transformations of such unimodular matrices and a natural number $n$, decide if there is a product of $\leq n$ (not necessarily different) members of $S$ that takes $X$ to the identity matrix.

[1]  Ramarathnam Venkatesan,et al.  Average case intractability of matrix and diophantine problems (extended abstract) , 1992, STOC '92.

[2]  Leonid A. Levin,et al.  Average Case Complete Problems , 1986, SIAM J. Comput..

[3]  Leonid A. Levin,et al.  Random instances of a graph coloring problem are hard , 1988, STOC '88.

[4]  E. Wright,et al.  An Introduction to the Theory of Numbers , 1939 .

[5]  Yuri Gurevich,et al.  Average Case Complexity ∗ , 1998 .

[6]  Samuel Eilenberg,et al.  Automata, languages, and machines. A , 1974, Pure and applied mathematics.

[7]  Yuri Gurevich.  Matrix decomposition problem is complete for the average case , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[8]  Saharon Shelah,et al.  Expected Computation Time for Hamiltonian Path Problem , 1987, SIAM J. Comput..

[9]  Yuri Gurevich,et al.  Average Case Completeness , 1991, J. Comput. Syst. Sci..

[10]  Donald E. Knuth,et al.  Big Omicron and big Omega and big Theta , 1976, SIGA.

[11]  E. T..  An Introduction to the Theory of Numbers , 1946, Nature.

[12]  Donald Ervin Knuth,et al.  The Art of Computer Programming , 1968 .

[13]  Andreas Blass,et al.  On the Reduction Theory for Average Case Complexity , 1990, CSL.

[14]  Andrew Chi-Chih Yao,et al.  Analysis of the subtractive algorithm for greatest common divisors , 1975, SIGS.

[15]  A. Yao,et al.  Analysis of the subtractive algorithm for greatest common divisors , 1975 .

[16]  Oded Goldreich,et al.  On the Theory of Average Case Complexity (abstract) , 1989, SCT.

[17]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[18]  Leonid A. Levin,et al.  No better ways to generate hard NP instances than picking uniformly at random , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[19]  Andreas Blass,et al.  Randomizing Reductions of Search Problems , 1993, SIAM J. Comput..

[20]  Hofreiter.  Gruppen von linearen Transformationen , 1935 .

[21]  Rolf Nevanlinna.  Gruppen von linearen Transformationen , 1967 .

[22]  Charles F. Miller.  Decision Problems in Algebraic Classes of Groups (A Survey) , 1973 .

[23]  室 章治郎.  Michael R.Garey/David S.Johnson 著, "COMPUTERS AND INTRACTABILITY A guide to the Theory of NP-Completeness", FREEMAN, A5判変形判, 338+xii, \5,217, 1979 , 1980 .

[24]  Oded Goldreich,et al.  On the theory of average case complexity , 1989, STOC '89.