47 July 2000
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Proceedings 15th Annual IEEE Conference on Computational Complexity
Publication Year: 2000 PDF (178 KB) 
The complexity of tensor calculus
Publication Year: 2000, Page(s):70  86Tensor calculus over semirings is shown relevant to complexity theory in unexpected ways. First, evaluating well formed tensor formulas with explicit tensor entries is shown complete for /spl oplus/P, for NP, and for #P as the semiring varies. Indeed the permanent of a matrix is shown expressible as the value of a tensor formula in much the same way that Berkowitz' theorem expresses its determinan... View full abstract»

Average case complexity of unbounded fanin circuits
Publication Year: 2000, Page(s):170  185Several authors have shown that the PARITYfunction cannot be computed by unbounded fanin circuits of small depth and polynomial size. Even more, constant depth k circuits of size exp(n/sup /spl ominus/(1/k)/) give wrong results for PARITY for almost half of all inputs. We generalize these results in two directions. First, we obtain similar tight lower bounds for the average case complexity of cir... View full abstract»

Author index
Publication Year: 2000, Page(s): 279 PDF (56 KB) 
On the complexity of some problems on groups input as multiplication tables
Publication Year: 2000, Page(s):62  69
Cited by: Papers (1)The Cayley group membership problem (CGM) is to input a groupoid (binary algebra) G given as a multiplication table, a subset X of G, and an element t of G, and to determine whether t can be expressed as a product of elements of X. For general groupoids CGM is Pcomplete, and for associative algebras (semigroups) it is NLcomplete. Here we investigate CGM for particular classes of groups. The prob... View full abstract»

Combinatorial interpretation of Kolmogorov complexity
Publication Year: 2000, Page(s):131  137
Cited by: Papers (1)The very first Kolmogorov's paper on algorithmic information theory was entitled “Three approaches to the definition of the quantity of information”. These three approaches were called combinatorial, probabilistic and algorithmic. Trying to establish formal connections between combinatorial and algorithmic approaches, we prove that every linear inequality including Kolmogorov complexit... View full abstract»

The query complexity of orderfinding
Publication Year: 2000, Page(s):54  59We consider the problem where π is an unknown permutation on (0, 1,..., 2n1), γ0∈(0, 1,..., 2n 1), and the goal is to determine the minimum r>0 such that πr(y0)=y0. Information about π is available only via queries that yield πx(y) from any x∈(0, 1,..., 2n1) and γ&is... View full abstract»

Timespace lower bounds for SAT on uniform and nonuniform machines
Publication Year: 2000, Page(s):22  33
Cited by: Papers (3)The arguments used by R. Kannan (1984), L. Fortnow (1997), and LiptonViglas (1999) are generalized and combined with a new argument for diagonalizing over machines taking n bits of advice on inputs of length n to obtain the first nontrivial timespace lower bounds for SAT on nonuniform machines. In particular we show that for any a <√2 and any b<1, SAT cannot be computed by a random... View full abstract»

New bounds for the language compression problem
Publication Year: 2000, Page(s):126  130
Cited by: Papers (5)The CD complexity of a string x is the length of the shortest polynomial time program which accepts only the string x. The language compression problem consists of giving an upper bound on the CD(A⩽n) complexity of all strings x in some set A. The best known upper bound for this problem is 2log(A⩽n)+O(log(n)), due to Buhrman and Fortnow. We show that the constant ... View full abstract»

Three approaches to the quantitative definition of information in an individual pure quantum state
Publication Year: 2000, Page(s):263  270
Cited by: Papers (2)In analogy of classical Kolmogorov complexity we develop a theory of the algorithmic information in bits contained in any one of continuously many pure quantum states: quantum Kolmogorov complexity. Classical Kolmogorov complexity coincides with the new quantum Kolmogorov complexity restricted to the classical domain. Quantum Kolmogorov complexity is upper bounded and can be effectively approximat... View full abstract»

The complexity of verifying the characteristic polynomial and testing similarity
Publication Year: 2000, Page(s):87  95
Cited by: Papers (1)We investigate the computational complexity of some important problems in linear algebra. 1. The problem of verifying the characteristic polynomial of a matrix is known to be in the complexity class C=L (Exact Counting in Logspace). We show that it is complete for C=L under logspace manyone reductions. 2. The problem of deciding whether two matrices are similar is known to b... View full abstract»

The communication complexity of enumeration, elimination, and selection
Publication Year: 2000, Page(s):44  53
Cited by: Papers (1)Let f:{0, 1}n×{0, 1}n→{0, 1}. Assume Alice has x1, ..., xk∈{0, 1}n , Bob has y1, ..., yk∈{0, 1}n, and they want to compute f(x1, y1)···f(xk, yk) communicating as few bits as possible. The Direct Sum Conjecture of Karchmer, Raz... View full abstract»

A lower bound for the shortest path problem
Publication Year: 2000, Page(s):14  21
Cited by: Papers (1)We show that the shortest path problem cannot be solved in o(log n) time on an unbounded fanin PRAM without bit operations using poly(n) processors even when the bitlengths of the weights on the edges are restricted to be of size O(log3 n). This shows that the matrixbased repeated squaring algorithm for the shortest path problem is optimal in the unbounded fanin PRAM model without b... View full abstract»

An application of matroid theory to the SAT problem
Publication Year: 2000, Page(s):116  124
Cited by: Papers (18)We consider the deficiency δ(F):=c(F)n(F) and the maximal deficiency δ*(F):=maxF'⊆Fδ(F) of a clauseset F (a conjunctive normal form), where c(F) is the number of clauses in F and n(F) is the number of variables. Combining ideas from matching and matroid theory with techniques from the area of resolution refutations, we prove that for clausesets F wit... View full abstract»

On the complexity of quantum ACC
Publication Year: 2000, Page(s):250  262
Cited by: Papers (1)For any q>1, let MODq be a quantum gate that determines if the number of 1's in the input is divisible by q. We show that for any q, t>1, MODP is equivalent to MODt (up to constant depth). Based on the case q=2, C. Moore (1999) has shown that quantum analogs of AC(0), ACC[q], and ACC, denoted QACwf (0) QACC[2], QACC respectively, define ... View full abstract»

Dimension in complexity classes
Publication Year: 2000, Page(s):158  169
Cited by: Papers (6)A theory of resourcebounded dimension is developed using gales, which are natural generalizations of martingales. When the resource bound Δ(a parameter of the theory) is unrestricted, the resulting dimension is precisely the classical Haludolff dimension (sometimes called “fractal dimension”). Other choices of the parameter Δ yield internal dimension theories in E, E... View full abstract»

On the complexity of intersecting finite state automata
Publication Year: 2000, Page(s):229  234
Cited by: Papers (1)We consider the problem of testing whether the intersection of a collection of k automata is empty. The straightforward algorithm for solving this problem runs in time σk where a is the size of the automata. In this work we prove that the assumption that there exists a better algorithm solving the FSA intersection emptiness problem implies that nondeterministic time is in subexpon... View full abstract»

Independent minimum length programs to translate between given strings
Publication Year: 2000, Page(s):138  144
Cited by: Papers (2)A string p is called a program to compute y given x if U(p, x)=y, where U denotes universal programming language. Kolmogorov complexity K(yz) of y relative to x is defined as minimum length of a program to compute y given x. Let K(x) denote K(xempty string) (Kolmogorov complexity of x) and let I(x: y)=K(x)+K(y)K(⟨x, y⟩) (the amount of mutual information in x, y). In the present paper ... View full abstract»

BP(f)=O(L(f)^{1+ε})
Publication Year: 2000, Page(s):36  43Any B2formula of size L can be transformed into a branching program of size O(eL1+ε)for arbitrary E>0. The presented proof is based on a technique due to R. Cleve (1991) to simulate balanced algebraic formulas of size s by algebraic straightline programs that employ a constant number of registers and have length O(s1+ε). The best previously known sim... View full abstract»

Characterization of nondeterministic quantum query and quantum communication complexity
Publication Year: 2000, Page(s):271  278
Cited by: Papers (2)It is known that the classical and quantum query complexities of a total Boolean function f are polynomially related to the degree of its representing polynomial, but the optimal exponents in these relations are unknown. We show that the nondeterministic quantum query complexity of f is linearly related to the degree of a “nondeterministic” polynomial for f. We also prove a quantumc... View full abstract»

Timespace tradeoffs for nondeterministic computation
Publication Year: 2000, Page(s):2  13
Cited by: Papers (11)We show new tradeoffs for satisfiability and nondeterministic linear time. Satisfiability cannot be solved on general purpose randomaccess Turing machines in time n1.618 and space no(1). This improves recent results of Fortnow and of Lipton and Viglas. In general, for any constant a less than the golden ratio, we prove that satisfiability cannot be solved in time na View full abstract»

A dual version of Reimer's inequality and a proof of Rudich's conjecture
Publication Year: 2000, Page(s):98  103
Cited by: Papers (7)We prove a dual version of the celebrated inequality of D. Reimer (a.k.a. the van den BergKesten conjecture). We use the dual inequality to prove a combinatorial conjecture of S. Rudich motivated by questions in cryptographic complexity. One consequence of Rudich's Conjecture is that there is an oracle relative to which oneway functions exist but oneway permutations do not. The dual inequality ... View full abstract»

On the complexity of the monotonicity verification
Publication Year: 2000, Page(s):235  238We present a sequence {S_{n}} of circuits of functional elements that check the monotonicity of input discrete functions depending on n variables and represented as value column vectors. The complexity of circuits equals O(N√log log N). This is the lowest asymptotical current complexity View full abstract»

A survey of optimal PCP characterizations of NP
Publication Year: 2000, Page(s):146  148Probabilistically checkable proofs (PCPs) define a model of computation that is quite interesting in its own right, and that is an extremely powerful tool to study the complexity of finding approximate solutions for combinatorial optimization. Since U. Feige et al. (1996) suggested a connection between proofchecking and approximation, this connection has been generalized and exploited to an amazi... View full abstract»

Computational complexity and phase transitions
Publication Year: 2000, Page(s):104  115
Cited by: Papers (3)Phase transitions in combinatorial problems have recently been shown to be useful in locating “hard” instances of combinatorial problems. The connection between computational complexity and the existence of phase transitions has been addressed in statistical mechanics and artificial intelligence, but not studied rigorously. We take a first step in this direction by investigating the ex... View full abstract»