Starting with Cook's pioneering work on NP-completeness in 1970, polynomial complexity theory, the study of polynomial-time com putability, has quickly emerged as the new foundation of algorithms. On the one hand, it bridges the gap between the abstract approach of recursive function theory and the concrete approach of analysis of algorithms. It extends the notions and tools of the theory of computability to provide a solid theoretical foundation for the study of computational complexity of practical problems. In addition, the theoretical studies of the notion of polynomial-time tractability some times also yield interesting new practical algorithms. A typical exam ple is the application of the ellipsoid algorithm to combinatorial op timization problems (see, for example, Lovasz [1986]). On the other hand, it has a strong influence on many different branches of mathe matics, including combinatorial optimization, graph theory, number theory and cryptography. As a consequence, many researchers have begun to re-examine various branches of classical mathematics from the complexity point of view. For a given nonconstructive existence theorem in classical mathematics, one would like to find a construc tive proof which admits a polynomial-time algorithm for the solution. One of the examples is the recent work on algorithmic theory of per mutation groups. In the area of numerical computation, there are also two tradi tionally independent approaches: recursive analysis and numerical analysis."

Complexity theory of real functions

We present exponential lower bounds on the size of depth-k Boolean circuits for computing certain functions. These results imply that there exists an oracle set A such that, relative to A, all the levels in the polynomial-time hierarchy are distinct, i.e., ΣkP,A is properly contained in Σk+1P,A for all k.

Separating the polynomial-time hierarchy by oracles

This is the second volume of a systematic two-volume presentation of the various areas of research in the field of structural complexity. The mathematical theory of computation has developed into a broad and rich discipline within which the theory of algorithmic complexity can be approached from several points of view. This volume is addressed to graduate students and researchers and assumes knowledge of the topics treated in the first volume but is otherwise nearly self-contained. Topics covered include vector machines, parallel computation, alternation, uniform circuit complexity, isomorphism, biimmunity and complexity cores, relativization and positive relativization, the low and high hierarchies, Kolmogorov complexity and probability classes. Numerous exercises and references are given.

Structural Complexity II

Separating the Polynomial-Time Hierarchy by Oracles (Preliminary Version)

Metric learning algorithms can provide useful distance functions for a variety of domains, and recent work has shown good accuracy for problems where the learner can access all distance constraints at once. However, in many real applications, constraints are only available incrementally, thus necessitating methods that can perform online updates to the learned metric. Existing online algorithms offer bounds on worst-case performance, but typically do not perform well in practice as compared to their offline counterparts. We present a new online metric learning algorithm that updates a learned Mahalanobis metric based on LogDet regularization and gradient descent. We prove theoretical worst-case performance bounds, and empirically compare the proposed method against existing online metric learning algorithms. To further boost the practicality of our approach, we develop an online locality-sensitive hashing scheme which leads to efficient updates to data structures used for fast approximate similarity search. We demonstrate our algorithm on multiple datasets and show that it outperforms relevant baselines.

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Online Metric Learning and Fast Similarity Search

Consider the following open problems: (i) ${\text{P}} = ? {\text{ NP}}$ (ii) ${\text{NP}} = ? {\text{ co-NP}}$; (iii) ${\text{P}} = ? {\text{ PSPACE}}$; (iv) ${\text{NP}} = ? {\text{ PSPACE}}$. In this paper we study these four problems from a particular point of view. To illustrate our approach, consider the first problem. It is known that there exist recursive sets A and B such that ${\text{P}}(A) = {\text{NP}}(A)$ and ${\text{P}}(B) \ne {\text{NP}}(B)$. We study restrictions R on both the deterministic and also the nondeterministic polynomial time-bounded oracle machines such that the following holds: ${\text{P}} = {\text{NP}}$ if and only if for every set A, ${\text{P}}_R (A) = {\text{NP}}_R (A)$. The restrictions are “quantitative” in the sense that the size of the set of strings queried by the oracle in computations of a machine on an input is bounded by a polynomial in the length of the input. We study several different ways of specifying such quantitative restrictions, each of which has the desire...

Quantitative relativizations of complexity classes

A restricted relativization of NP, denoted ${\text{NP}}_B ( \cdot )$,was introduced in [SIAM J. Comput., 13 (1984), pp. 461– 487] in the study of positive relativizations on the ${\text{P}} = ?{\text{ NP}}$ question. The ${\text{NP}}_B ( \cdot )$ restriction allows nondeterministic polynomial-time oracle machines to query only polynomially many strings in its computation trees. In this paper we compare the language classes ${\text{NP}}_B (A)$, relative to arbitrary sets A, with the language classes $\text{NP}(S)$, relative to sparse sets S, showing that it is not always possible to obtain a class specified by ${\text{NP}}_B (A)$ as an $\text{NP}(S)$ class and vice versa. As a corollary to these results, we prove that there is a sparse set S such that for all tally sets T, $S \not\equiv_T^{\text{SN}} T$. This implies that the relationship established in [5] that for every sparse set S there is a tally set T such that $S \equiv _T^{\text{NP}} T$ cannot be improved to any of the strong nondeterministic polyn...

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Erratum: On restricting the size of Oracles compared with restricting access to Oracles

A note on sparse oracles for NP

It is proved that there exists, for every set B e NP − co-NP, a set A ∋ NP ⌣ co-NP such that A and B have the same deterministic time complexity. Note that B is easier, nondeterministically, than both A and AĀ in that B e NP while A ∋ NP ⌣ co-NP. We also prove there exists, for each B ∋ NP − co-NP, a set A ∋ NP ⌣ co-NP such that the deterministic time complexity of A is less than that of B on some infinite set. In this case then, A is easier than B deterministically while B is easier than A nondeterministically.

Timothy J. Long

Papers

Quantitative relativizations of complexity classes

Erratum: On restricting the size of Oracles compared with restricting access to Oracles

A note on sparse oracles for NP

A note on deterministic and nondeterministic time complexity

Controlled relativizations of p and np.