Steven Homer

Oracle-dependent properties of the lattice of NP sets

Abstract We consider under the assumption P ≠ NP questions concerning the structure of the lattice of NP sets together with the sublattice P. We show that two questions which are slightly more complex than the known splitting properties of this lattice cannot be settled by arguments which relativize. The two questions which we consider are whether every infinite NP set contains an infinite P subset and whether there exists an NP-simple set. We construct several oracles, all of which make P ≠ NP, and which in addition make the above-mentioned statements either true or false. In particular we give a positive answer to the question, raised by Bennett and Gill (1981), whether an oracle B exists making P B ≠ NP B and such that every infinite set in NP B has an infinite subset in P B . The constructions of the oracles are finite injury priority arguments.

Design and Performance of Parallel and Distributed Approximation Algorithms for Maxcut

We develop and experiment with a new parallel algorithm to approximate the maximum weight cut in a weighted undirected graph. Our implementation starts with the recent (serial) algorithm of Goemans and Williamson for this problem. We consider several different versions of this algorithm, varying the interior-point part of the algorithm in order to optimize the parallel efficiency of our method. Our work aims for an efficient, practical formulation of the algorithm with close-to-optimal parallelization. We analyze our parallel algorithm in the LogP model and predict linear speedup for a wide range of the parameters. We have implemented the algorithm using the message passing interface (MPI) and run it on several parallel machines. In particular, we present performance measurements on the IBM SP2, the Connection Machine CM5, and a cluster of workstations. We observe that the measured speedups are predicted well by our analysis in the LogP model. Finally, we test our implementation on several large graphs (up to 13,000 vertices), particularly on large instances of the Ising model.

Superpolynomial Circuits, Almost Sparse Oracles and the Exponential Hierarchy

Several problems concerning superpolynomial size circuits and superpolynomial-time advice classes are investigated. First we consider the implications of NP (and other fundamental complexity classes) having circuits slighter bigger than polynomial. We prove that if such circuits exist, for example if NP has n logn size circuits, the exponential hierarchy collapses to the second level. Next we consider the consequences of the bottom levels of the exponential hierarchy being contained in small advice classes. Again various collapses result. For example, if EXP NP \(\subseteq\)EXP/poly then EXP NP =EXP.

Complete problems and strong polynomial reducibilities

A set A is m-reducible to a set B if and only if there is a polynomial-time computable function f such that for all x,

On 1-truth-table-hard languages

x \in A \Leftrightarrow f(x) \in B

Finding a Hidden Code by Asking Questions

. A set C is m-complete for a class S if

Structural properties of nondeterministic complete sets

C \in S

Almost-everywhere complexity hierarchies for nondeterministic time

and all sets in S are m-reducible to C. One-reducibility and one-completeness can be defined by requiring f to be one–one. Two sets A and B are p-isomorphic if the function f can be taken one-to-one, onto, and polynomially invertible. In this paper it is shown that all the m-complete sets are one–one complete for

On reductions of NP sets to sparse sets

{\operatorname{DTIME}}(2^{\mathcal{O}(n)} )

Completeness for nondeterministic complexity classes

,