Deborah Joseph

Some remarks on witness functions for nonpolynomial and noncomplete sets in NP

Abstract We present two results about witness functions for sets in NP and coNP. First, any set that has a polynomially computable function which witnesses that it is not in coNP must be at least NP-hard. It follows from this result that any set in NP-coNP that has a polynomially computable function which witnesses this fact must already be complete for NP. Second, if B is any set for which there is a polynomially computable function which witnesses that it is not complete for NP by witnessing that some fixed set in NP is not in P B , then B must already be in NP ⊃ coNP. Thus, for two sets in NP-coNP there are no polynomially computable functions which witness that one is not polynomially reducible to the other. In proving the first result we introduce the notion of a k -creative set and prove that all k -creative sets are NP-complete. Since these sets seem not to be all polynomially isomorphic, we counter the conjecture of Berman and Hartmanis that all NP-complete sets are isomorphic to SAT with our own conjecture that not all k -creative sets are isomorphic to SAT. The proofs we give are recursion-theoretic in style, but straightforward.

Movement problems for 2-dimensional linkages

This paper is motivated by questions concerning the planning of motion in robotics. In particular, it is concerned with the motion of planar linkages from the complexity point of view. There are two main results. First, a planar linkage can be constrained to stay inside a bounded region whose boundary consists of straight lines by the addition of a polynomial number of new links. Second, the question of whether a planar linkage in some initial configuration can be moved so that a designated joint reaches a given point in the plane is PSPACE-hard.

Which Triangulations Approximate the Complete Graph

Chew and Dobkin et. al. have shown that the Delaunay triangulation and its variants are sparse approximations of the complete graph, in that the shortest distance between two sites within the triangulation is bounded by a constant multiple of their Euclidean separation. In this paper, we show that other classical triangulation algorithms, such as the greedy triangulation, and more notably, the minimum weight triangulation, also approximate the complete graph in this sense. We also design an algorithm for constructing extremely sparse (nontriangular) planar graphs that approximate the complete graph.

On the Movement of Robot Arms in 2-Dimensional Bounded Regions

The classical movers problem is the following: can a rigid object in 3-dimensional space be moved from one given position to another while avoiding obstacles? It is known that a more general version of this problem involving objects with movable joints is PSPACE-complete, even for a simple tree-like structure. In this paper, we investigate a 2-dimensional movers problem in which the object being moved is a robot arm with an arbitrary number of joints. We reduce the movers problem for arms constrained to move within bounded regions whose boundaries are made up of straight lines to the movers problem for a more complex linkage that is not constrained. We prove that the latter problem is PSPACE-hard even in 2-dimensional space and then turn to special cases of the movers problem for arms. In particular, we give a polynomial time algorithm for moving an arm confined within a circle from one given configuration to another. We also give a polynomial time algorithm for moving the arm from its initial position to a position in which the end of the arm reaches a given point within the circle.

Determining DNA Sequence Similarity Using Maximum Independent Set Algorithms for Interval Graphs

Motivated by the problem of finding similarities in DNA and amino acid sequences, we study a particular class of two dimensional interval graphs and present an algorithm that finds a maximum weight “increasing” independent set for this class. Our class of interval graphs is a subclass of the graphs with interval number 2. The algorithm we present runs in O(n log n) time, where n is the number of nodes, and its implementation provides a practical solution to a common problem in genetic sequence comparison.

Generating Sparse Spanners for Weighted Graphs

Given a graph G, a subgraph G′ is a t-spanner of G, if for every u, v ∈ V, the distance from u to v in G′ is at most t times longer than the distance in G. In this paper we give a very simple algorithm for constructing sparse spanners for arbitrary weighted graphs. We then apply this algorithm to obtain specific results for planar graphs and Euclidean graphs. We discuss the optimality of our results and present several nearly matching lower bounds.

On the movement of robot arms in 2-dimensional bounded regions

The classical movers problem is the following: can a rigid object in 3-dimensional space be moved from one given position to another while avoiding obstacles? It is known that a more general version of this problem involving objects with movable joints is PSPACE-complete, even for a simple tree-like structure. In this paper, we investigate a 2-dimensional movers problem in which the object being moved is a robot arm with an arbitrary number of joints. We reduce the movers problem for arms constrained to move within bounded regions whose boundaries are made up of straight lines to the movers problem for a more complex linkage that is not constrained. We prove that the latter problem is PSPACE-hard even in 2-dimensional space and then turn to special cases of the movers problem for arms. In particular, we give a polynomial time algorithm for moving an arm confined within a circle from one given configuration to another. We also give a polynomial time algorithm for moving the arm from its initial position to a position in which the end of the arm reaches a given point within the circle.

Three results on the polynomial isomorphism of complete sets

This paper proves three results relating to the isomorphism question for NP-complete sets. Result 1: We construct an oracle A such that SATA is ≤mP- complete for NPA and all ≤mP-complete sets for NPA are pA- isomorphic to SATA. Result 2: We construct a time function T(n) such that DTIME(T(n)) contains btt-complete sets, which are many-one equivalent, but are not p-isomorphic. The proof of this result has two corollaries: 1) There is an oracle, D, such that NPD contains non-p-isomorophic ≤m(D),P-complete sets. 2) There is a ≤mP-degree that contains non-p-isomorphic sets. Result 3: We show that no simple modification of the diagonalization argument used by Ko, Long and Du can be used to produce sets that are both EXPtime-complete w.r.t, polynomial many-one reducibility and not p-isomorphic.

Near-testable sets

In this paper a new property of sets, near-testability, is introduced. A set S is near-testable

Self-Reducibility: Effects of Internal Structure on Computational Complexity

(S \in NT)