A caterpillar C is a tree having a path that contains all vertices of C of degree at least 3. We show in this article that every balanced caterpillar with maximum degree 3 and 2(n) vertices is a subgraph of the n-dimensional hypercube. This solves a long-standing open problem and generalizes a result of Havel and Liebl (1986), who considered only such caterpillars that have a path containing all vertices of degree at least 2.

We show that computing the lexicographically first four-coloring for planar graphs is Δ2p-hard. This result optimally improves upon a result of Khuller and Vazirani who prove this problem NP-hard, and conclude that it is not self-reducible in the sense of Schnorr, assuming P ≠ NP. We discuss this application to non-self-reducibility and provide a general related result. We also discuss when raising a problem's NP-hardness lower bound to Δ2p-hardness can be valuable.

This short survey of properties of complexity classes (CC's for short) does not pretend to be complete. We rather confine ourselves to the illustration of important features by typical examples. Simultaneously an attempt is made to find a reasonable systematization of the vast variety of papers contributing to our topic. Among the chosen examples there are four so far unpublished statements (numbered (5), (6), (19) and (35)) about the return complexity [70] and a new measure A for nondeterministic Turing machines (NDTM) which is similar to the return complexity.

Let S be a set of n given points in 3-dimensional space. We present parallel algorithms for the construction of the convex hull and for triangulation of S on a CREW-PRAM. For 3-dim. convex hull our algorithm is time-optimal and uses time O(1/ε· log(n)) with O(n 1+e) processors. By duality parallel convex hull algorithms induce new ones for Voronoidiagrams in the plane, using the same time and processor bounds. A second parallel algorithm for Voronoi-diagrams presented here uses time O(log(n) 2) with O(n) processors. For 3-dim. triangulation of S we give the first parallel algorithm for the generalized problem, using time O(log(n) 2) with O(n 1+e) processors. For the tangential-plane problem we give a parallel algorithm, needing time O(log(n)) with O(n) processors.

By defining a general max and a general min operator for complexity classes we obtain that there are other interesting classes of optimization functions besides Krentel's class OptP. We investigate the behavior of these operators on the polynomial hierarchy, in particular we study the inclusion structure of the classes max · P, max · NP, max · coNP, min · P, min · NP, and min · coNP. It turns out that our operators when applied to the polynomial hierarchy yield a refinement of Krentel's hierarchy of optimization functions. We prove that this refinement is strict unless the polynomial hierarchy collapses and show that the refinement is useful to exactly classify optimization functions. Moreover, our investigations shed new light on Krentel's result that every function from some level of the polynomial hierarchy can be characterized in terms of an optimization function.

By endowing usual nondeterministic Turing machines with new modes of acceptance we introduce new machines whose computational power is bounded by that of alternating Turing machines making only one alternation. The polynomial time classes of these machines are exactly the levels of the Boolean closure of NP which can be defined in a natural way. For all these classes natural problems can be found which are proved to be ≤ m P -complete in these classes.

Can easy sets only have easy certificate schemes? In this paper, we study the class of sets that, for all NP certificate schemes (i.e., NP machines), always have easy acceptance certificates (i.e., accepting paths) that can be computed in polynomial time. We also study the class of sets that, for all NP certificate schemes, infinitely often have easy acceptance certificates. We give structural conditions that control the size of these classes. We also provide negative results showing that some of our positive claims are optimal. Our negative results are proven using a novel observation: The classic “wide spacing” oracle construction technique instantly yields non-bi-immunity results. Easy certificate classes are also a useful notion in the study of whether one-way functions exist. This is one of the most important open questions in cryptology. We extend the results of Grollmann and Selman [GS88] by obtaining a complete characterization regarding the existence of a certain type of one-way function—(partial) one-way permutations—in terms of easy certificate classes. By Grädel's recent results about one-way functions [Grä94], this also links statements about easy certificates of NP sets with statements in finite model theory. In addition, we give a condition necessary and sufficient for the existence of (total) one-way permutations.

The complexity classification of problems defined by restricting NP-complets problems to those instances having unique solutions requires still finer hierarchies within BC(NP) (the Boolean closure of NP) than that introduced in [Wec 85] (see also [WeWa 85], [GuWe 85], [CaHe 85] and [KöSc 85]) which will be called the Hausdorff hierarchy generated by NP. In this paper an extremely fine hierarchy within BC(NP) is proposed. The classes of this hierarchy are characterized by nondeter-ministic polynomial time Turing machines with suitably modified acceptance notions (Section 2). Complete sets for the classes of the hierarchy are presented in Section 5. The hierarchy is studied under relativizations (Sections 3 and 5). Section 4 yields more insight in the structure of the hierarchy.

The operators min· ,m ax· ,a nd #· translate classes of the polynomial-time hierarchy to function classes. Although the inclusion relationships between these func- tion classes have been studied in depth, some questions concerning separations re- mained open. We provide oracle constructions that answer most of these open questions in the rel- ativized case. As a typical instance for the type of results of this paper, we construct a relativized world where min·P � #·NP, thus giving evidence for the hardness of proving min·P ⊆ #·NP in the unrelativized case. The strongest results, proved in the paper, are the constructions of oracles D and E, such that min·coNP D ⊆ #·P D ∧ NP DoNP D and UP E =N P E ∧ min·P E � #·P E.

We prove that computing a single pair of vertices that are mapped onto each other by an isomorphism φ between two isomorphic graphs is as hard as computing φ itself. This result optimally improves upon a result of Gál, Halevi, Lipton, and Petrank. We establish a similar, albeit slightly weaker, result about computing complete Hamiltonian cycles of a graph from partial Hamiltonian cycles.

More than a quarter of a century ago, the question of the complexity of determining whether a given Boolean formula is minimal motivated Meyer and Stockmeyer to define the polynomial hierarchy. This problem (in the standard formalized version---that of Garey and Johnson) has been known for decades to be coNP-hard and in NPNP, and yet no one had even been able to establish (many-one) NP-hardness. In this paper, we show that and more: The problem in fact is (many-one) hard for parallel access to NP.

We show that computing the lexicographically first four-coloring for planar graphs is P^{NP}-hard. This result optimally improves upon a result of Khuller and Vazirani who prove this problem to be NP-hard, and conclude that it is not self-reducible in the sense of Schnorr, assuming P \neq NP. We discuss this application to non-self-reducibility and provide a general related result.

We prove that computing a single pair of vertices that are mapped onto each other by an isomorphism ϕ between two isomorphic graphs is as hard as computing ϕ itself.Th is result optimally improves upon a result of Gál et al.W e establish a similar, albeit slightly weaker, result about computing complete Hamiltonian cycles of a graph from partial Hamiltonian cycles.W e also show that computing the lexicographically first four-coloring for planar graphs is δ 2 p -hard. This result optimally improves upon a result of Khuller and Vazirani who prove this problem to be NP-hard, and conclude that it is not self-reducible in the sense of Schnorr, assuming P ≠ NP. W e discuss this application to non-self-reducibility and provide a general related result.

We study whether one can prune solutions from NP functions. Though it is known that, unless surprising complexity class collapses occur, one cannot reduce the number of accepting paths of NP machines [17], we nonetheless show that it often is possible to reduce the number of solutions of NP functions. For finite cardinality types, we give a sufficient condition for such solution reduction. We also give absolute and conditional necessary conditions for solution reduction, and in particular we show that in many cases solution reduction is impossible unless the polynomial hierarchy collapses.