Arfst Nickelsen

Counting, Selecting, adn Sorting by Query-Bounded Machines

We study the query-complexity of counting, selecting, and sorting functions. That is, for a given set A and a positive integer k, we ask, how many queries to an arbitrary oracle does a polynomial-time machine on input (x1, x2,..., x k ) need to determine how many strings of the input are in A. We also ask how many queries are necessary to select a string in A from the input (x1, x2,..., x k ) if such a string exists and to sort the input (x1, x2,..., x k ) with respect to the ordering x ≼ y if and only if x ∈ A ⇒ y ∈ A. We obtain optimal query-bounds for these problems, and show that sets for which these functions have a low query-complexity must be easy in some sense. For such sets we obtain optimal placements in the extended low hierarchy. We also show that in the case of NP-complete sets the lower bounds for counting and selecting hold unless P=NP. Finally, we relate these notions to cheatability and p-superterseness. Our results yield as corollaries extensions of previously know results.

On Polynomially D-Verbose Sets

A general framework is presente for the study of complexity classes that are defined via polynomial time algorithms that compute partial information about the characteristic function of a given input. Given n ∈ N and a family D of sets D⊂-{0,1}*, a language A is polynomially D-verbose (or: A ∈ P [D]) iff there is a polynomial time algorithm that on input (x1,..., x n ) outputs a D ∈D such that the characteristic string χa(x1,..., x n ) is in D. Also the variant where only pairwise distinct input words are allowed is studied. p-selective sets, p-verbose sets, easily p-countable sets, sets that allow a polynomial time frequency computation, and cheatable sets are special cases of this definition. It is shown that it suffices to study families that are in a certain normal form. An algorithm is presented that decides for given families D1, D2 whether P [D1] ⊂-P [D2]. The classes P [D] are, except for trivial cases, not closed under union, intersection or join. The classes closed under complement are characterized, as well as those closed under ≤ m p - and ≤ 1−tt p -reductions. For a given family of sets D the class of polynomially D-verbose languages contains non-recursive languages iff it contains all p-selective languages. The families D for which a D-verbose set can be non-recursive are fully characterized by a simple combinatorial property. It is also shown that for fixed n the classes form a distributive lattice. A diagram that shows this lattice for n=2 is presented.

The Complexity of Finding Paths in Graphs with Bounded Independence Number

We study the problem of finding a path between two vertices in finite directed graphs whose independence number is bounded by some constant k. The independence number of a graph is the largest number of vertices that can be picked such that there is no edge between any two of them. The complexity of this problem depends on the exact question we ask: Do we wish only to tell whether a path exists? Do we also wish to construct such a path? Are we required to construct the shortest one? Concerning the first question, we show that the reachability problem is first-order definable for all k and that its succinct version is

Partial information classes

\Pi_2^{\mathrm{P}}

Algebraic Properties for Selector Functions

-complete for all k. In contrast, the reachability problems for many other types of finite graphs, including dags and trees, are not first-order definable, and their succinct versions are PSPACE-complete. Concerning the second question, we show not only that we can construct paths in logarithmic space, but that there even exists a logspace approximation scheme for this problem. The scheme gets a ratio r > 1 as additional input and outputs a path that is at most r times as long as the shortest path. Concerning the third question, we show that even telling whether the shortest path has a certain length is NL-complete and thus is as difficult as for arbitrary directed graphs.

On Reachability in Graphs with Bounded Independence Number

In this survey we present partial information classes, which have been studied under different names and in different contexts in the literature. They are defined in terms of partial information algorithms. Such algorithms take a word tuple as input and yield a small set of possibilities for its characteristic string as output. We define a unified framework for the study of partial information classes and show how previous notions fit into the framework. The framework allows us to study the relationship of a large variety of partial information classes in a uniform way. We survey how partial information classes are related to other complexity theoretic notions like advice classes, lowness, bi-immunity, NP-completeness, and decidability.

Closure of Polynomial Time Partial Information Classes under Polynomial Time Reductions

The nondeterministic advice complexity of the P-selective sets is known to be exactly linear. Regarding the deterministic advice complexity of the P-selective sets---i.e., the amount of Karp--Lipton advice needed for polynomial-time machines to recognize them in general---the best current upper bound is quadratic [K. Ko, J. Comput. System Sci., 26 (1983), pp. 209--221] and the best current lower bound is linear [L. Hemaspaandra and L. Torenvliet, Theoret. Comput. Sci., 154 (1996), pp. 367--377]. We prove that every associatively P-selective set is commutatively, associatively P-selective. Using this, we establish an algebraic sufficient condition for the P-selective sets to have a linear upper bound (which thus would match the existing lower bound) on their deterministic advice complexity: If all P-selective sets are associatively P-selective, then the deterministic advice complexity of the P-selective sets is linear. The weakest previously known sufficient condition was P = NP. We also establish related results for algebraic properties of, and advice complexity of, the nondeterministically selective sets.

Algebraic Properties for P-Selectivity

We study the reachability problem for finite directed graphs whose independence number is bounded by some constant k. This problem is a generalisation of the reachability problem for tournaments. We show that the problem is first-order definable for all k. In contrast, the reachability problems for many other types of finite graphs, including dags and trees, are not first-order definable. Also in contrast, first-order definability does not carry over to the infinite version of the problem. We prove that the number of strongly connected components in a graph with bounded independence number can be computed using TC0-circuits, but cannot be computed using AC0-circuits. We also study the succinct version of the problem and show that it is ?P2 -complete for all k.

Partial Information and Special Case Algorithms

Polynomial time partial information classes are extensions of the class P of languages decidable in polynomial time. A partial information algorithm for a language A computes, for fixed n ∈ N, on input of words x1, . . . , xn a set P of bitstrings, called a pool, such that XA(x1, . . . , xn) ∈ P, where P is chosen from a family D of pools. A language A is in P[D], if there is a polynomial time partial information algorithm which for all inputs (x1, . . . , xn) outputs a pool P ∈ D with XA(x1, . . . , xn) ∈ P. Manyextensions of P studied in the literature, including approximable languages, cheatability, p-selectivity and frequency computations, form a class P[D] for an appropriate family D. We characterise those families D for which P[D] is closed under certain polynomial time reductions, namely bounded truth-table, truth-table, and Turing reductions. We also treat positive reductions. A class P[D] is presented which strictlycon tains the class P-sel of p-selective languages and is closed under positive truth-table reductions.

One query reducibilities between partial information classes

Karp and Lipton, in their seminal 1980 paper, introduced the notion of advice (nonuniform) complexity, which since has been of central importance in complexity theory. Nonetheless, much remains unknown about the optimal advice complexity of classes having polynomial advice complexity. In particular, let P-sel denote the class of all P-selective sets [23] For the nondeterministic advice complexity of P-sel, linear upper and lower bounds are known [10]. However, for the deterministic advice complexity of P-sel, the best known upper bound is quadratic [13], and the best known lower bound is the linear lower bound inherited from the nondeterministic case. This paper establishes an algebraic sufficient condition for P-sel to have a linear upper bound: If all P-selective sets are associatively P-selective then the deterministic advice complexity of P-sel is linear. (The weakest previously known sufficient condition was P = NP.) Relatedly, we prove that every associatively P-selective set is commutatively, associatively P-selective.