Rainer Schuler

A Probabilistic 3-SAT Algorithm Further Improved

In [Sch99], Schoning proposed a simple yet efficient randomized algorithm for solving the k-SAT problem. In the case of 3-SAT, the algorithm has an expected running time of poly(n) ? (4/3)n = O(1.3334n) when given a formula F on n variables. This was the up to now best running time known for an algorithm solving 3-SAT. Here, we describe an algorithm which improves upon this time bound by combining an improved version of the above randomized algorithm with other randomized algorithms. Our new expected time bound for 3-SAT is O(1.3302n).

DNA-based Parallel Computation of Simple Arithmetic

We propose a model for representing and manipulating binary numbers on a DNA chip which allows parallel execution of simple arithmetic. As an example we describe how addition of large binary numbers can be done by using a DNA chip. The number of steps is independent of the size (bits) of the numbers. However, the time for some biochemical reactions is still large, and increases with the size of the sequences to be assembled.

Strategies for the development of a peptide computer

MOTIVATION We devise a computational model using protein-protein interactions. RESULTS Peptide-antibody interactions can be used to perform a large number of small logical operations in parallel. We show for example how a sequence of operations can be used to compare the number of occurrences of an element in two sets and how to estimate the number of occurrences of an element in a set. Similar to DNA-computing, these techniques could in principle be extended to solve instances of NP-complete problems. We give as an example a procedure to solve examples of the satisfiability problem.

Improving a Probabilistic 3-SAT Algorithm by Dynamic Search and Independent Clause Pairs

The satisfiability problem of Boolean Formulae in 3-CNF (3-SAT) is a well known NP-complete problem and the development of faster (moderately exponential time) algorithms has received much interest in recent years. We show that the 3-SAT problem can be solved by a probabilistic algorithm in expected time O(1,3290 n ). Our approach is based on Schoning’s random walk algorithm for k-SAT, modified in two ways.

If NP has polynomial-size circuits, then MA=AM

Abstract It is shown that the assumption of NP having polynomial-size circuits implies (apart from a collapse of the polynomial-time hierarchy as shown by Karp and Lipton) that the classes AM and MA of Babais Arthur-Merlin hierarchy coincide. This means that also a certain inner collapse of the remaining classes of the polynomial-time hierarchy occurs.

The Quantum Query Complexity of 0-1 Knapsack and Associated Claw Problems

We give an O(2 n/3) quantum algorithm for the 0-1 Knapsack problem with n variables and an O(2 n/3 n d ) quantum algorithm for 0-1 Integer Linear Programs with n variables and d inequalities. To investigate lower bounds we formulate a symmetric claw problem corresponding to 0-1 Knapsack. For this problem we establish a lower bound of O(2 n/4) for its quantum query complexity and an O(2 n/3) upper bound. We also give a 2(1 − α)n/2 quantum algorithm for satisfiability of CNF formulas with no restrictions on clause size, but with the number of clauses bounded by cn for a constant c, where n is the number of variables. Here α is a constant depending on c.

Towards average-case complexity analysis of NP optimization problems

For the worst-case complexity measure, if P=NP, then P=OptP, i.e., all NP optimization problems are polynomial-time solvable. On the other hand, it is not clear whether a similar relation holds when considering average-case complexity. We investigate the relationship between the complexity of NP decision problems and that of NP optimization problems under polynomial-time computable distributions, and study what makes them (seemingly) different. It is shown that the difference between P/sub tt//sup NP/-samplable and P/sup NP/-samplable distributions is crucial.

Sets Computable in Polynomial Time on Average

In this paper, we discuss the complexity and properties of the sets which are computable in polynomial-time on average. This study is motivated by Levins question of whether all sets in NP are solvable in polynomial-time on average for every reasonable (i.e., polynomial-time computable) distribution on the instances. Let PP-comp denote the class of all those sets which are computable in polynomial-time on average for every polynomial-time computable distribution on the instances. It is known that P ⊂ PP-comp ⊂ E. In this paper, we show that PP-comp is not contained in DTIME(2cn) for any constant c and that it lacks some basic structural properties: for example, it is not closed under many-one reducibility or for the existential operator. From these results, it follows that PP-comp contains P-immune sets but no P-bi-immune sets; it is not included in P/cn for any constant c; and it is different from most of the well-known complexity classes, such as UP, NP, BPP, and PP. Finally, we show that, relative to a random oracle, NP is not included in PP-comp and PP-comp is not in PSPACE with probability 1.

On the complexity of intersecting multiple circles for graphical display

Many experiments in the biomedical field generate vast amounts of data. This is especially true for microarray experiments which measure the expression levels of thousands of genes simultaneously. In this context the display of functional information attributed to the individual gene is important to obtain an overview of the major processes involved. This set data can be displayed as Euler/Venn diagrams in which the circle size corresponds to the cardinality of the set. Efficient algorithms for the calculation of intersections of circles and their resulting boundary have not been published so far. We present two algorithms (one optimal) for intersecting these different sized circles to display set relationships.

ON HELPING AND INTERACTIVE PROOF SYSTEMS

We investigate the complexity of honest provers in interactive proof systems. This corresponds precisely to the complexity of oracles helping the computation of robust probabilistic oracle machines. We obtain upper bounds for languages in FewEXP and for sparse sets in NP. Further, interactive protocols with provers that are reducible to sets of low information content are considered. Specifically, if the verifier communicates only with provers in P/poly, then the accepted language is low for . In the case that the provers are polynomial-time reducible to log*-sparse sets or to sets in strong-P/log then the protocol can be simulated by the verifier even without the help of provers. As a consequence we obtain new collapse results under the assumption that intractable sets reduce to sets with low information content.