Till Tantau

Algorithmic Meta Theorems for Circuit Classes of Constant and Logarithmic Depth

An algorithmic meta theorem for a logic and a class C of structures states that all problems expressible in this logic can be solved eciently for inputs from C. The prime example is Courcelle’s Theorem, which states that monadic second-order (mso) definable problems are linear-time solvable on graphs of bounded tree width. We contribute new algorithmic meta theorems, which state that mso-definable problems are (a) solvable by uniform constant-depth circuit families (AC 0 for decision problems and TC 0 for counting problems) when restricted to input structures of bounded tree depth and (b) solvable by uniform logarithmic-depth circuit families (NC 1 for decision problems and #NC 1 for counting problems) when a tree decomposition of bounded width in term representation is part of the input. Applications of our theorems include a TC 0 -completeness proof for the unary version of integer linear programming with a fixed number of equations and extensions of a recent result that counting the number of accepting paths of a visible pushdown automaton lies in #NC 1 . Our main technical contributions are a new tree automata model for unordered, unranked, labeled trees; a method for representing the tree automata’s computations algebraically using convolution circuits; and a lemma on computing balanced width-3 tree decompositions of trees in TC 0 , which encapsulates most of the technical diculties surrounding earlier results connecting tree automata and NC 1 . 1998 ACM Subject Classification F.1.3 Complexity Measures and Classes

Logspace algorithms for computing shortest and longest paths in series-parallel graphs

For many types of graphs, including directed acyclic graphs, undirected graphs, tournament graphs, and graphs with bounded independence number, the shortest path problem is NL-complete. The longest path problem is even NP-complete for many types of graphs, including undirected K5-minor-free graphs and planar graphs. In the present paper we present logspace algorithms for computing shortest and longest paths in series-parallel graphs where the edges can be directed arbitrarily. The class of series-parallel graphs that we study can be characterized alternatively as the class of K4-minor-free graphs and also as the class of graphs of tree-width 2. It is well-known that for graphs of bounded tree-width many intractable problems can be solved efficiently, but previous work was focused on finding algorithms with low parallel or sequential time complexity. In contrast, our results concern the space complexity of shortest and longest path problems. In particular, our results imply that for directed graphs of tree-width 2 these problems are L-complete.

Haplotyping with missing data via perfect path phylogenies

Computational methods for inferring haplotype information from genotype data are used in studying the association between genomic variation and medical condition. Recently, Gusfield proposed a haplotype inference method that is based on perfect phylogeny principles. A fundamental problem arises when one tries to apply this approach in the presence of missing genotype data, which is common in practice. We show that the resulting theoretical problem is NP-hard even in very restricted cases. To cope with missing data, we introduce a variant of haplotyping via perfect phylogeny in which a path phylogeny is sought. Searching for perfect path phylogenies is strongly motivated by the characteristics of human genotype data: 70% of real instances that admit a perfect phylogeny also admit a perfect path phylogeny. Our main result is a fixed-parameter algorithm for haplotyping with missing data via perfect path phylogenies. We also present a simple linear-time algorithm for the problem on complete data.

On the space complexity of parameterized problems

Parameterized complexity theory measures the complexity of computational problems predominantly in terms of their parameterized time complexity. The purpose of the present paper is to demonstrate that the study of parameterized space complexity can give new insights into the complexity of well-studied parameterized problems like the feedback vertex set problem. We show that the undirected and the directed feedback vertex set problems have different parameterized space complexities, unless L = NL; which explains why the two problem variants seem to necessitate different algorithmic approaches even though their parameterized time complexity is the same. For a number of further natural parameterized problems, including the longest common subsequence problem and the acceptance problem for multi-head automata, we show that they lie in or are complete for different parameterized space classes; which explains why previous attempts at proving completeness of these problems for parameterized time classes have failed.

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

Towards a Cardinality Theorem for Finite Automata

\Pi_2^{\mathrm{P}}

Smoothed analysis of left-to-right maxima with applications

-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 the Space and Circuit Complexity of Parameterized Problems: Classes and Completeness

Kummers cardinalitytheorem states that a language is recursive if a Turing machine can exclude for any n words one of the n + 1 possibilities for the number of words in the language. This paper gathers evidence that the cardinalitytheorem might also hold for finite automata. Three reasons are given. First, Beigels nonspeedup theorem also holds for finite automata. Second, the cardinalitytheorem for finite automata holds for n = 2. Third, the restricted cardinalitytheorem for finite automata holds for all n.

Partial information classes

A left-to-right maximum in a sequence of <i>n</i> numbers <i>s</i>₁, …, <i>s_n</i> is a number that is strictly larger than all preceding numbers. In this article we present a smoothed analysis of the number of left-to-right maxima in the presence of additive random noise. We show that for every sequence of <i>n</i> numbers <i>s_i</i> ∈ [0,1] that are perturbed by uniform noise from the interval [-ε,ε], the expected number of left-to-right maxima is Θ(&sqrt;<i>n</i>/ε + log <i>n</i>) for ε>1/<i>n</i>. For Gaussian noise with standard deviation σ we obtain a bound of <i>O</i>((log^3/2 <i>n</i>)/σ + log <i>n</i>). We apply our results to the analysis of the smoothed height of binary search trees and the smoothed number of comparisons in the quicksort algorithm and prove bounds of Θ(&sqrt;<i>n</i>/ε + log <i>n</i>) and Θ(<i>n</i>/ε+1&sqrt;<i>n</i>/ε + <i>n</i> log <i>n</i>), respectively, for uniform random noise from the interval [-ε,ε]. Our results can also be applied to bound the smoothed number of points on a convex hull of points in the two-dimensional plane and to <i>smoothed motion complexity,</i> a concept we describe in this article. We bound how often one needs to update a data structure storing the smallest axis-aligned box enclosing a set of points moving in <i>d</i>-dimensional space.

Logspace Optimization Problems and Their Approximability Properties

The parameterized complexity of a problem is generally considered “settled” once it has been shown to be fixed-parameter tractable or to be complete for a class in a parameterized hierarchy such as the weft hierarchy. Several natural parameterized problems have, however, resisted such a classification. In the present paper we argue that, in some cases, this is due to the fact that the parameterized complexity of these problems can be better understood in terms of their parameterized space or parameterized circuit complexity. This includes well-studied, natural problems like the feedback vertex set problem, the associative generability problem, or the longest common subsequence problem. We show that these problems lie in and may even be complete for different parameterized space classes, leading to new insights into the problems’ complexity. The classes we study are defined in terms of different forms of bounded nondeterminism and simultaneous time–space bounds.