Sebastian Seibert

Translating Regular Expressions into Small ε-Free Nondeterministic Finite Automata

We prove that every regular expression of size n can be converted into an equivalent nondeterministic ?-free finite automaton (NFA) with O(n(logn)2) transitions in time O(n2logn). The best previously known conversions result in NFAs of worst-case size ?(n2). We complement our result by proving an almost matching lower bound. We exhibit a sequence of regular expressions of size O(n) and show the number of transitions required in equivalent NFAs is ?(nlogn). This also proves there does not exist a linear-size conversion from regular expressions to NFAs.

Monadic second-order logic over rectangular pictures and recognizability by tiling systems

Abstract It is shown that a set of pictures (rectangular arrays of symbols) is recognized by a finite tiling system iff it is definable in existential monadic second-order logic. As a consequence, finite tiling systems constitute a notion of recognizability over two-dimensional inputs which at the same time generalizes finite-state recognizability over strings and also matches a natural logic. The proof is based on the Ehrenfeucht–Fraisse technique for first-order logic and an implementation of “threshold counting” within tiling systems.

Communication Complexity Method for Measuring Nondeterminism in Finite Automata

While deterministic finite automata seem to be well understood, surprisingly many important problems concerning nondeterministic finite automata (nfas) remain open. One such problem area is the study of different measures of nondeterminism in finite automata and the estimation of the sizes of minimal nondeterministic finite automata. In this paper the concept of communication complexity is applied in order to achieve progress in this problem area. The main results are as follows:(1) Deterministic communication complexity provides lower bounds on the size of nfas with bounded unambiguity. Applying this fact, the proofs of several results about nfas with limited ambiguity can be simplified and presented in a uniform way. (2) There is a family of languages KONk2 with an exponential size gap between nfas with polynomial leaf number/ambiguity and nfas with ambiguity k. This partially provides an answer to the open problem posed by B. Ravikumar and O. Ibarra (1989, SIAM J. Comput. 18, 1263-1282) and H. Leung (1998, SIAM J. Comput. 27, 1073-1082).

Towards the notion of stability of approximation for hard optimization tasks and the traveling salesman problem

The investigation of the possibility to efficiently compute approximations of hard optimization problems is one of the central and most fruitful areas of current algorithm and complexity theory. The aim of this paper is twofold. First, we introduce the notion of stability of approximation algorithms. This notion is shown to be of practical as well as of theoretical importance, especially for the real understanding of the applicability of approximation algorithms and for the determination of the border between easy instances and hard instances of optimization problems that do not admit polynomial-time approximation. Secondly, we apply our concept to the study of the traveling salesman problem (TSP). We show how to modify the Christofides algorithm for Δ-TSP to obtain efficient approximation algorithms with constant approximation ratio for every instance of TSP that violates the triangle inequality by a multiplicative constant factor. This improves the result of Andreae and Bandelt (SIAM J. Discrete Math. 8 (1995) 1).

An Improved Lower Bound on the Approximability of Metric TSP and Approximation Algorithms for the TSP with Sharpened Triangle Inequality

The traveling salesman problem (TSP) is one of the hardest optimization problems in NPO because it does not admit any polynomial time approximation algorithm (unless P = NP). On the other hand we have a polynomial time approximation scheme (PTAS) for the Euclidean TSP and the 3/2 -approximation algorithm of Christofides for TSP instances satisfying the triangle inequality. The main contributions of this paper are the following: (i) We essentially modify the method of Engebretsen [En99] in order to get a lower bound of 3813/3812-Ɛ on the polynomial-time approximability of the metric TSP for any Ɛ > 0. This is an improvement over the lower bound of 5381/5380 -Ɛ in [En99]. Using this approach we moreover prove a lower bound δβ on the approximability of Δβ-TSP for 1/2 < β < 1, where Δβ-TSP is a subproblem of the TSP whose input instances satisfy the β-sharpened triangle inequality cost({u, v}) ≤ β ċ (cost({u, v}) + cost({x, v})) for all vertices u, v, x. (ii) We present three different methods for the design of polynomial-time approximation algorithms for Δβ-TSP with 1/2 < β < 1, where the approximation ratio lies between 1 and 3/2, depending on β.

Improved Lower Bounds on the Approximability of the Traveling Salesman Problem

This paper deals with lower bounds on the approximability of different subproblems of the Traveling Salesman Problem (TSP) which is known not to admit any polynomial time approximation algorithm in general (unless P = NP). First of all, we present an improved lower bound for the Traveling Salesman Problem with Triangle Inequality, Δ-TSP for short. Moreover our technique, an extension of the method of Engebretsen [11], also applies to the case of relaxed and sharpened triangle inequality, respectively, denoted Δ β -TSP for an appropriate β. In case of the Δ-TSP, we obtain a lower bound of 3813/3812 - e on the polynomial-time approximability (for any small e > 0), compared to the previous bound of 5381/5380 - e in [11] In case of the Δ β -TSP, for the relaxed case (β > 1) we present a lower bound of 3803-10β/3804+8β - e, and for the sharpened triangle inequality (1/2 < β < 1), the lower bound is 7611+10β 2 +5β/7612+8β 2 +4β - e The latter result is of interest especially since it shows that the TSP is APX-hard even if one comes arbitrarily close to the trivial case that all edges have the same cost.

Translating Regular Expressions into Small epsilon-Free Nondeterministic Finite Automata

It is proved that every regular expression of size n can be converted into an equivalent nondeterministic finite automaton (NFA) of size O(n(log n)2) in polynomial time. The best previous conversions result in NFAs of worst case size Θ(n2). Moreover, the nonexistence of any linear conversion is proved: we give a language L n described by a regular expression of size O(n) such that every NFA accepting L n is of size Ω(n log n).

Advice Complexity of the Online Coloring Problem

We study online algorithms with advice for the problem of coloring graphs which come as input vertex by vertex. We consider the class of all 3-colorable graphs and its sub-classes of chordal and maximal outerplanar graphs, respectively.

On the hardness of constructing minimal 2-connected spanning subgraphs in complete graphs with sharpened triangle inequality

In this paper we investigate the problem of finding a 2-connected spanning subgraph of minimal cost in a complete and weighted graph G. This problem is known to be APX-hard, for both the edge and the vertex connectivity case. Here we prove that the APX-hardness still holds even if one restricts the edge costs to an interval [1, 1 + e], for an arbitrary small e > 0. This result implies the first explicit lower bound on the approximability of the general version (i.e., for arbitrary graphs) of the problem. On the other hand, if the input graph satisfies the sharpened β-triangle inequality, then a (2/3 + 1/3 ċ β/1-β)-approximation algorithm is designed. This ratio tends to 1 with β tending to ½, and it improves the previous known bound of 3/2, holding for graphs satisfying the triangle inequality, as soon as β < 5/7Furthermore, a generalized problem of increasing to 2 the edge-connectivity of any spanning subgraph of G by means of a set of edges of minimum cost is considered. This problem is known to admit a 2-approximation algorithm. Here we show that whenever the input graph satisfies the sharpened β-triangle inequality with β < 2/3, then this ratio can be improved to β/1-β.

A 1.5-approximation of the minimal manhattan network problem

Given a set of points in the plane, the Minimal Manhattan Network Problem asks for an axis-parallel network that connects every pair of points by a shortest path under L1-norm (Manhattan metric). The goal is to minimize the overall length of the network. We present an approximation algorithm that provides a solution of length at most 1.5 times the optimum. Previously, the best known algorithm has given only a 2-approximation.