We prove that the graph isomorphism problem restricted to trees and to colored graphs with color multiplicities 2 and 3 is many-one complete for several complexity classes within NC². In particular we show that tree isomorphism, when trees are encoded as strings, is NC¹-hard under AC0-reductions. NC¹-completeness thus follows from Buss's NC¹ upper bound. By contrast, we prove that testing isomorphism of two trees encoded as pointer lists is L-complete. Concerning colored graphs we show that the isomorphism problem for graphs with color multiplicities 2 and 3 is complete for symmetric logarithmic space SL under many-one reductions. This result improves the existing upper bounds for the problem. We also show that the graph automorphism problem for colored graphs with color classes of size 2 is equivalent to deciding whether a graph has more than a single connected component and we prove that for color classes of size 3 the graph automorphism problem is contained in SL.

The problem of canonically labeling a graph is studied. Within the general framework of backtracking algorithms based on individualization and refinement, data structures, subroutines, and pruning heuristics especially for fast handling of large and sparse graphs are developed. Experiments indicate that the algorithm implementation in most cases clearly outperforms existing state-of-the-art tools.

A Boolean constraint satisfaction instance is a conjunction of constraint applications, where the allowed constraints are drawn from a fi xed set C of Boolean functions. We consider the problem of determining whether two given constraint satisfaction instances are equivalent in the sense that they possess the same sets of satisfying assignments. We prove a Dichotomy Theorem by showing that for all sets C of allowed constraints, this problem is either polynomial-time solvable or coNP-complete, and we give a simple criterion to determine which case holds. Another equivalence problem...

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We prove that the graph isomorphism problem restricted to colored graphs with color multiplicities 2 and 3 is complete for symmetric logarithmic space SL under many-one reductions. This result improves the existing upper bounds for the problem. We also

Abstract. Graph isomorphism (GI) is one of the few remaining problems in NP whose complexity status couldn’t be solved by classifying it as being either NP-complete or solvable in P. Nevertheless, efficient (polynomial-time or even NC) algorithms for restricted versions of GI have been found over the last four decades. Depending on the graph class, the design and analysis of algorithms for GI use tools from various fields, such as combinatorics, algebra and logic. In this paper, we collect several complexity results on graph isomorphism testing and related algorithmic problems for restricted graph classes from the literature. Further, we provide some new complexity bounds (as well as easier proofs of some known results) and highlight some open questions. 1

Gire and Hoang introduce a xed-point logic with a `symmetric ' choice operator that makes a nondeterministic choice from a de nable set of tuples at each stage in the inductive construction of a relation, as long as the set of tuples is an automorphism class of the structure. We present a clean de nition of the syntax and semantics of this logic and investigate its expressive power. We extend the logic of Gire and Hoang with parameterized and nested xed points and rst-order combinations of xed points. We show that the ability to supply parameters to xed points strictly increases the power of the logic. Our logic can express the graph isomorphism problem and we show that, on almost all structures, it captures P , the class of problems decidable in polynomial time by a deterministic Turing machine with an oracle for graph isomorphism.

This paper describes a software tool that utilizes the Tablet PC’s natural user interface to provide interactivity between an instructor and her students in an organic chemistry course. The instructor and the students are equipped with electronic tablets and wireless access to the Internet. The software tool, called OrganicPad, enables an instructor to engage her students in class by sending them problems to solve. The students develop answers to the problems guided by tips and hints provided by the software. All communication is through the web, using the HTTP communication protocol. At any time, the instructor can evaluate student submissions and can quickly understand where students are having difficulty. The instructor can hide students ’ identities and anonymously use the submissions pedagogically. Finally, the instructor can develop tutorial exercises that students can work on outside of class. We present how we have used OrganicPad and how we plan to extend its functionality in the future. 1.

In this paper we consider privacy problems with anonymized transaction databases, i.e., transaction databases where the items are renamed in order to hide sensitive information. In particular, we show how an anonymized transaction database can be deanonymized using non-anonymized frequent itemsets. We describe how the problem can be formulated as an integer programming task, study the computational complexity of the problem, discuss how the computations could be done more e#ciently in practice and experimentally examine the feasibility of the proposed approach.

Abstract. Using logspace counting classes we study the computational complexity of hypergraph and graph isomorphism where the vertex sets have bounded color classes for certain specific bounds. We also give a polynomial-time algorithm for hypergraph isomorphism for bounded color classes of arbitrary size. 1