Sebastian Kuhnert

Interval Graphs: Canonical Representations in Logspace

We present a logspace algorithm for computing a canonical labeling, in fact, a canonical interval representation, for interval graphs. To achieve this, we compute canonical interval representations of interval hypergraphs. This approach also yields a canonical labeling of convex graphs. As a consequence, the isomorphism and automorphism problems for these graph classes are solvable in logspace. For proper interval graphs we also design logspace algorithms computing their canonical representations by proper and by unit interval systems.

The Isomorphism Problem for k-Trees Is Complete for Logspace

We show that k-tree isomorphism can be decided in logarithmic space by giving a logspace canonical labeling algorithm. This improves over the previous StUL upper bound and matches the lower bound. As a consequence, the isomorphism, the automorphism, as well as the canonization problem for k-trees are all complete for deterministic logspace. We also show that even simple structural properties of k-trees are complete for logspace.

Approximate graph isomorphism

We study optimization versions of Graph Isomorphism. Given two graphs G1,G2, we are interested in finding a bijection π from V(G1) to V(G2) that maximizes the number of matches (edges mapped to edges or non-edges mapped to non-edges). We give an nO(logn) time approximation scheme that for any constant factor α<1, computes an α-approximation. We prove this by combining the nO(logn) time additive error approximation algorithm of Arora et al. [Math. Program., 92, 2002] with a simple averaging algorithm. We also consider the corresponding minimization problem (of mismatches) and prove that it is NP-hard to α-approximate for any constant factor α. Further, we show that it is also NP-hard to approximate the maximum number of edges mapped to edges beyond a factor of 0.94. We also explore these optimization problems for bounded color class graphs which is a well studied tractable special case of Graph Isomorphism. Surprisingly, the bounded color class case turns out to be harder than the uncolored case in the approximate setting.

Solving the canonical representation and Star System Problems for proper circular-arc graphs in logspace☆

Abstract We present a logspace algorithm that constructs a canonical intersection model for a given proper circular-arc graph, where canonical means that isomorphic graphs receive identical models. This implies that the recognition and the isomorphism problems for these graphs are solvable in logspace. For the broader class of concave-round graphs, which still possess (not necessarily proper) circular-arc models, we show that a canonical circular-arc model can also be constructed in logspace. As a building block for these results, we design a logspace algorithm for computing canonical circular-arc models of circular-arc hypergraphs. This class of hypergraphs corresponds to matrices with the circular ones property , which play an important role in computational genomics. Our results imply that there is a logspace algorithm that decides whether a given matrix has this property. Furthermore, we consider the Star System Problem that consists in reconstructing a graph from its closed neighborhood hypergraph. We show that this problem is solvable in logarithmic space for the classes of proper circular-arc, concave-round, and co-convex graphs. Note that solving a problem in logspace implies that it is solvable by a parallel algorithm of the class AC 1 . For the problems under consideration, at most AC 2 algorithms were known earlier.

Efficient Edge-Finding on Unary Resources with Optional Activities

Unary resources play a central role in modelling scheduling problems. Edge-finding is one of the most popular techniques to deal with unary resources in constraint programming environments. Often it depends on external factors if an activity will be included in the final schedule, making the activity optional. Currently known edge-finding algorithms cannot take optional activities into account. This paper introduces an edge-finding algorithm that finds restrictions for enabled and optional activities. The performance of this new algorithm is studied for modified job-shop and random-placement problems.

Helly Circular-Arc Graph Isomorphism Is in Logspace

We present logspace algorithms for the canonical labeling problem and the representation problem of Helly circular-arc (HCA) graphs. The first step is a reduction to canonical labeling and representation of interval intersection matrices. In a second step, the Δ trees employed in McConnell’s linear time representation algorithm for interval matrices are adapted to the logspace setting and endowed with additional information to allow canonization. As a consequence, the isomorphism and recognition problems for HCA graphs turn out to be logspace complete.

Interval Graph Representation with Given Interval and Intersection Lengths

We consider the problem of finding interval representations of graphs that additionally respect given interval lengths and/or pairwise intersection lengths, which are represented as weight functions on the vertices and edges, respectively. Pe’er and Shamir proved that the problem is \(\text{\upshape\textsf{NP}}\)-complete if only the former are given [SIAM J. Discr. Math. 10.4, 1997]. We give both a linear-time and a logspace algorithm for the case when both are given, and both an \(\ensuremath{\mathcal{O}}(n\cdot m)\) time and a logspace algorithm when only the latter are given. We also show that the resulting interval systems are unique up to isomorphism.

Solving the Canonical Representation and Star System Problems for Proper Circular-Arc Graphs in Logspace

We present a logspace algorithm that constructs a canonical intersection model for a given proper circular-arc graph, where canonical means that isomorphic graphs receive identical models. This implies that the recognition and the isomorphism problems for these graphs are solvable in logspace. For the broader class of concave-round graphs, which still possess (not necessarily proper) circular-arc models, we show that a canonical circular-arc model can also be constructed in logspace. As a building block for these results, we design a logspace algorithm for computing canonical circular-arc models of circular-arc hypergraphs; this important class of hypergraphs corresponds to matrices with the circular ones property. Furthermore, we consider the Star System Problem that consists in reconstructing a graph from its closed neighborhood hypergraph. We show that this problem is solvable in logarithmic space for the classes of proper circular-arc, concave-round, and co-convex graphs.

Interval graphs: canonical representation in logspace

We present a logspace algorithm for computing a canonical labeling, in fact a canonical interval representation, for interval graphs. As a consequence, the isomorphism and automorphism problems for interval graphs are solvable in logspace.

On the isomorphism problem for decision trees and decision lists

We study the complexity of isomorphism testing for boolean functions that are represented by decision trees or decision lists. Our results are the following:Isomorphism testing of rank 1 decision trees is complete for logspace.For any constant r ? 2 , isomorphism testing for rank r decision trees is polynomial-time equivalent to Graph Isomorphism. As a consequence of our reduction, we obtain our main result for decision trees: A 2 n ( log ? s ) O ( 1 ) time algorithm for isomorphism testing of decision trees of size s over n variables.The isomorphism problem for decision lists admits a Schaefer-type trichotomy: depending on the class of base functions, the isomorphism problem is either in L , or polynomial-time equivalent to Graph Isomorphism, or coNP -hard.