17 research outputs found
Towards an Isomorphism Dichotomy for Hereditary Graph Classes
In this paper we resolve the complexity of the isomorphism problem on all but
finitely many of the graph classes characterized by two forbidden induced
subgraphs. To this end we develop new techniques applicable for the structural
and algorithmic analysis of graphs. First, we develop a methodology to show
isomorphism completeness of the isomorphism problem on graph classes by
providing a general framework unifying various reduction techniques. Second, we
generalize the concept of the modular decomposition to colored graphs, allowing
for non-standard decompositions. We show that, given a suitable decomposition
functor, the graph isomorphism problem reduces to checking isomorphism of
colored prime graphs. Third, we extend the techniques of bounded color valence
and hypergraph isomorphism on hypergraphs of bounded color size as follows. We
say a colored graph has generalized color valence at most k if, after removing
all vertices in color classes of size at most k, for each color class C every
vertex has at most k neighbors in C or at most k non-neighbors in C. We show
that isomorphism of graphs of bounded generalized color valence can be solved
in polynomial time.Comment: 37 pages, 4 figure
Strong isomorphism reductions in complexity theory
We give the first systematic study of strong isomorphism reductions, a notion of reduction more appropriate than polynomial time reduction when, for example, comparing the computational complexity of the isomorphim problem for different classes of structures. We show that the partial ordering of its degrees is quite rich. We analyze its relationship to a further type of reduction between classes of structures based on purely comparing for every n the number of nonisomorphic structures of cardinality at most n in both classes. Furthermore, in a more general setting we address the question of the existence of a maximal element in the partial ordering of the degrees
The Iteration Number of Colour Refinement
The Colour Refinement procedure and its generalisation to higher dimensions, the Weisfeiler-Leman algorithm, are central subroutines in approaches to the graph isomorphism problem. In an iterative fashion, Colour Refinement computes a colouring of the vertices of its input graph.
A trivial upper bound on the iteration number of Colour Refinement on graphs of order n is n-1. We show that this bound is tight. More precisely, we prove via explicit constructions that there are infinitely many graphs G on which Colour Refinement takes |G|-1 iterations to stabilise. Modifying the infinite families that we present, we show that for every natural number n ? 10, there are graphs on n vertices on which Colour Refinement requires at least n-2 iterations to reach stabilisation
Logarithmic Weisfeiler-Leman Identifies All Planar Graphs
The Weisfeiler-Leman (WL) algorithm is a well-known combinatorial procedure for detecting symmetries in graphs and it is widely used in graph-isomorphism tests. It proceeds by iteratively refining a colouring of vertex tuples. The number of iterations needed to obtain the final output is crucial for the parallelisability of the algorithm.
We show that there is a constant k such that every planar graph can be identified (that is, distinguished from every non-isomorphic graph) by the k-dimensional WL algorithm within a logarithmic number of iterations. This generalises a result due to Verbitsky (STACS 2007), who proved the same for 3-connected planar graphs.
The number of iterations needed by the k-dimensional WL algorithm to identify a graph corresponds to the quantifier depth of a sentence that defines the graph in the (k+1)-variable fragment C^{k+1} of first-order logic with counting quantifiers. Thus, our result implies that every planar graph is definable with a C^{k+1}-sentence of logarithmic quantifier depth
Witnessed Symmetric Choice and Interpretations in Fixed-Point Logic with Counting
At the core of the quest for a logic for Ptime is a mismatch between algorithms making arbitrary choices and isomorphism-invariant logics. One approach to tackle this problem is witnessed symmetric choice. It allows for choices from definable orbits certified by definable witnessing automorphisms.
We consider the extension of fixed-point logic with counting (IFPC) with witnessed symmetric choice (IFPC+WSC) and a further extension with an interpretation operator (IFPC+WSC+I). The latter operator evaluates a subformula in the structure defined by an interpretation. When similarly extending pure fixed-point logic (IFP), IFP+WSC+I simulates counting which IFP+WSC fails to do. For IFPC+WSC, it is unknown whether the interpretation operator increases expressiveness and thus allows studying the relation between WSC and interpretations beyond counting.
In this paper, we separate IFPC+WSC from IFPC+WSC+I by showing that IFPC+WSC is not closed under FO-interpretations. By the same argument, we answer an open question of Dawar and Richerby regarding non-witnessed symmetric choice in IFP. Additionally, we prove that nesting WSC-operators increases the expressiveness using the so-called CFI graphs. We show that if IFPC+WSC+I canonizes a particular class of base graphs, then it also canonizes the corresponding CFI graphs. This differs from various other logics, where CFI graphs provide difficult instances
Witnessed Symmetric Choice and Interpretations in Fixed-Point Logic with Counting
At the core of the quest for a logic for PTime is a mismatch between
algorithms making arbitrary choices and isomorphism-invariant logics. One
approach to overcome this problem is witnessed symmetric choice. It allows for
choices from definable orbits which are certified by definable witnessing
automorphisms.
We consider the extension of fixed-point logic with counting (IFPC) with
witnessed symmetric choice (IFPC+WSC) and a further extension with an
interpretation operator (IFPC+WSC+I). The latter operator evaluates a
subformula in the structure defined by an interpretation. This structure
possibly has other automorphisms exploitable by the WSC-operator. For similar
extensions of pure fixed-point logic (IFP) it is known that IFP+WSCI simulates
counting which IFP+WSC fails to do. For IFPC+WSC it is unknown whether the
interpretation operator increases expressiveness and thus allows studying the
relation between WSC and interpretations beyond counting.
We separate IFPC+WSC from IFPC+WSCI by showing that IFPC+WSC is not closed
under FO-interpretations. By the same argument, we answer an open question of
Dawar and Richerby regarding non-witnessed symmetric choice in IFP.
Additionally, we prove that nesting WSC-operators increases the expressiveness
using the so-called CFI graphs. We show that if IFPC+WSC+I canonizes a
particular class of base graphs, then it also canonizes the corresponding CFI
graphs. This differs from various other logics, where CFI graphs provide
difficult instances.Comment: 46 pages, 5 figures, [v2] and [v3] Corrected minor mistakes and added
figure
Choiceless Polynomial Time with Witnessed Symmetric Choice
We extend Choiceless Polynomial Time (CPT), the currently only remaining
promising candidate in the quest for a logic capturing PTime, so that this
extended logic has the following property: for every class of structures for
which isomorphism is definable, the logic automatically captures PTime.
For the construction of this logic we extend CPT by a witnessed symmetric
choice operator. This operator allows for choices from definable orbits. But,
to ensure polynomial time evaluation, automorphisms have to be provided to
certify that the choice set is indeed an orbit.
We argue that, in this logic, definable isomorphism implies definable
canonization. Thereby, our construction removes the non-trivial step of
extending isomorphism definability results to canonization. This step was a
part of proofs that show that CPT or other logics capture PTime on a particular
class of structures. The step typically required substantial extra effort.Comment: 65 pages. Full version of a paper to appear at LICS 22. v2: corrected
typos and small mistake