15,642 research outputs found
The Graph Isomorphism Problem and approximate categories
It is unknown whether two graphs can be tested for isomorphism in polynomial
time. A classical approach to the Graph Isomorphism Problem is the
d-dimensional Weisfeiler-Lehman algorithm. The d-dimensional WL-algorithm can
distinguish many pairs of graphs, but the pairs of non-isomorphic graphs
constructed by Cai, Furer and Immerman it cannot distinguish. If d is fixed,
then the WL-algorithm runs in polynomial time. We will formulate the Graph
Isomorphism Problem as an Orbit Problem: Given a representation V of an
algebraic group G and two elements v_1,v_2 in V, decide whether v_1 and v_2 lie
in the same G-orbit. Then we attack the Orbit Problem by constructing certain
approximate categories C_d(V), d=1,2,3,... whose objects include the elements
of V. We show that v_1 and v_2 are not in the same orbit by showing that they
are not isomorphic in the category C_d(V) for some d. For every d this gives us
an algorithm for isomorphism testing. We will show that the WL-algorithms
reduce to our algorithms, but that our algorithms cannot be reduced to the
WL-algorithms. Unlike the Weisfeiler-Lehman algorithm, our algorithm can
distinguish the Cai-Furer-Immerman graphs in polynomial time.Comment: 29 page
Graph Similarity and Approximate Isomorphism
The graph similarity problem, also known as approximate graph isomorphism or graph matching problem, has been extensively studied in the machine learning community, but has not received much attention in the algorithms community: Given two graphs G,H of the same order n with adjacency matrices A_G,A_H, a well-studied measure of similarity is the Frobenius distance dist(G,H):=min_{pi}|A_G^{pi}-A_H|_F, where pi ranges over all permutations of the vertex set of G, where A_G^pi denotes the matrix obtained from A_G by permuting rows and columns according to pi, and where |M |_F is the Frobenius norm of a matrix M. The (weighted) graph similarity problem, denoted by GSim (WSim), is the problem of computing this distance for two graphs of same order. This problem is closely related to the notoriously hard quadratic assignment problem (QAP), which is known to be NP-hard even for severely restricted cases.
It is known that GSim (WSim) is NP-hard; we strengthen this hardness result by showing that the problem remains NP-hard even for the class of trees. Identifying the boundary of tractability for WSim is best done in the framework of linear algebra. We show that WSim is NP-hard as long as one of the matrices has unbounded rank or negative eigenvalues: hence, the realm of tractability is restricted to positive semi-definite matrices of bounded rank. Our main result is a polynomial time algorithm for the special case where the associated (weighted) adjacency graph for one of the matrices has a bounded number of twin equivalence classes. The key parameter underlying our algorithm is the clustering number of a graph; this parameter arises in context of the spectral graph drawing machinery
Theoretically Expressive and Edge-aware Graph Learning
We propose a new Graph Neural Network that combines recent advancements in
the field. We give theoretical contributions by proving that the model is
strictly more general than the Graph Isomorphism Network and the Gated Graph
Neural Network, as it can approximate the same functions and deal with
arbitrary edge values. Then, we show how a single node information can flow
through the graph unchanged
Expectation-Complete Graph Representations with Homomorphisms
We investigate novel random graph embeddings that can be computed in expected
polynomial time and that are able to distinguish all non-isomorphic graphs in
expectation. Previous graph embeddings have limited expressiveness and either
cannot distinguish all graphs or cannot be computed efficiently for every
graph. To be able to approximate arbitrary functions on graphs, we are
interested in efficient alternatives that become arbitrarily expressive with
increasing resources. Our approach is based on Lov\'asz' characterisation of
graph isomorphism through an infinite dimensional vector of homomorphism
counts. Our empirical evaluation shows competitive results on several benchmark
graph learning tasks.Comment: accepted for publication at ICML 202
Projected Power Iteration for Network Alignment
The network alignment problem asks for the best correspondence between two
given graphs, so that the largest possible number of edges are matched. This
problem appears in many scientific problems (like the study of protein-protein
interactions) and it is very closely related to the quadratic assignment
problem which has graph isomorphism, traveling salesman and minimum bisection
problems as particular cases. The graph matching problem is NP-hard in general.
However, under some restrictive models for the graphs, algorithms can
approximate the alignment efficiently. In that spirit the recent work by Feizi
and collaborators introduce EigenAlign, a fast spectral method with convergence
guarantees for Erd\H{o}s-Reny\'i graphs. In this work we propose the algorithm
Projected Power Alignment, which is a projected power iteration version of
EigenAlign. We numerically show it improves the recovery rates of EigenAlign
and we describe the theory that may be used to provide performance guarantees
for Projected Power Alignment.Comment: 8 page
Flexible graph matching and graph edit distance using answer set programming
The graph isomorphism, subgraph isomorphism, and graph edit distance problems
are combinatorial problems with many applications. Heuristic exact and
approximate algorithms for each of these problems have been developed for
different kinds of graphs: directed, undirected, labeled, etc. However,
additional work is often needed to adapt such algorithms to different classes
of graphs, for example to accommodate both labels and property annotations on
nodes and edges. In this paper, we propose an approach based on answer set
programming. We show how each of these problems can be defined for a general
class of property graphs with directed edges, and labels and key-value
properties annotating both nodes and edges. We evaluate this approach on a
variety of synthetic and realistic graphs, demonstrating that it is feasible as
a rapid prototyping approach.Comment: To appear, PADL 202
On the Lattice Distortion Problem
We introduce and study the \emph{Lattice Distortion Problem} (LDP). LDP asks
how "similar" two lattices are. I.e., what is the minimal distortion of a
linear bijection between the two lattices? LDP generalizes the Lattice
Isomorphism Problem (the lattice analogue of Graph Isomorphism), which simply
asks whether the minimal distortion is one.
As our first contribution, we show that the distortion between any two
lattices is approximated up to a factor by a simple function of
their successive minima. Our methods are constructive, allowing us to compute
low-distortion mappings that are within a factor
of optimal in polynomial time and within a factor of optimal in
singly exponential time. Our algorithms rely on a notion of basis reduction
introduced by Seysen (Combinatorica 1993), which we show is intimately related
to lattice distortion. Lastly, we show that LDP is NP-hard to approximate to
within any constant factor (under randomized reductions), by a reduction from
the Shortest Vector Problem.Comment: This is the full version of a paper that appeared in ESA 201
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