120 research outputs found
Maximum matching width: new characterizations and a fast algorithm for dominating set
We give alternative definitions for maximum matching width, e.g. a graph
has if and only if it is a subgraph of a chordal
graph and for every maximal clique of there exists with and such that any subset of
that is a minimal separator of is a subset of either or .
Treewidth and branchwidth have alternative definitions through intersections of
subtrees, where treewidth focuses on nodes and branchwidth focuses on edges. We
show that mm-width combines both aspects, focusing on nodes and on edges. Based
on this we prove that given a graph and a branch decomposition of mm-width
we can solve Dominating Set in time , thereby beating
whenever . Note that and these inequalities are
tight. Given only the graph and using the best known algorithms to find
decompositions, maximum matching width will be better for solving Dominating
Set whenever
Finding branch-decompositions of matroids, hypergraphs, and more
Given subspaces of a finite-dimensional vector space over a fixed finite
field , we wish to find a "branch-decomposition" of these subspaces
of width at most , that is a subcubic tree with leaves mapped
bijectively to the subspaces such that for every edge of , the sum of
subspaces associated with leaves in one component of and the sum of
subspaces associated with leaves in the other component have the intersection
of dimension at most . This problem includes the problems of computing
branch-width of -represented matroids, rank-width of graphs,
branch-width of hypergraphs, and carving-width of graphs.
We present a fixed-parameter algorithm to construct such a
branch-decomposition of width at most , if it exists, for input subspaces of
a finite-dimensional vector space over . Our algorithm is analogous
to the algorithm of Bodlaender and Kloks (1996) on tree-width of graphs. To
extend their framework to branch-decompositions of vector spaces, we developed
highly generic tools for branch-decompositions on vector spaces. The only known
previous fixed-parameter algorithm for branch-width of -represented
matroids was due to Hlin\v{e}n\'y and Oum (2008) that runs in time
where is the number of elements of the input -represented
matroid. But their method is highly indirect. Their algorithm uses the
non-trivial fact by Geelen et al. (2003) that the number of forbidden minors is
finite and uses the algorithm of Hlin\v{e}n\'y (2005) on checking monadic
second-order formulas on -represented matroids of small
branch-width. Our result does not depend on such a fact and is completely
self-contained, and yet matches their asymptotic running time for each fixed
.Comment: 73 pages, 10 figure
Deciding whether there are infinitely many prime graphs with forbidden induced subgraphs
A homogeneous set of a graph G is a set X of vertices such that 2≤|X|V(G)| and no vertex in V(G)−X has both a neighbor and a non-neighbor in X. A graph is prime if it has no homogeneous set. We present an algorithm to decide whether a class of graphs given by a finite set of forbidden induced subgraphs contains infinitely many non-isomorphic prime graphs
Scaling Law for Recommendation Models: Towards General-purpose User Representations
Recent advancement of large-scale pretrained models such as BERT, GPT-3,
CLIP, and Gopher, has shown astonishing achievements across various task
domains. Unlike vision recognition and language models, studies on
general-purpose user representation at scale still remain underexplored. Here
we explore the possibility of general-purpose user representation learning by
training a universal user encoder at large scales. We demonstrate that the
scaling law is present in user representation learning areas, where the
training error scales as a power-law with the amount of computation. Our
Contrastive Learning User Encoder (CLUE), optimizes task-agnostic objectives,
and the resulting user embeddings stretch our expectation of what is possible
to do in various downstream tasks. CLUE also shows great transferability to
other domains and companies, as performances on an online experiment shows
significant improvements in Click-Through-Rate (CTR). Furthermore, we also
investigate how the model performance is influenced by the scale factors, such
as training data size, model capacity, sequence length, and batch size.
Finally, we discuss the broader impacts of CLUE in general.Comment: Accepted at AAAI 2023. This version includes the technical appendi
Deformable Graph Transformer
Transformer-based models have recently shown success in representation
learning on graph-structured data beyond natural language processing and
computer vision. However, the success is limited to small-scale graphs due to
the drawbacks of full dot-product attention on graphs such as the quadratic
complexity with respect to the number of nodes and message aggregation from
enormous irrelevant nodes. To address these issues, we propose Deformable Graph
Transformer (DGT) that performs sparse attention via dynamically sampled
relevant nodes for efficiently handling large-scale graphs with a linear
complexity in the number of nodes. Specifically, our framework first constructs
multiple node sequences with various criteria to consider both structural and
semantic proximity. Then, combining with our learnable Katz Positional
Encodings, the sparse attention is applied to the node sequences for learning
node representations with a significantly reduced computational cost. Extensive
experiments demonstrate that our DGT achieves state-of-the-art performance on 7
graph benchmark datasets with 2.5 - 449 times less computational cost compared
to transformer-based graph models with full attention.Comment: 16 pages, 3 figure
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