665 research outputs found
Simplicial and Cellular Trees
Much information about a graph can be obtained by studying its spanning
trees. On the other hand, a graph can be regarded as a 1-dimensional cell
complex, raising the question of developing a theory of trees in higher
dimension. As observed first by Bolker, Kalai and Adin, and more recently by
numerous authors, the fundamental topological properties of a tree --- namely
acyclicity and connectedness --- can be generalized to arbitrary dimension as
the vanishing of certain cellular homology groups. This point of view is
consistent with the matroid-theoretic approach to graphs, and yields
higher-dimensional analogues of classical enumerative results including
Cayley's formula and the matrix-tree theorem. A subtlety of the
higher-dimensional case is that enumeration must account for the possibility of
torsion homology in trees, which is always trivial for graphs. Cellular trees
are the starting point for further high-dimensional extensions of concepts from
algebraic graph theory including the critical group, cut and flow spaces, and
discrete dynamical systems such as the abelian sandpile model.Comment: 39 pages (including 5-page bibliography); 5 figures. Chapter for
forthcoming IMA volume "Recent Trends in Combinatorics
FPT algorithms for finding near-cliques in c-closed graphs
Finding large cliques or cliques missing a few edges is a fundamental algorithmic task in the study of real-world graphs, with applications in community detection, pattern recognition, and clustering. A number of effective backtracking-based heuristics for these problems have emerged from recent empirical work in social network analysis. Given the NP-hardness of variants of clique counting, these results raise a challenge for beyond worst-case analysis of these problems. Inspired by the triadic closure of real-world graphs, Fox et al. (SICOMP 2020) introduced the notion of c-closed graphs and proved that maximal clique enumeration is fixed-parameter tractable with respect to c. In practice, due to noise in data, one wishes to actually discover “near-cliques”, which can be characterized as cliques with a sparse subgraph removed. In this work, we prove that many different kinds of maximal near-cliques can be enumerated in polynomial time (and FPT in c) for c-closed graphs. We study various established notions of such substructures, including k-plexes, complements of bounded-degeneracy and bounded-treewidth graphs. Interestingly, our algorithms follow relatively simple backtracking procedures, analogous to what is done in practice. Our results underscore the significance of the c-closed graph class for theoretical understanding of social network analysis
Computing Dense and Sparse Subgraphs of Weakly Closed Graphs
A graph is weakly -closed if every induced subgraph of
contains one vertex such that for each non-neighbor of it holds
that . The weak closure of a graph,
recently introduced by Fox et al. [SIAM J. Comp. 2020], is the smallest number
such that is weakly -closed. This graph parameter is never larger
than the degeneracy (plus one) and can be significantly smaller. Extending the
work of Fox et al. [SIAM J. Comp. 2020] on clique enumeration, we show that
several problems related to finding dense subgraphs, such as the enumeration of
bicliques and -plexes, are fixed-parameter tractable with respect to
. Moreover, we show that the problem of determining whether a weakly
-closed graph has a subgraph on at least vertices that belongs
to a graph class which is closed under taking subgraphs admits a
kernel with at most vertices. Finally, we provide fixed-parameter
algorithms for Independent Dominating Set and Dominating Clique when
parameterized by where is the solution size.Comment: Appeared in ISAAC '2
Geometry of rank tests
We study partitions of the symmetric group which have desirable geometric
properties. The statistical tests defined by such partitions involve counting
all permutations in the equivalence classes. These permutations are the linear
extensions of partially ordered sets specified by the data. Our methods refine
rank tests of non-parametric statistics, such as the sign test and the runs
test, and are useful for the exploratory analysis of ordinal data. Convex rank
tests correspond to probabilistic conditional independence structures known as
semi-graphoids. Submodular rank tests are classified by the faces of the cone
of submodular functions, or by Minkowski summands of the permutohedron. We
enumerate all small instances of such rank tests. Graphical tests correspond to
both graphical models and to graph associahedra, and they have excellent
statistical and algorithmic properties.Comment: 8 pages, 4 figures. See also http://bio.math.berkeley.edu/ranktests/.
v2: Expanded proofs, revised after reviewer comment
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