5,244 research outputs found
Counting Subgraphs in Somewhere Dense Graphs
We study the problems of counting copies and induced copies of a small
pattern graph in a large host graph . Recent work fully classified the
complexity of those problems according to structural restrictions on the
patterns . In this work, we address the more challenging task of analysing
the complexity for restricted patterns and restricted hosts. Specifically we
ask which families of allowed patterns and hosts imply fixed-parameter
tractability, i.e., the existence of an algorithm running in time for some computable function . Our main results present
exhaustive and explicit complexity classifications for families that satisfy
natural closure properties. Among others, we identify the problems of counting
small matchings and independent sets in subgraph-closed graph classes
as our central objects of study and establish the following crisp
dichotomies as consequences of the Exponential Time Hypothesis: (1) Counting
-matchings in a graph is fixed-parameter tractable if and
only if is nowhere dense. (2) Counting -independent sets in a
graph is fixed-parameter tractable if and only if
is nowhere dense. Moreover, we obtain almost tight conditional
lower bounds if is somewhere dense, i.e., not nowhere dense.
These base cases of our classifications subsume a wide variety of previous
results on the matching and independent set problem, such as counting
-matchings in bipartite graphs (Curticapean, Marx; FOCS 14), in
-colourable graphs (Roth, Wellnitz; SODA 20), and in degenerate graphs
(Bressan, Roth; FOCS 21), as well as counting -independent sets in bipartite
graphs (Curticapean et al.; Algorithmica 19).Comment: 35 pages, 3 figures, 4 tables, abstract shortened due to ArXiv
requirement
Independent sets, matchings, and occupancy fractions
We prove tight upper bounds on the logarithmic derivative of the independence and matching polynomials of d-regular graphs. For independent sets, this theorem is a strengthening of Kahn's result that a disjoint union of copies of Kd;d maximizes the number of independent sets of a bipartite d-regular graph, Galvin and Tetali's result that the independence polynomial is maximized by the same, and Zhao's extension of both results to all d-regular graphs. For matchings, this shows that the matching polynomial and the total number of matchings of a d-regular graph are maximized by a union of copies of Kd;d. Using this we prove the asymptotic upper matching conjecture of Friedland, Krop, Lundow, and Markstrom. In probabilistic language, our main theorems state that for all d-regular graphs and all �, the occupancy fraction of the hard-core model and the edge occupancy fraction of the monomer-dimer model with fugacity � are maximized by Kd;d. Our method involves constrained optimization problems over distributions of random variables and applies to all d-regular graphs directly, without a reduction to the bipartite case. Using a variant of the method we prove a lower bound on the occupancy fraction of the hard-core model on any d-regular, vertex-transitive, bipartite graph: the occupancy fraction of such a graph is strictly greater than the occupancy fraction of the unique translationinvariant hard-core measure on the infinite d-regular tre
Linear-Time Algorithms for Maximum-Weight Induced Matchings and Minimum Chain Covers in Convex Bipartite Graphs
A bipartite graph is convex if the vertices in can be
linearly ordered such that for each vertex , the neighbors of are
consecutive in the ordering of . An induced matching of is a
matching such that no edge of connects endpoints of two different edges of
. We show that in a convex bipartite graph with vertices and
weighted edges, an induced matching of maximum total weight can be computed in
time. An unweighted convex bipartite graph has a representation of
size that records for each vertex the first and last neighbor
in the ordering of . Given such a compact representation, we compute an
induced matching of maximum cardinality in time.
In convex bipartite graphs, maximum-cardinality induced matchings are dual to
minimum chain covers. A chain cover is a covering of the edge set by chain
subgraphs, that is, subgraphs that do not contain induced matchings of more
than one edge. Given a compact representation, we compute a representation of a
minimum chain cover in time. If no compact representation is given, the
cover can be computed in time.
All of our algorithms achieve optimal running time for the respective problem
and model. Previous algorithms considered only the unweighted case, and the
best algorithm for computing a maximum-cardinality induced matching or a
minimum chain cover in a convex bipartite graph had a running time of
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