517 research outputs found
Even Delta-Matroids and the Complexity of Planar Boolean CSPs
The main result of this paper is a generalization of the classical blossom
algorithm for finding perfect matchings. Our algorithm can efficiently solve
Boolean CSPs where each variable appears in exactly two constraints (we call it
edge CSP) and all constraints are even -matroid relations (represented
by lists of tuples). As a consequence of this, we settle the complexity
classification of planar Boolean CSPs started by Dvorak and Kupec.
Using a reduction to even -matroids, we then extend the tractability
result to larger classes of -matroids that we call efficiently
coverable. It properly includes classes that were known to be tractable before,
namely co-independent, compact, local, linear and binary, with the following
caveat: we represent -matroids by lists of tuples, while the last two
use a representation by matrices. Since an matrix can represent
exponentially many tuples, our tractability result is not strictly stronger
than the known algorithm for linear and binary -matroids.Comment: 33 pages, 9 figure
A Hall-type theorem with algorithmic consequences in planar graphs
Given a graph , for a vertex set , let denote
the set of vertices in that have a neighbor in . Extending the concept
of binding number of graphs by Woodall~(1973), for a vertex set , we define the binding number of , denoted by \bind(X), as the maximum
number such that for every where it holds
that . Given this definition, we prove that if a graph
contains a subset with \bind(X)= 1/k where is an integer, then
possesses a matching of size at least . Using this statement, we
derive tight bounds for the estimators of the matching size in planar graphs.
These estimators are previously used in designing sublinear space algorithms
for approximating the maching size in the data stream model of computation. In
particular, we show that the number of locally superior vertices is a
factor approximation of the matching size in planar graphs. The previous
analysis by Jowhari (2023) proved a approximation factor. As another
application, we show a simple variant of an estimator by Esfandiari \etal
(2015) achieves factor approximation of the matching size in planar graphs.
Namely, let be the number of edges with both endpoints having degree at
most and let be the number of vertices with degree at least . We
prove that when the graph is planar, the size of matching is at least
. This result generalizes a known fact that every planar graph on
vertices with minimum degree has a matching of size at least .Comment: 9 page
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