14 research outputs found
Encoding CSP into CCS
We study encodings from CSP into asynchronous CCS with name passing and
matching, so in fact, the asynchronous pi-calculus. By doing so, we discuss two
different ways to map the multi-way synchronisation mechanism of CSP into the
two-way synchronisation mechanism of CCS. Both encodings satisfy the criteria
of Gorla except for compositionality, as both use an additional top-level
context. Following the work of Parrow and Sj\"odin, the first encoding uses a
centralised coordinator and establishes a variant of weak bisimilarity between
source terms and their translations. The second encoding is decentralised, and
thus more efficient, but ensures only a form of coupled similarity between
source terms and their translations.Comment: In Proceedings EXPRESS/SOS 2015, arXiv:1508.0634
Cyclewidth and the Grid Theorem for Perfect Matching Width of Bipartite Graphs
A connected graph G is called matching covered if every edge of G is
contained in a perfect matching. Perfect matching width is a width parameter
for matching covered graphs based on a branch decomposition. It was introduced
by Norine and intended as a tool for the structural study of matching covered
graphs, especially in the context of Pfaffian orientations. Norine conjectured
that graphs of high perfect matching width would contain a large grid as a
matching minor, similar to the result on treewidth by Robertson and Seymour. In
this paper we obtain the first results on perfect matching width since its
introduction. For the restricted case of bipartite graphs, we show that perfect
matching width is equivalent to directed treewidth and thus the Directed Grid
Theorem by Kawarabayashi and Kreutzer for directed \treewidth implies Norine's
conjecture.Comment: Manuscrip
Constant Congestion Brambles
A bramble in an undirected graph is a family of connected subgraphs of
such that for every two subgraphs and in the bramble either
or there is an edge of with one
endpoint in and the second endpoint in . The order of the
bramble is the minimum size of a vertex set that intersects all elements of a
bramble.
Brambles are objects dual to treewidth: As shown by Seymour and Thomas, the
maximum order of a bramble in an undirected graph equals one plus the
treewidth of . However, as shown by Grohe and Marx, brambles of high order
may necessarily be of exponential size: In a constant-degree -vertex
expander a bramble of order requires size exponential
in for any fixed . On the
other hand, the combination of results of Grohe and Marx and Chekuri and
Chuzhoy shows that a graph of treewidth admits a bramble of order
and size .
( and hide polylogarithmic
factors and divisors, respectively.)
In this note, we first sharpen the second bound by proving that every graph
of treewidth at least contains a bramble of order
and congestion , i.e., every vertex of is
contained in at most two elements of the bramble (thus the bramble is of size
linear in its order). Second, we provide a tight upper bound for the lower
bound of Grohe and Marx: For every , every graph
of treewidth at least contains a bramble of order
and size
Graphs with at most two moplexes
A moplex is a natural graph structure that arises when lifting Dirac's
classical theorem from chordal graphs to general graphs. However, while every
non-complete graph has at least two moplexes, little is known about structural
properties of graphs with a bounded number of moplexes. The study of these
graphs is motivated by the parallel between moplexes in general graphs and
simplicial modules in chordal graphs: Unlike in the moplex setting, properties
of chordal graphs with a bounded number of simplicial modules are well
understood. For instance, chordal graphs having at most two simplicial modules
are interval. In this work we initiate an investigation of -moplex graphs,
which are defined as graphs containing at most moplexes. Of particular
interest is the smallest nontrivial case , which forms a counterpart to
the class of interval graphs. As our main structural result, we show that the
class of connected -moplex graphs is sandwiched between the classes of
proper interval graphs and cocomparability graphs; moreover, both inclusions
are tight for hereditary classes. From a complexity theoretic viewpoint, this
leads to the natural question of whether the presence of at most two moplexes
guarantees a sufficient amount of structure to efficiently solve problems that
are known to be intractable on cocomparability graphs, but not on proper
interval graphs. We develop new reductions that answer this question negatively
for two prominent problems fitting this profile, namely Graph Isomorphism and
Max-Cut. On the other hand, we prove that every connected -moplex graph
contains a Hamiltonian path, generalising the same property of connected proper
interval graphs. Furthermore, for graphs with a higher number of moplexes, we
lift the previously known result that graphs without asteroidal triples have at
most two moplexes to the more general setting of larger asteroidal sets
Tuza's conjecture for threshold graphs
Tuza famously conjectured in 1981 that in a graph without k+1 edge-disjoint triangles, it suffices to delete at most 2k edges to obtain a triangle-free graph. The conjecture holds for graphs with small treewidth or small maximum average degree, including planar graphs. However, for dense graphs that are neither cliques nor 4-colorable, only asymptotic results are known. Here, we confirm the conjecture for threshold graphs, i.e. graphs that are both split graphs and cographs, and for co-chain graphs with both sides of the same size divisible by 4
Fixed-parameter tractability of Directed Multicut with three terminal pairs parameterized by the size of the cutset: twin-width meets flow-augmentation
We show fixed-parameter tractability of the Directed Multicut problem with
three terminal pairs (with a randomized algorithm). This problem, given a
directed graph , pairs of vertices (called terminals) ,
, and , and an integer , asks to find a set of at most
non-terminal vertices in that intersect all -paths, all
-paths, and all -paths. The parameterized complexity of this
case has been open since Chitnis, Cygan, Hajiaghayi, and Marx proved
fixed-parameter tractability of the 2-terminal-pairs case at SODA 2012, and
Pilipczuk and Wahlstr\"{o}m proved the W[1]-hardness of the 4-terminal-pairs
case at SODA 2016.
On the technical side, we use two recent developments in parameterized
algorithms. Using the technique of directed flow-augmentation [Kim, Kratsch,
Pilipczuk, Wahlstr\"{o}m, STOC 2022] we cast the problem as a CSP problem with
few variables and constraints over a large ordered domain.We observe that this
problem can be in turn encoded as an FO model-checking task over a structure
consisting of a few 0-1 matrices. We look at this problem through the lenses of
twin-width, a recently introduced structural parameter [Bonnet, Kim,
Thomass\'{e}, Watrigant, FOCS 2020]: By a recent characterization [Bonnet,
Giocanti, Ossona de Mendes, Simon, Thomass\'{e}, Toru\'{n}czyk, STOC 2022] the
said FO model-checking task can be done in FPT time if the said matrices have
bounded grid rank. To complete the proof, we show an irrelevant vertex rule: If
any of the matrices in the said encoding has a large grid minor, a vertex
corresponding to the ``middle'' box in the grid minor can be proclaimed
irrelevant -- not contained in the sought solution -- and thus reduced