7,759 research outputs found
An FPT 2-Approximation for Tree-Cut Decomposition
The tree-cut width of a graph is a graph parameter defined by Wollan [J.
Comb. Theory, Ser. B, 110:47-66, 2015] with the help of tree-cut
decompositions. In certain cases, tree-cut width appears to be more adequate
than treewidth as an invariant that, when bounded, can accelerate the
resolution of intractable problems. While designing algorithms for problems
with bounded tree-cut width, it is important to have a parametrically tractable
way to compute the exact value of this parameter or, at least, some constant
approximation of it. In this paper we give a parameterized 2-approximation
algorithm for the computation of tree-cut width; for an input -vertex graph
and an integer , our algorithm either confirms that the tree-cut width
of is more than or returns a tree-cut decomposition of certifying
that its tree-cut width is at most , in time .
Prior to this work, no constructive parameterized algorithms, even approximated
ones, existed for computing the tree-cut width of a graph. As a consequence of
the Graph Minors series by Robertson and Seymour, only the existence of a
decision algorithm was known.Comment: 17 pages, 3 figure
How the structure of precedence constraints may change the complexity class of scheduling problems
This survey aims at demonstrating that the structure of precedence
constraints plays a tremendous role on the complexity of scheduling problems.
Indeed many problems can be NP-hard when considering general precedence
constraints, while they become polynomially solvable for particular precedence
constraints. We also show that there still are many very exciting challenges in
this research area
On The Power of Tree Projections: Structural Tractability of Enumerating CSP Solutions
The problem of deciding whether CSP instances admit solutions has been deeply
studied in the literature, and several structural tractability results have
been derived so far. However, constraint satisfaction comes in practice as a
computation problem where the focus is either on finding one solution, or on
enumerating all solutions, possibly projected to some given set of output
variables. The paper investigates the structural tractability of the problem of
enumerating (possibly projected) solutions, where tractability means here
computable with polynomial delay (WPD), since in general exponentially many
solutions may be computed. A general framework based on the notion of tree
projection of hypergraphs is considered, which generalizes all known
decomposition methods. Tractability results have been obtained both for classes
of structures where output variables are part of their specification, and for
classes of structures where computability WPD must be ensured for any possible
set of output variables. These results are shown to be tight, by exhibiting
dichotomies for classes of structures having bounded arity and where the tree
decomposition method is considered
A Logic of Reachable Patterns in Linked Data-Structures
We define a new decidable logic for expressing and checking invariants of
programs that manipulate dynamically-allocated objects via pointers and
destructive pointer updates. The main feature of this logic is the ability to
limit the neighborhood of a node that is reachable via a regular expression
from a designated node. The logic is closed under boolean operations
(entailment, negation) and has a finite model property. The key technical
result is the proof of decidability. We show how to express precondition,
postconditions, and loop invariants for some interesting programs. It is also
possible to express properties such as disjointness of data-structures, and
low-level heap mutations. Moreover, our logic can express properties of
arbitrary data-structures and of an arbitrary number of pointer fields. The
latter provides a way to naturally specify postconditions that relate the
fields on entry to a procedure to the fields on exit. Therefore, it is possible
to use the logic to automatically prove partial correctness of programs
performing low-level heap mutations
Bidimensionality of Geometric Intersection Graphs
Let B be a finite collection of geometric (not necessarily convex) bodies in
the plane. Clearly, this class of geometric objects naturally generalizes the
class of disks, lines, ellipsoids, and even convex polygons. We consider
geometric intersection graphs GB where each body of the collection B is
represented by a vertex, and two vertices of GB are adjacent if the
intersection of the corresponding bodies is non-empty. For such graph classes
and under natural restrictions on their maximum degree or subgraph exclusion,
we prove that the relation between their treewidth and the maximum size of a
grid minor is linear. These combinatorial results vastly extend the
applicability of all the meta-algorithmic results of the bidimensionality
theory to geometrically defined graph classes
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