7 research outputs found
Edge exploration of temporal graphs
We introduce a natural temporal analogue of Eulerian circuits and prove that,
in contrast with the static case, it is NP-hard to determine whether a given
temporal graph is temporally Eulerian even if strong restrictions are placed on
the structure of the underlying graph and each edge is active at only three
times. However, we do obtain an FPT-algorithm with respect to a new parameter
called interval-membership-width which restricts the times assigned to
different edges; we believe that this parameter will be of independent interest
for other temporal graph problems. Our techniques also allow us to resolve two
open question of Akrida, Mertzios and Spirakis [CIAC 2019] concerning a related
problem of exploring temporal stars. Furthermore, we introduce a vertex-variant
of interval-membership-width (which can be arbitrarily larger than its
edge-counterpart) and use it to obtain an FPT-time algorithm for a natural
vertex-exploration problem that remains hard even when
interval-membership-width is bounded.Comment: Extended abstract of this paper appeared in IWOCA 2021: Combinatorial
Algorithms pp 107-121 (doi: https://doi.org/10.1007/978-3-030-79987-8_8
Generalizing graph decompositions
The Latin aphorism ‘divide et impera’ conveys a simple, but central idea in mathematics and computer science: ‘split your problem recursively into smaller parts, attack the parts, and conquer the whole’. There is a vast literature on how to do this on graphs. But often we need to compute on other structures (decorated graphs or perhaps algebraic objects such as groups) for which we do not have a wealth of decomposition methods. This thesis attacks this problem head on: we propose new decomposition methods in a variety of settings.
In the setting of directed graphs, we introduce a new tree-width analogue called directed branch-width. We show that parameterizing by directed branch-width allows us to obtain linear-time algorithms for problems such as directed Hamilton Path and Max-Cut which are intractable by any other known directed analogue of tree-width. In fact, the algorithmic success of our new measure is more far-reaching: by proving algorithmic meta-theorems parameterized by directed branch-width, we deduce linear-time algorithms for all problems expressable in a variant of monadic second-order logic.
Moving on from directed graphs, we then provide a meta-answer to the broader question of obtaining tree-width analogues for objects other than simple graphs. We do so introducing the theory of spined categories and triangualtion functors which constitutes a vast category-theoretic abstraction of a definition of tree-width due to Halin. Our theory acts as a black box for the definition and discovery of tree-width-like parameters in new settings: given a spined category as input, it yields an appropriate tree-width analogue as output.
Finally we study temporal graphs: these are graphs whose edges appear and disappear over time. Many problems on temporal graphs are intractable even when their topology is severely restricted (such as being a tree or even a star); thus, to be able to conquer, we need decompositions that take temporal information into account. We take these considerations to heart and define a purely temporal width measure called interval-membership-width which allows us to employ dynamic programming (i.e. divide and conquer) techniques on temporal graphs whose times are sufficiently well-structured, regardless of the underlying topology
Compositional Algorithms on Compositional Data: Deciding Sheaves on Presheaves
Algorithmicists are well-aware that fast dynamic programming algorithms are
very often the correct choice when computing on compositional (or even
recursive) graphs. Here we initiate the study of how to generalize this
folklore intuition to mathematical structures writ large. We achieve this
horizontal generality by adopting a categorial perspective which allows us to
show that: (1) structured decompositions (a recent, abstract generalization of
many graph decompositions) define Grothendieck topologies on categories of data
(adhesive categories) and that (2) any computational problem which can be
represented as a sheaf with respect to these topologies can be decided in
linear time on classes of inputs which admit decompositions of bounded width
and whose decomposition shapes have bounded feedback vertex number. This
immediately leads to algorithms on objects of any C-set category; these include
-- to name but a few examples -- structures such as: symmetric graphs, directed
graphs, directed multigraphs, hypergraphs, directed hypergraphs, databases,
simplicial complexes, circular port graphs and half-edge graphs.
Thus we initiate the bridging of tools from sheaf theory, structural graph
theory and parameterized complexity theory; we believe this to be a very
fruitful approach for a general, algebraic theory of dynamic programming
algorithms. Finally we pair our theoretical results with concrete
implementations of our main algorithmic contribution in the AlgebraicJulia
ecosystem.Comment: Revised and simplified notation and improved exposition. The
companion code can be found here:
https://github.com/AlgebraicJulia/StructuredDecompositions.j
Structured Decompositions: Structural and Algorithmic Compositionality
We introduce structured decompositions: category-theoretic generalizations of
many combinatorial invariants -- including tree-width, layered tree-width,
co-tree-width and graph decomposition width -- which have played a central role
in the study of structural and algorithmic compositionality in both graph
theory and parameterized complexity. Structured decompositions allow us to
generalize combinatorial invariants to new settings (for example decompositions
of matroids) in which they describe algorithmically useful structural
compositionality. As an application of our theory we prove an algorithmic meta
theorem for the Sub_P-composition problem which, when instantiated in the
category of graphs, yields compositional algorithms for NP-hard problems such
as: Maximum Bipartite Subgraph, Maximum Planar Subgraph and Longest Path
Directed branch-width: A directed analogue of tree-width
We introduce a new digraph width measure called directed branch-width. To do
this, we generalize a characterization of graph classes of bounded tree-width
in terms of their line graphs to digraphs.
Under parameterizations by directed branch-width we obtain linear time
algorithms for many problems, such as directed Hamilton path and Max-Cut, which
are hard when parameterized by other known directed width measures. More
generally, we obtain an algorithmic meta-theorem for the model-checking problem
for a restricted variant of MSO_2-logic on classes of bounded directed
branch-width
Search-Space Reduction via Essential Vertices
We investigate preprocessing for vertex-subset problems on graphs. While the notion of kernelization, originating in parameterized complexity theory, is a formalization of provably effective preprocessing aimed at reducing the total instance size, our focus is on finding a non-empty vertex set that belongs to an optimal solution. This decreases the size of the remaining part of the solution which still has to be found, and therefore shrinks the search space of fixed-parameter tractable algorithms for parameterizations based on the solution size. We introduce the notion of a c-essential vertex as one that is contained in all c-approximate solutions. For several classic combinatorial problems such as Odd Cycle Transversal and Directed Feedback Vertex Set, we show that under mild conditions a polynomial-time preprocessing algorithm can find a subset of an optimal solution that contains all 2-essential vertices, by exploiting packing/covering duality. This leads to FPT algorithms to solve these problems where the exponential term in the running time depends only on the number of non-essential vertices in the solution