49 research outputs found
Robust Hamiltonicity in families of Dirac graphs
A graph is called Dirac if its minimum degree is at least half of the number
of vertices in it. Joos and Kim showed that every collection
of Dirac graphs on the same vertex set of
size contains a Hamilton cycle transversal, i.e., a Hamilton cycle on
with a bijection such that
for every .
In this paper, we determine up to a multiplicative constant, the threshold
for the existence of a Hamilton cycle transversal in a collection of random
subgraphs of Dirac graphs in various settings. Our proofs rely on constructing
a spread measure on the set of Hamilton cycle transversals of a family of Dirac
graphs.
As a corollary, we obtain that every collection of Dirac graphs on
vertices contains at least different Hamilton cycle transversals
for some absolute constant . This is optimal up to the constant
. Finally, we show that if is sufficiently large, then every such
collection spans pairwise edge-disjoint Hamilton cycle transversals, and
this is best possible. These statements generalize classical counting results
of Hamilton cycles in a single Dirac graph
Increasing spanning forests in graphs and simplicial complexes
Let G be a graph with vertex set {1,...,n}. A spanning forest F of G is increasing if the
sequence of labels on any path starting at the minimum vertex of a tree of F forms an increasing
sequence. Hallam and Sagan showed that the generating function ISF(G, t) for increasing
spanning forests of G has all nonpositive integral roots. Furthermore they proved that, up to
a change of sign, this polynomial equals the chromatic polynomial of G precisely when 1,..., n
is a perfect elimination order for G. We give new, purely combinatorial proofs of these results
which permit us to generalize them in several ways. For example, we are able to bound the coef-
cients of ISF(G, t) using broken circuits. We are also able to extend these results to simplicial
complexes using the new notion of a cage-free complex. A generalization to labeled multigraphs
is also given. We observe that the de nition of an increasing spanning forest can be formulated
in terms of pattern avoidance, and we end by exploring spanning forests that avoid the patterns
231, 312 and 321
Exponential time algorithms via separators and random subsets
Exponential time algorithms for NP-hard problems is rich and diverse research area. This thesis aims to improve known problems with new algorithms and careful analysis of running times by extending on and using known techniques such as graph separators, and random subset selection.
We first present a polynomial-space algorithm that computes the number of independent sets of any input graph in time O(1.1389^n) for graphs with maximum degree 3 and in time O(1.2356^n) for general graphs, where n is the number of vertices. Together with the inclusion-exclusion approach of [Björklund, Husfeldt, and Koivisto 2009], this leads to a faster polynomial-space algorithm for the graph coloring problem with running time O(2.2356^n). As a byproduct, we also obtain an exponential-space O(1.2330^n) time algorithm for counting independent sets.
We also consider the family of Φ-Subset problems, where the input consists of an instance I of size N over a universe U_I of size n and the task is to check whether the universe contains a subset with property Φ (e.g., Φ could be the property of being a feedback vertex set for the input graph of size at most k). Our main tool is a simple randomized algorithm which solves Φ-Subset in time (1 + b − 1/c)^n N^O(1), provided that there is an algorithm for the Φ-Extension problem with b^{n−|X|}c^k N^O(1) running time. Here, the input for Φ-Extension is an instance I of size N over a universe UI of size n, a subset X ⊆ U_I , and an integer k, and the task is to check whether there is a set Y with X ⊆ Y ⊆ UI and |Y \ X| ≤ k with property Φ. We also derandomize this algorithm at the cost of increasing the running time by a subexponential factor in n, and we adapt it to the enumeration setting where we need to enumerate all subsets of the universe with property Φ.
Lastly we consider the application of random subset selection to approximation algorithms and improve on known approximation algorithms [Escoffier, Paschos and Tourniaire 2016] through a careful application of subset selection, and new analytic methods based on Monotone Local Search. This involves using as sub- routines Fixed Parameter Tractable Algorithms for an exponential time approximation algorithm, and we also investigate using parameterized approximation algorithms for subroutines from [Kulik and Shachnai 2020]
Exact Localisations of Feedback Sets
The feedback arc (vertex) set problem, shortened FASP (FVSP), is to transform
a given multi digraph into an acyclic graph by deleting as few arcs
(vertices) as possible. Due to the results of Richard M. Karp in 1972 it is one
of the classic NP-complete problems. An important contribution of this paper is
that the subgraphs , of all elementary
cycles or simple cycles running through some arc , can be computed in
and , respectively. We use
this fact and introduce the notion of the essential minor and isolated cycles,
which yield a priori problem size reductions and in the special case of so
called resolvable graphs an exact solution in . We show
that weighted versions of the FASP and FVSP possess a Bellman decomposition,
which yields exact solutions using a dynamic programming technique in times
and
, where , , respectively. The parameters can
be computed in , ,
respectively and denote the maximal dimension of the cycle space of all
appearing meta graphs, decoding the intersection behavior of the cycles.
Consequently, equal zero if all meta graphs are trees. Moreover, we
deliver several heuristics and discuss how to control their variation from the
optimum. Summarizing, the presented results allow us to suggest a strategy for
an implementation of a fast and accurate FASP/FVSP-SOLVER
Efficient Evaluation of Large Polynomials
In scientific computing, it is often required to evaluate a polynomial expression (or a matrix depending on some variables) at many points which are not known in advance or with coordinates containing “symbolic expressions”. In these circumstances, standard evaluation schemes, such as those based on Fast Fourier Transforms do not apply. Given a polynomial f expressed as the sum of its terms, we propose an algorithm which generates a representation of f optimizing the process of evaluating f at some points. In addition, this evaluation of f can be done efficiently in terms of data locality and parallelism. We have implemented our algorithm in the Cilk++ concurrency platform and our implementation achieves nearly linear speedup on 16 cores with large enough input. For some large polynomials, the generated schedule can be evaluated at least 10 times faster than the schedules produced by other available software solutions. Moreover, our code can handle much larger input polynomials
Recommended from our members
Combinatorics
Combinatorics is a fundamental mathematical discipline which focuses on the study of discrete objects and their properties. The current workshop brought together researchers from diverse fields such as Extremal and Probabilistic Combinatorics, Discrete Geometry, Graph theory, Combiantorial Optimization and Algebraic Combinatorics for a fruitful interaction. New results, methods and developments and future challenges were discussed. This is a report on the meeting containing abstracts of the presentations and a summary of the problem session