2,958 research outputs found
A Unifying Theory for Graph Transformation
The field of graph transformation studies the rule-based transformation of graphs. An important branch is the algebraic graph transformation tradition, in which approaches are defined and studied using the language of category theory. Most algebraic graph transformation approaches (such as DPO, SPO, SqPO, and AGREE) are opinionated about the local contexts that are allowed around matches for rules, and about how replacement in context should work exactly. The approaches also differ considerably in their underlying formal theories and their general expressiveness (e.g., not all frameworks allow duplication). This dissertation proposes an expressive algebraic graph transformation approach, called PBPO+, which is an adaptation of PBPO by Corradini et al. The central contribution is a proof that PBPO+ subsumes (under mild restrictions) DPO, SqPO, AGREE, and PBPO in the important categorical setting of quasitoposes. This result allows for a more unified study of graph transformation metatheory, methods, and tools. A concrete example of this is found in the second major contribution of this dissertation: a graph transformation termination method for PBPO+, based on decreasing interpretations, and defined for general categories. By applying the proposed encodings into PBPO+, this method can also be applied for DPO, SqPO, AGREE, and PBPO
Computational Analyses of Metagenomic Data
Metagenomics studies the collective microbial genomes extracted from a particular environment without requiring the culturing or isolation of individual genomes, addressing questions revolving around the composition, functionality, and dynamics of microbial communities. The intrinsic complexity of metagenomic data and the diversity of applications call for efficient and accurate computational methods in data handling. In this thesis, I present three primary projects that collectively focus on the computational analysis of metagenomic data, each addressing a distinct topic.
In the first project, I designed and implemented an algorithm named Mapbin for reference-free genomic binning of metagenomic assemblies. Binning aims to group a mixture of genomic fragments based on their genome origin. Mapbin enhances binning results by building a multilayer network that combines the initial binning, assembly graph, and read-pairing information from paired-end sequencing data. The network is further partitioned by the community-detection algorithm, Infomap, to yield a new binning result. Mapbin was tested on multiple simulated and real datasets. The results indicated an overall improvement in the common binning quality metrics.
The second and third projects are both derived from ImMiGeNe, a collaborative and multidisciplinary study investigating the interplay between gut microbiota, host genetics, and immunity in stem-cell transplantation (SCT) patients. In the second project, I conducted microbiome analyses for the metagenomic data. The workflow included the removal of contaminant reads and multiple taxonomic and functional profiling. The results revealed that the SCT recipients' samples yielded significantly fewer reads with heavy contamination of the host DNA, and their microbiomes displayed evident signs of dysbiosis. Finally, I discussed several inherent challenges posed by extremely low levels of target DNA and high levels of contamination in the recipient samples, which cannot be rectified solely through bioinformatics approaches.
The primary goal of the third project is to design a set of primers that can be used to cover bacterial flagellin genes present in the human gut microbiota. Considering the notable diversity of flagellins, I incorporated a method to select representative bacterial flagellin gene sequences, a heuristic approach based on established primer design methods to generate a degenerate primer set, and a selection method to filter genes unlikely to occur in the human gut microbiome. As a result, I successfully curated a reduced yet representative set of primers that would be practical for experimental implementation
Large cliques or cocliques in hypergraphs with forbidden order-size pairs
The well-known Erdős-Hajnal conjecture states that for any graph , there exists such that every -vertex graph that contains no induced copy of has a homogeneous set of size at least . We consider a variant of the Erdős-Hajnal problem for hypergraphs where we forbid a family of hypergraphs described by their orders and sizes. For graphs, we observe that if we forbid induced subgraphs on vertices and edges for any positive and , then we obtain large homogeneous sets. For triple systems, in the first nontrivial case , for every , we give bounds on the minimum size of a homogeneous set in a triple system where the number of edges spanned by every four vertices is not in . In most cases the bounds are essentially tight. We also determine, for all , whether the growth rate is polynomial or polylogarithmic. Some open problems remain
Algorithms and complexity for approximately counting hypergraph colourings and related problems
The past decade has witnessed advancements in designing efficient algorithms for approximating the number of solutions to constraint satisfaction problems (CSPs), especially in the local lemma regime. However, the phase transition for the computational tractability is not known. This thesis is dedicated to the prototypical problem of this kind of CSPs, the hypergraph colouring. Parameterised by the number of colours q, the arity of each hyperedge k, and the vertex maximum degree Δ, this problem falls into the regime of Lovász local lemma when Δ ≲ qᵏ. In prior, however, fast approximate counting algorithms exist when Δ ≲ qᵏ/³, and there is no known inapproximability result. In pursuit of this, our contribution is two-folded, stated as follows.
• When q, k ≥ 4 are evens and Δ ≥ 5·qᵏ/², approximating the number of hypergraph colourings is NP-hard.
• When the input hypergraph is linear and Δ ≲ qᵏ/², a fast approximate counting algorithm does exist
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
The Complexity of Recognizing Geometric Hypergraphs
As set systems, hypergraphs are omnipresent and have various representations
ranging from Euler and Venn diagrams to contact representations. In a geometric
representation of a hypergraph , each vertex is associated
with a point and each hyperedge is associated
with a connected set such that for all . We say that a given
hypergraph is representable by some (infinite) family of sets in
, if there exist and such
that is a geometric representation of . For a family F, we define
RECOGNITION(F) as the problem to determine if a given hypergraph is
representable by F. It is known that the RECOGNITION problem is
-hard for halfspaces in . We study the
families of translates of balls and ellipsoids in , as well as of
other convex sets, and show that their RECOGNITION problems are also
-complete. This means that these recognition problems are
equivalent to deciding whether a multivariate system of polynomial equations
with integer coefficients has a real solution.Comment: Appears in the Proceedings of the 31st International Symposium on
Graph Drawing and Network Visualization (GD 2023) 17 pages, 11 figure
Parameterized Complexity of Binary CSP: Vertex Cover, Treedepth, and Related Parameters
We investigate the parameterized complexity of Binary CSP parameterized by the vertex cover number and the treedepth of the constraint graph, as well as by a selection of related modulator-based parameters. The main findings are as follows:
- Binary CSP parameterized by the vertex cover number is W[3]-complete. More generally, for every positive integer d, Binary CSP parameterized by the size of a modulator to a treedepth-d graph is W[2d+1]-complete. This provides a new family of natural problems that are complete for odd levels of the W-hierarchy.
- We introduce a new complexity class XSLP, defined so that Binary CSP parameterized by treedepth is complete for this class. We provide two equivalent characterizations of XSLP: the first one relates XSLP to a model of an alternating Turing machine with certain restrictions on conondeterminism and space complexity, while the second one links XSLP to the problem of model-checking first-order logic with suitably restricted universal quantification. Interestingly, the proof of the machine characterization of XSLP uses the concept of universal trees, which are prominently featured in the recent work on parity games.
- We describe a new complexity hierarchy sandwiched between the W-hierarchy and the A-hierarchy: For every odd t, we introduce a parameterized complexity class S[t] with W[t] ? S[t] ? A[t], defined using a parameter that interpolates between the vertex cover number and the treedepth. We expect that many of the studied classes will be useful in the future for pinpointing the complexity of various structural parameterizations of graph problems
The success probability in Levine's hat problem, and independent sets in graphs
Lionel Levine's hat challenge has players, each with a (very large, or
infinite) stack of hats on their head, each hat independently colored at random
black or white. The players are allowed to coordinate before the random colors
are chosen, but not after. Each player sees all hats except for those on her
own head. They then proceed to simultaneously try and each pick a black hat
from their respective stacks. They are proclaimed successful only if they are
all correct. Levine's conjecture is that the success probability tends to zero
when the number of players grows. We prove that this success probability is
strictly decreasing in the number of players, and present some connections to
problems in graph theory: relating the size of the largest independent set in a
graph and in a random induced subgraph of it, and bounding the size of a set of
vertices intersecting every maximum-size independent set in a graph.Comment: arXiv admin note: substantial text overlap with arXiv:2103.01541,
arXiv:2103.0599
A precise condition for independent transversals in bipartite covers
Given a bipartite graph in which any vertex in
(resp. ) has degree at most (resp. ), suppose there is a
partition of that is a refinement of the bipartition such
that the parts in (resp. ) have size at least (resp. ).
We prove that the condition is sufficient for the
existence of an independent set of vertices of that is simultaneously
transversal to the partition, and show moreover that this condition is sharp.
This result is a bipartite refinement of two well-known results on independent
transversals, one due to the second author the other due to Szab\'o and Tardos.Comment: 10 pages, 2 figure
Borel versions of the Local Lemma and LOCAL algorithms for graphs of finite asymptotic separation index
Asymptotic separation index is a parameter that measures how easily a Borel
graph can be approximated by its subgraphs with finite components. In contrast
to the more classical notion of hyperfiniteness, asymptotic separation index is
well-suited for combinatorial applications in the Borel setting. The main
result of this paper is a Borel version of the Lov\'asz Local Lemma -- a
powerful general-purpose tool in probabilistic combinatorics -- under a finite
asymptotic separation index assumption. As a consequence, we show that locally
checkable labeling problems that are solvable by efficient randomized
distributed algorithms admit Borel solutions on bounded degree Borel graphs
with finite asymptotic separation index. From this we derive a number of
corollaries, for example a Borel version of Brooks's theorem for graphs with
finite asymptotic separation index
- …