1,927 research outputs found
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
Categorical Invariants of Graphs and Matroids
Graphs and matroids are two of the most important objects in combinatorics.We study invariants of graphs and matroids that behave well with respect to
certain morphisms by realizing these invariants as functors from a category of
graphs (resp. matroids).
For graphs, we study invariants that respect deletions and contractions ofedges. For an integer , we define a category of of graphs of genus at most
g where morphisms correspond to deletions and contractions. We prove that this
category is locally Noetherian and show that many graph invariants form finitely
generated modules over the category . This fact allows us to exihibit many
stabilization properties of these invariants. In particular we show that the torsion
that can occur in the homologies of the unordered configuration space of n points
in a graph and the matching complex of a graph are uniform over the entire family
of graphs with genus .
For matroids, we study valuative invariants of matroids. Given a matroid,one can define a corresponding polytope called the base polytope. Often, the base
polytope of a matroid can be decomposed into a cell complex made up of base
polytopes of other matroids. A valuative invariant of matroids is an invariant that
respects these polytope decompositions. We define a category of matroids
whose morphisms correspond to containment of base polytopes. We then define the
notion of a categorical matroid invariant which categorifies the notion of a valuative
invariant. Finally, we prove that the functor sending a matroid to its Orlik-Solomon
algebra is a categorical valuative invariant. This allows us to derive relations among
the Orlik-Solomon algebras of a matroid and matroids that decompose its base
polytope viewed as representations of any group whose action is compatible with
the polytope decomposition.
This dissertation includes previously unpublished co-authored material
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Connectivity gaps among matroids with the same enumerative invariants
Many important enumerative invariants of a matroid can be obtained from its
Tutte polynomial, and many more are determined by two stronger invariants, the
-invariant and the configuration of the matroid. We show that the
same is not true of the most basic connectivity invariants. Specifically, we
show that for any positive integer , there are pairs of matroids that have
the same configuration (and so the same -invariant and the same
Tutte polynomial) but the difference between their Tutte connectivities exceeds
, and likewise for vertical connectivity and branch-width. The examples that
we use to show this, which we construct using an operation that we introduce,
are transversal matroids that are also positroids
Dynamic Pricing Schemes in Combinatorial Markets
In combinatorial markets where buyers are self-interested, the buyers may make purchases that lead to suboptimal item allocations. As a central coordinator, our goal is to impose prices on the items of the market so that its buyers are incentivized to exclusively make optimal purchases. In this thesis, we study the question of whether dynamic pricing schemes can achieve the optimal social welfare in multi-demand combinatorial markets. This well-motivated question has been the topic of some study, but has remained mostly open, and to date, positive results are only known for extremal cases.
In this thesis, we present the current results for unit-demand, bi-demand and tri-demand markets. In the context of these results, we discuss the significance of not having a deficiency of items, which is known as the (OPT) condition. We outline an approach for handling an item deficiency, and we expose barriers to extending the known techniques to markets of larger demand
Selfadhesivity in Gaussian conditional independence structures
Selfadhesivity is a property of entropic polymatroids which guarantees that
the polymatroid can be glued to an identical copy of itself along arbitrary
restrictions such that the two pieces are independent given the common
restriction. We show that positive definite matrices satisfy this condition as
well and examine consequences for Gaussian conditional independence structures.
New axioms of Gaussian CI are obtained by applying selfadhesivity to the
previously known axioms of structural semigraphoids and orientable gaussoids.Comment: 13 pages; v3: minor revisio
Algebraic combinatorial optimization on the degree of determinants of noncommutative symbolic matrices
We address the computation of the degrees of minors of a noncommutative
symbolic matrix of form where are matrices over a
field , are noncommutative variables, are integer
weights, and is a commuting variable specifying the degree. This problem
extends noncommutative Edmonds' problem (Ivanyos et al. 2017), and can
formulate various combinatorial optimization problems. Extending the study by
Hirai 2018, and Hirai, Ikeda 2022, we provide novel duality theorems and
polyhedral characterization for the maximum degrees of minors of of all
sizes, and develop a strongly polynomial-time algorithm for computing them.
This algorithm is viewed as a unified algebraization of the classical Hungarian
method for bipartite matching and the weight-splitting algorithm for linear
matroid intersection. As applications, we provide polynomial-time algorithms
for weighted fractional linear matroid matching and linear optimization over
rank-2 Brascamp-Lieb polytopes
Polyhedral Geometry in OSCAR
OSCAR is an innovative new computer algebra system which combines and extends
the power of its four cornerstone systems - GAP (group theory), Singular
(algebra and algebraic geometry), Polymake (polyhedral geometry), and Antic
(number theory). Here, we give an introduction to polyhedral geometry
computations in OSCAR, as a chapter of the upcoming OSCAR book. In particular,
we define polytopes, polyhedra, and polyhedral fans, and we give a brief
overview about computing convex hulls and solving linear programs. Three
detailed case studies are concerned with face numbers of random polytopes,
constructions and properties of Gelfand-Tsetlin polytopes, and secondary
polytopes.Comment: 19 pages, 8 figure
Inequalities for totally nonnegative matrices: Gantmacher--Krein, Karlin, and Laplace
A real linear combination of products of minors which is nonnegative over all
totally nonnegative (TN) matrices is called a determinantal inequality for
these matrices. It is referred to as multiplicative when it compares two
collections of products of minors and additive otherwise. Set theoretic
operations preserving the class of TN matrices naturally translate into
operations preserving determinantal inequalities in this class. We introduce
set row/column operations that act directly on all determinantal inequalities
for TN matrices, and yield further inequalities for these matrices. These
operations assist in revealing novel additive inequalities for TN matrices
embedded in the classical identities due to Laplace Mem Acad Sciences
Paris and Karlin In particular, for any square TN matrix
these derived inequalities generalize -- to every i^{\mbox{th}} row of
and j^{\mbox{th}} column of -- the classical Gantmacher--Krein
fluctuating inequalities for Furthermore, our row/column
operations reveal additional undiscovered fluctuating inequalities for TN
matrices.
The introduced set row/column operations naturally birth an algorithm that
can detect certain determinantal expressions that do not form an inequality for
TN matrices. However, the algorithm completely characterizes the multiplicative
inequalities comparing products of pairs of minors. Moreover, the underlying
row/column operations add that these inequalities are offshoots of certain
''complementary/higher'' ones. These novel results seem very natural, and in
addition thoroughly describe and enrich the classification of these
multiplicative inequalities due to Fallat--Gekhtman--Johnson Adv Appl
Math and later Skandera J Algebraic Comb Comment: 2 figures, 39 pages, and minor adjustments in the expositio
-Module Techniques for Solving Differential Equations in the Context of Feynman Integrals
Feynman integrals are solutions to linear partial differential equations with
polynomial coefficients. Using a triangle integral with general exponents as a
case in point, we compare -module methods to dedicated methods developed for
solving differential equations appearing in the context of Feynman integrals,
and provide a dictionary of the relevant concepts. In particular, we implement
an algorithm due to Saito, Sturmfels, and Takayama to derive canonical series
solutions of regular holonomic -ideals, and compare them to asymptotic
series derived by the respective Fuchsian systems.Comment: 35 pages, 2 figures, 2 appendices; comments welcom
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