389,293 research outputs found
The Triangle Closure is a Polyhedron
Recently, cutting planes derived from maximal lattice-free convex sets have
been studied intensively by the integer programming community. An important
question in this research area has been to decide whether the closures
associated with certain families of lattice-free sets are polyhedra. For a long
time, the only result known was the celebrated theorem of Cook, Kannan and
Schrijver who showed that the split closure is a polyhedron. Although some
fairly general results were obtained by Andersen, Louveaux and Weismantel [ An
analysis of mixed integer linear sets based on lattice point free convex sets,
Math. Oper. Res. 35 (2010), 233--256] and Averkov [On finitely generated
closures in the theory of cutting planes, Discrete Optimization 9 (2012), no.
4, 209--215], some basic questions have remained unresolved. For example,
maximal lattice-free triangles are the natural family to study beyond the
family of splits and it has been a standing open problem to decide whether the
triangle closure is a polyhedron. In this paper, we show that when the number
of integer variables the triangle closure is indeed a polyhedron and its
number of facets can be bounded by a polynomial in the size of the input data.
The techniques of this proof are also used to give a refinement of necessary
conditions for valid inequalities being facet-defining due to Cornu\'ejols and
Margot [On the facets of mixed integer programs with two integer variables and
two constraints, Mathematical Programming 120 (2009), 429--456] and obtain
polynomial complexity results about the mixed integer hull.Comment: 39 pages; made self-contained by merging material from
arXiv:1107.5068v
Ranking Sets of Objects: The Complexity of Avoiding Impossibility Results
The problem of lifting a preference order on a set of objects to a preference
order on a family of subsets of this set is a fundamental problem with a wide
variety of applications in AI. The process is often guided by axioms
postulating properties the lifted order should have. Well-known impossibility
results by Kannai and Peleg and by Barber\`a and Pattanaik tell us that some
desirable axioms - namely dominance and (strict) independence - are not jointly
satisfiable for any linear order on the objects if all non-empty sets of
objects are to be ordered. On the other hand, if not all non-empty sets of
objects are to be ordered, the axioms are jointly satisfiable for all linear
orders on the objects for some families of sets. Such families are very
important for applications as they allow for the use of lifted orders, for
example, in combinatorial voting. In this paper, we determine the computational
complexity of recognizing such families. We show that it is -complete
to decide for a given family of subsets whether dominance and independence or
dominance and strict independence are jointly satisfiable for all linear orders
on the objects if the lifted order needs to be total. Furthermore, we show that
the problem remains coNP-complete if the lifted order can be incomplete.
Additionally, we show that the complexity of these problem can increase
exponentially if the family of sets is not given explicitly but via a succinct
domain restriction. Finally, we show that it is NP-complete to decide for
family of subsets whether dominance and independence or dominance and strict
independence are jointly satisfiable for at least one linear orders on the
objects
Beyond the Existential Theory of the Reals
We show that completeness at higher levels of the theory of the reals is a
robust notion (under changing the signature and bounding the domain of the
quantifiers). This mends recognized gaps in the hierarchy, and leads to
stronger completeness results for various computational problems. We exhibit
several families of complete problems which can be used for future completeness
results in the real hierarchy. As an application we sharpen some results by
B\"{u}rgisser and Cucker on the complexity of properties of semialgebraic sets,
including the Hausdorff distance problem also studied by Jungeblut, Kleist, and
Miltzow
Counting Subgraphs in Somewhere Dense Graphs
We study the problems of counting copies and induced copies of a small
pattern graph in a large host graph . Recent work fully classified the
complexity of those problems according to structural restrictions on the
patterns . In this work, we address the more challenging task of analysing
the complexity for restricted patterns and restricted hosts. Specifically we
ask which families of allowed patterns and hosts imply fixed-parameter
tractability, i.e., the existence of an algorithm running in time for some computable function . Our main results present
exhaustive and explicit complexity classifications for families that satisfy
natural closure properties. Among others, we identify the problems of counting
small matchings and independent sets in subgraph-closed graph classes
as our central objects of study and establish the following crisp
dichotomies as consequences of the Exponential Time Hypothesis: (1) Counting
-matchings in a graph is fixed-parameter tractable if and
only if is nowhere dense. (2) Counting -independent sets in a
graph is fixed-parameter tractable if and only if
is nowhere dense. Moreover, we obtain almost tight conditional
lower bounds if is somewhere dense, i.e., not nowhere dense.
These base cases of our classifications subsume a wide variety of previous
results on the matching and independent set problem, such as counting
-matchings in bipartite graphs (Curticapean, Marx; FOCS 14), in
-colourable graphs (Roth, Wellnitz; SODA 20), and in degenerate graphs
(Bressan, Roth; FOCS 21), as well as counting -independent sets in bipartite
graphs (Curticapean et al.; Algorithmica 19).Comment: 35 pages, 3 figures, 4 tables, abstract shortened due to ArXiv
requirement
Counting Subgraphs in Somewhere Dense Graphs
We study the problems of counting copies and induced copies of a small pattern graph H in a large host graph G. Recent work fully classified the complexity of those problems according to structural restrictions on the patterns H. In this work, we address the more challenging task of analysing the complexity for restricted patterns and restricted hosts. Specifically we ask which families of allowed patterns and hosts imply fixed-parameter tractability, i.e., the existence of an algorithm running in time f(H)?|G|^O(1) for some computable function f. Our main results present exhaustive and explicit complexity classifications for families that satisfy natural closure properties. Among others, we identify the problems of counting small matchings and independent sets in subgraph-closed graph classes ? as our central objects of study and establish the following crisp dichotomies as consequences of the Exponential Time Hypothesis:
- Counting k-matchings in a graph G ? ? is fixed-parameter tractable if and only if ? is nowhere dense.
- Counting k-independent sets in a graph G ? ? is fixed-parameter tractable if and only if ? is nowhere dense. Moreover, we obtain almost tight conditional lower bounds if ? is somewhere dense, i.e., not nowhere dense. These base cases of our classifications subsume a wide variety of previous results on the matching and independent set problem, such as counting k-matchings in bipartite graphs (Curticapean, Marx; FOCS 14), in F-colourable graphs (Roth, Wellnitz; SODA 20), and in degenerate graphs (Bressan, Roth; FOCS 21), as well as counting k-independent sets in bipartite graphs (Curticapean et al.; Algorithmica 19).
At the same time our proofs are much simpler: using structural characterisations of somewhere dense graphs, we show that a colourful version of a recent breakthrough technique for analysing pattern counting problems (Curticapean, Dell, Marx; STOC 17) applies to any subgraph-closed somewhere dense class of graphs, yielding a unified view of our current understanding of the complexity of subgraph counting
Efficient approximate unitary t-designs from partially invertible universal sets and their application to quantum speedup
At its core a -design is a method for sampling from a set of unitaries in
a way which mimics sampling randomly from the Haar measure on the unitary
group, with applications across quantum information processing and physics. We
construct new families of quantum circuits on -qubits giving rise to
-approximate unitary -designs efficiently in
depth. These quantum circuits are based on a relaxation of technical
requirements in previous constructions. In particular, the construction of
circuits which give efficient approximate -designs by Brandao, Harrow, and
Horodecki (F.G.S.L Brandao, A.W Harrow, and M. Horodecki, Commun. Math. Phys.
(2016).) required choosing gates from ensembles which contained inverses for
all elements, and that the entries of the unitaries are algebraic. We reduce
these requirements, to sets that contain elements without inverses in the set,
and non-algebraic entries, which we dub partially invertible universal sets. We
then adapt this circuit construction to the framework of measurement based
quantum computation(MBQC) and give new explicit examples of -qubit graph
states with fixed assignments of measurements (graph gadgets) giving rise to
unitary -designs based on partially invertible universal sets, in a natural
way. We further show that these graph gadgets demonstrate a quantum speedup, up
to standard complexity theoretic conjectures. We provide numerical and
analytical evidence that almost any assignment of fixed measurement angles on
an -qubit cluster state give efficient -designs and demonstrate a quantum
speedup.Comment: 25 pages,7 figures. Comments are welcome. Some typos corrected in
newest version. new References added.Proofs unchanged. Results unchange
Descriptive Set Theory and Applications
The systematic study of Polish spaces within the scope of Descriptive Set Theory furnishes the working mathematician with powerful techniques and illuminating insights. In this thesis, we start with a concise recapitulation of some classical aspects of Descriptive Set Theory which is followed by a succint review of topological groups, measures and some of their associated algebras.The main application of these techniques contained in this thesis is the study of two families of closed subsets of a locally compact Polish groupG, namely U(G) - closed sets of uniqueness - and U0(G) - closed sets of extended uniqueness. We locate the descriptive set theoretic complexityof these families, proving in particular that U(G) is \Pi_1^1-complete whenever G/\overline{[G,G]} is non-discrete, thereby extending the existing literature regarding the abelian case. En route, we establish some preservation results concerning sets of (extended) uniqueness and their operator theoretic counterparts. These results constitute a pivotal part in the arguments used and entail alternative proofs regarding the computation of the complexity of U(G) and U0(G) in some classes of the abelian case.We study U(G) as a calibrated \Pi_1^1 \sigma-ideal of F(G) - for G amenable - and prove some criteria concerning necessary conditions for the inexistence of a Borel basis for U(G). These criteria allow us to retrieve information about G after examination of its subgroups or quotients. Furthermore, a sufficient condition for the inexistence of a Borel basis for U(G) is proven for the case when G is a product of compact (abelian or not) Polish groupssatisfying certain conditions.\ua0Finally, we study objects associated with the point spectrum of linear bounded operators T\in L(X) acting on a separable Banach space X. We provide a characterization of reflexivity for Banach spaces with an unconditional basis : indeed such space X is reflexive if and only if for all closed subspaces Y\subset X;Z\subset X^{\ast} and T\in 2 L(Y); S\in 2 L(Z) it holds that the point spectra \sigma_p(T); \sigma_p(S) are Borel. We study the complexity of sets prescribed by eigenvalues and prove a stability criterion for Jamison sequences
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