61,541 research outputs found
Parameterized Complexity Dichotomy for Steiner Multicut
The Steiner Multicut problem asks, given an undirected graph G, terminals
sets T1,...,Tt V(G) of size at most p, and an integer k, whether
there is a set S of at most k edges or nodes s.t. of each set Ti at least one
pair of terminals is in different connected components of G \ S. This problem
generalizes several graph cut problems, in particular the Multicut problem (the
case p = 2), which is fixed-parameter tractable for the parameter k [Marx and
Razgon, Bousquet et al., STOC 2011].
We provide a dichotomy of the parameterized complexity of Steiner Multicut.
That is, for any combination of k, t, p, and the treewidth tw(G) as constant,
parameter, or unbounded, and for all versions of the problem (edge deletion and
node deletion with and without deletable terminals), we prove either that the
problem is fixed-parameter tractable or that the problem is hard (W[1]-hard or
even (para-)NP-complete). We highlight that:
- The edge deletion version of Steiner Multicut is fixed-parameter tractable
for the parameter k+t on general graphs (but has no polynomial kernel, even on
trees). We present two proofs: one using the randomized contractions technique
of Chitnis et al, and one relying on new structural lemmas that decompose the
Steiner cut into important separators and minimal s-t cuts.
- In contrast, both node deletion versions of Steiner Multicut are W[1]-hard
for the parameter k+t on general graphs.
- All versions of Steiner Multicut are W[1]-hard for the parameter k, even
when p=3 and the graph is a tree plus one node. Hence, the results of Marx and
Razgon, and Bousquet et al. do not generalize to Steiner Multicut.
Since we allow k, t, p, and tw(G) to be any constants, our characterization
includes a dichotomy for Steiner Multicut on trees (for tw(G) = 1), and a
polynomial time versus NP-hardness dichotomy (by restricting k,t,p,tw(G) to
constant or unbounded).Comment: As submitted to journal. This version also adds a proof of
fixed-parameter tractability for parameter k+t using the technique of
randomized contraction
Nonlinear sigma model approach for phase disorder transitions and the pseudogap phase in chiral Gross-Neveu, Nambu-Jona-Lasinio models and strong-coupling superconductors
We briefly review the nonlinear sigma model approach for the subject of
increasing interest: "two-step" phase transitions in the Gross-Neveu and the
modified Nambu-Jona-Lasinio models at low and condensation from pseudogap
phase in strong-coupling superconductors. Recent success in describing
"Bose-type" superconductors that possess two characterstic temperatures and a
pseudogap above is the development approximately comparable with the BCS
theory. One can expect that it should have influence on high-energy physics,
similar to impact of the BCS theory on this subject. Although first
generalizations of this concept to particle physics were made recently, these
results were not systematized. In this review we summarize this development and
discuss similarities and differences of the appearence of the pseudogap phase
in superconductors and the Gross-Neveu and Nambu-Jona-Lasinio - like models. We
discuss its possible relevance for chiral phase transition in QCD and color
superconductors. This paper is organized in three parts: in the first section
we briefly review the separation of temperatures of pair formation and pair
condensation in strong - coupling and low carrier density superconductors (i.e.
the formation of the {\it pseudogap phase}).
Second part is a review of nonlinear sigma model approach to an analogous
phenomenon in the Chiral Gross-Neveu model at small N. In the third section we
discuss the modified Nambu-Jona-Lasinio model where the chiral phase transition
is accompanied by a formation of a phase analogous to the pseudogap phase.Comment: A brief review. Replaced with journal version (some grammatical
corrections). The latest updates of this and related papers are also
available at the author home page http://www.teorfys.uu.se/PEOPLE/egor
Fast Construction of Nets in Low Dimensional Metrics, and Their Applications
We present a near linear time algorithm for constructing hierarchical nets in
finite metric spaces with constant doubling dimension. This data-structure is
then applied to obtain improved algorithms for the following problems:
Approximate nearest neighbor search, well-separated pair decomposition, compact
representation scheme, doubling measure, and computation of the (approximate)
Lipschitz constant of a function. In all cases, the running (preprocessing)
time is near-linear and the space being used is linear.Comment: 41 pages. Extensive clean-up of minor English error
Graphical models for marked point processes based on local independence
A new class of graphical models capturing the dependence structure of events
that occur in time is proposed. The graphs represent so-called local
independences, meaning that the intensities of certain types of events are
independent of some (but not necessarily all) events in the past. This dynamic
concept of independence is asymmetric, similar to Granger non-causality, so
that the corresponding local independence graphs differ considerably from
classical graphical models. Hence a new notion of graph separation, called
delta-separation, is introduced and implications for the underlying model as
well as for likelihood inference are explored. Benefits regarding facilitation
of reasoning about and understanding of dynamic dependencies as well as
computational simplifications are discussed.Comment: To appear in the Journal of the Royal Statistical Society Series
The Median Class and Superrigidity of Actions on CAT(0) Cube Complexes
We define a bounded cohomology class, called the {\em median class}, in the
second bounded cohomology -- with appropriate coefficients --of the
automorphism group of a finite dimensional CAT(0) cube complex X. The median
class of X behaves naturally with respect to taking products and appropriate
subcomplexes and defines in turn the {\em median class of an action} by
automorphisms of X.
We show that the median class of a non-elementary action by automorphisms
does not vanish and we show to which extent it does vanish if the action is
elementary. We obtain as a corollary a superrigidity result and show for
example that any irreducible lattice in the product of at least two locally
compact connected groups acts on a finite dimensional CAT(0) cube complex X
with a finite orbit in the Roller compactification of X. In the case of a
product of Lie groups, the Appendix by Caprace allows us to deduce that the
fixed point is in fact inside the complex X.
In the course of the proof we construct a \Gamma-equivariant measurable map
from a Poisson boundary of \Gamma with values in the non-terminating
ultrafilters on the Roller boundary of X.Comment: Minor changes that clarify some confusion have been made. Some
figures have been adde
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