1,714 research outputs found
Covering Partial Cubes with Zones
A partial cube is a graph having an isometric embedding in a hypercube.
Partial cubes are characterized by a natural equivalence relation on the edges,
whose classes are called zones. The number of zones determines the minimal
dimension of a hypercube in which the graph can be embedded. We consider the
problem of covering the vertices of a partial cube with the minimum number of
zones. The problem admits several special cases, among which are the problem of
covering the cells of a line arrangement with a minimum number of lines, and
the problem of finding a minimum-size fibre in a bipartite poset. For several
such special cases, we give upper and lower bounds on the minimum size of a
covering by zones. We also consider the computational complexity of those
problems, and establish some hardness results
On Parameterized Complexity of Group Activity Selection Problems on Social Networks
In Group Activity Selection Problem (GASP), players form coalitions to
participate in activities and have preferences over pairs of the form
(activity, group size). Recently, Igarashi et al. have initiated the study of
group activity selection problems on social networks (gGASP): a group of
players can engage in the same activity if the members of the group form a
connected subset of the underlying communication structure. Igarashi et al.
have primarily focused on Nash stable outcomes, and showed that many associated
algorithmic questions are computationally hard even for very simple networks.
In this paper we study the parameterized complexity of gGASP with respect to
the number of activities as well as with respect to the number of players, for
several solution concepts such as Nash stability, individual stability and core
stability. The first parameter we consider in the number of activities. For
this parameter, we propose an FPT algorithm for Nash stability for the case
where the social network is acyclic and obtain a W[1]-hardness result for
cliques (i.e., for classic GASP); similar results hold for individual
stability. In contrast, finding a core stable outcome is hard even if the
number of activities is bounded by a small constant, both for classic GASP and
when the social network is a star. Another parameter we study is the number of
players. While all solution concepts we consider become polynomial-time
computable when this parameter is bounded by a constant, we prove W[1]-hardness
results for cliques (i.e., for classic GASP).Comment: 9 pages, long version of accepted AAMAS-17 pape
Transiently Consistent SDN Updates: Being Greedy is Hard
The software-defined networking paradigm introduces interesting opportunities
to operate networks in a more flexible, optimized, yet formally verifiable
manner. Despite the logically centralized control, however, a Software-Defined
Network (SDN) is still a distributed system, with inherent delays between the
switches and the controller. Especially the problem of changing network
configurations in a consistent manner, also known as the consistent network
update problem, has received much attention over the last years. In particular,
it has been shown that there exists an inherent tradeoff between update
consistency and speed. This paper revisits the problem of updating an SDN in a
transiently consistent, loop-free manner. First, we rigorously prove that
computing a maximum (greedy) loop-free network update is generally NP-hard;
this result has implications for the classic maximum acyclic subgraph problem
(the dual feedback arc set problem) as well. Second, we show that for special
problem instances, fast and good approximation algorithms exist
Positivity for Gaussian graphical models
Gaussian graphical models are parametric statistical models for jointly
normal random variables whose dependence structure is determined by a graph. In
previous work, we introduced trek separation, which gives a necessary and
sufficient condition in terms of the graph for when a subdeterminant is zero
for all covariance matrices that belong to the Gaussian graphical model. Here
we extend this result to give explicit cancellation-free formulas for the
expansions of nonzero subdeterminants.Comment: 16 pages, 3 figure
Testing Consumer Rationality using Perfect Graphs and Oriented Discs
Given a consumer data-set, the axioms of revealed preference proffer a binary
test for rational behaviour. A natural (non-binary) measure of the degree of
rationality exhibited by the consumer is the minimum number of data points
whose removal induces a rationalisable data-set.We study the computational
complexity of the resultant consumer rationality problem in this paper. This
problem is, in the worst case, equivalent (in terms of approximation) to the
directed feedback vertex set problem. Our main result is to obtain an exact
threshold on the number of commodities that separates easy cases and hard
cases. Specifically, for two-commodity markets the consumer rationality problem
is polynomial time solvable; we prove this via a reduction to the vertex cover
problem on perfect graphs. For three-commodity markets, however, the problem is
NP-complete; we prove thisusing a reduction from planar 3-SAT that is based
upon oriented-disc drawings
Meta-Kernelization using Well-Structured Modulators
Kernelization investigates exact preprocessing algorithms with performance
guarantees. The most prevalent type of parameters used in kernelization is the
solution size for optimization problems; however, also structural parameters
have been successfully used to obtain polynomial kernels for a wide range of
problems. Many of these parameters can be defined as the size of a smallest
modulator of the given graph into a fixed graph class (i.e., a set of vertices
whose deletion puts the graph into the graph class). Such parameters admit the
construction of polynomial kernels even when the solution size is large or not
applicable. This work follows up on the research on meta-kernelization
frameworks in terms of structural parameters.
We develop a class of parameters which are based on a more general view on
modulators: instead of size, the parameters employ a combination of rank-width
and split decompositions to measure structure inside the modulator. This allows
us to lift kernelization results from modulator-size to more general
parameters, hence providing smaller kernels. We show (i) how such large but
well-structured modulators can be efficiently approximated, (ii) how they can
be used to obtain polynomial kernels for any graph problem expressible in
Monadic Second Order logic, and (iii) how they allow the extension of previous
results in the area of structural meta-kernelization
On Directed Feedback Vertex Set parameterized by treewidth
We study the Directed Feedback Vertex Set problem parameterized by the
treewidth of the input graph. We prove that unless the Exponential Time
Hypothesis fails, the problem cannot be solved in time on general directed graphs, where is the treewidth of
the underlying undirected graph. This is matched by a dynamic programming
algorithm with running time .
On the other hand, we show that if the input digraph is planar, then the
running time can be improved to .Comment: 20
The Rotor-Router Model on Regular Trees
The rotor-router model is a deterministic analogue of random walk. It can be
used to define a deterministic growth model analogous to internal DLA. We show
that the set of occupied sites for this model on an infinite regular tree is a
perfect ball whenever it can be, provided the initial rotor configuration is
acyclic (that is, no two neighboring vertices have rotors pointing to one
another). This is proved by defining the rotor-router group of a graph, which
we show is isomorphic to the sandpile group. We also address the question of
recurrence and transience: We give two rotor configurations on the infinite
ternary tree, one for which chips exactly alternate escaping to infinity with
returning to the origin, and one for which every chip returns to the origin.
Further, we characterize the possible "escape sequences" for the ternary tree,
that is, binary words a_1 ... a_n for which there exists a rotor configuration
so that the k-th chip escapes to infinity if and only if a_k=1.Comment: v2 incorporates referee comments, clarifies that the results of
section 2 apply also to multigraph
Linearly edge-reinforced random walks
We review results on linearly edge-reinforced random walks. On finite graphs,
the process has the same distribution as a mixture of reversible Markov chains.
This has applications in Bayesian statistics and it has been used in studying
the random walk on infinite graphs. On trees, one has a representation as a
random walk in an independent random environment. We review recent results for
the random walk on ladders: recurrence, a representation as a random walk in a
random environment, and estimates for the position of the random walker.Comment: Published at http://dx.doi.org/10.1214/074921706000000103 in the IMS
Lecture Notes--Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org
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