791 research outputs found
The Complexity of Finding Effectors
The NP-hard EFFECTORS problem on directed graphs is motivated by applications
in network mining, particularly concerning the analysis of probabilistic
information-propagation processes in social networks. In the corresponding
model the arcs carry probabilities and there is a probabilistic diffusion
process activating nodes by neighboring activated nodes with probabilities as
specified by the arcs. The point is to explain a given network activation state
as well as possible by using a minimum number of "effector nodes"; these are
selected before the activation process starts.
We correct, complement, and extend previous work from the data mining
community by a more thorough computational complexity analysis of EFFECTORS,
identifying both tractable and intractable cases. To this end, we also exploit
a parameterization measuring the "degree of randomness" (the number of "really"
probabilistic arcs) which might prove useful for analyzing other probabilistic
network diffusion problems as well.Comment: 28 page
Hitting and Harvesting Pumpkins
The "c-pumpkin" is the graph with two vertices linked by c>0 parallel edges.
A c-pumpkin-model in a graph G is a pair A,B of disjoint subsets of vertices of
G, each inducing a connected subgraph of G, such that there are at least c
edges in G between A and B. We focus on covering and packing c-pumpkin-models
in a given graph: On the one hand, we provide an FPT algorithm running in time
2^O(k) n^O(1) deciding, for any fixed c>0, whether all c-pumpkin-models can be
covered by at most k vertices. This generalizes known single-exponential FPT
algorithms for Vertex Cover and Feedback Vertex Set, which correspond to the
cases c=1,2 respectively. On the other hand, we present a O(log
n)-approximation algorithm for both the problems of covering all
c-pumpkin-models with a smallest number of vertices, and packing a maximum
number of vertex-disjoint c-pumpkin-models.Comment: v2: several minor change
The Roles of Internal Representation and Processing in Problem Solving Involving Insight: A Computational Complexity Perspective
In human problem solving, there is a wide variation between individuals in problem solution time and success rate, regardless of whether or not this problem solving involves insight. In this paper, we apply computational and parameterized analysis to a plausible formalization of extended representation change theory (eRCT), an integration of problem solving by problem space search and insight as problem restructuring which proposes that this variation may be explainable by individuals having different problem representations and search heuristic choices. Our analyses establish not only the intractability of eRCT in general, but also sets of restrictions under which eRCT-based problem solving can and cannot be done quickly. As such, our analyses both prove that several conjectures about what makes problem solving under eRCT possible in practice are incomplete, in the sense that not all factors in the model whose restriction is responsible for efficient solvability are part of the explanation, and provide several new explanations that are complete
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
On unrooted and root-uncertain variants of several well-known phylogenetic network problems
The hybridization number problem requires us to embed a set of binary rooted
phylogenetic trees into a binary rooted phylogenetic network such that the
number of nodes with indegree two is minimized. However, from a biological
point of view accurately inferring the root location in a phylogenetic tree is
notoriously difficult and poor root placement can artificially inflate the
hybridization number. To this end we study a number of relaxed variants of this
problem. We start by showing that the fundamental problem of determining
whether an \emph{unrooted} phylogenetic network displays (i.e. embeds) an
\emph{unrooted} phylogenetic tree, is NP-hard. On the positive side we show
that this problem is FPT in reticulation number. In the rooted case the
corresponding FPT result is trivial, but here we require more subtle
argumentation. Next we show that the hybridization number problem for unrooted
networks (when given two unrooted trees) is equivalent to the problem of
computing the Tree Bisection and Reconnect (TBR) distance of the two unrooted
trees. In the third part of the paper we consider the "root uncertain" variant
of hybridization number. Here we are free to choose the root location in each
of a set of unrooted input trees such that the hybridization number of the
resulting rooted trees is minimized. On the negative side we show that this
problem is APX-hard. On the positive side, we show that the problem is FPT in
the hybridization number, via kernelization, for any number of input trees.Comment: 28 pages, 8 Figure
A Time Hierarchy Theorem for the LOCAL Model
The celebrated Time Hierarchy Theorem for Turing machines states, informally,
that more problems can be solved given more time. The extent to which a time
hierarchy-type theorem holds in the distributed LOCAL model has been open for
many years. It is consistent with previous results that all natural problems in
the LOCAL model can be classified according to a small constant number of
complexities, such as , etc.
In this paper we establish the first time hierarchy theorem for the LOCAL
model and prove that several gaps exist in the LOCAL time hierarchy.
1. We define an infinite set of simple coloring problems called Hierarchical
-Coloring}. A correctly colored graph can be confirmed by simply
checking the neighborhood of each vertex, so this problem fits into the class
of locally checkable labeling (LCL) problems. However, the complexity of the
-level Hierarchical -Coloring problem is ,
for . The upper and lower bounds hold for both general graphs
and trees, and for both randomized and deterministic algorithms.
2. Consider any LCL problem on bounded degree trees. We prove an
automatic-speedup theorem that states that any randomized -time
algorithm solving the LCL can be transformed into a deterministic -time algorithm. Together with a previous result, this establishes that on
trees, there are no natural deterministic complexities in the ranges
--- or ---.
3. We expose a gap in the randomized time hierarchy on general graphs. Any
randomized algorithm that solves an LCL problem in sublogarithmic time can be
sped up to run in time, which is the complexity of the distributed
Lovasz local lemma problem, currently known to be and
A note on hierarchical hubbing for a generalization of the VPN problem
Robust network design refers to a class of optimization problems that occur
when designing networks to efficiently handle variable demands. The notion of
"hierarchical hubbing" was introduced (in the narrow context of a specific
robust network design question), by Olver and Shepherd [2010]. Hierarchical
hubbing allows for routings with a multiplicity of "hubs" which are connected
to the terminals and to each other in a treelike fashion. Recently, Fr\'echette
et al. [2013] explored this notion much more generally, focusing on its
applicability to an extension of the well-studied hose model that allows for
upper bounds on individual point-to-point demands. In this paper, we consider
hierarchical hubbing in the context of a previously studied (and extremely
natural) generalization of the hose model, and prove that the optimal
hierarchical hubbing solution can be found efficiently. This result is relevant
to a recently proposed generalization of the "VPN Conjecture".Comment: 14 pages, 1 figur
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