105 research outputs found
Counting edge-injective homomorphisms and matchings on restricted graph classes
We consider the -hard problem of counting all matchings with
exactly edges in a given input graph ; we prove that it remains
-hard on graphs that are line graphs or bipartite graphs
with degree on one side. In our proofs, we use that -matchings in line
graphs can be equivalently viewed as edge-injective homomorphisms from the
disjoint union of length- paths into (arbitrary) host graphs. Here, a
homomorphism from to is edge-injective if it maps any two distinct
edges of to distinct edges in . We show that edge-injective
homomorphisms from a pattern graph can be counted in polynomial time if
has bounded vertex-cover number after removing isolated edges. For hereditary
classes of pattern graphs, we complement this result: If the
graphs in have unbounded vertex-cover number even after deleting
isolated edges, then counting edge-injective homomorphisms with patterns from
is -hard. Our proofs rely on an edge-colored
variant of Holant problems and a delicate interpolation argument; both may be
of independent interest.Comment: 35 pages, 9 figure
Phase Transition of the 2-Choices Dynamics on Core-Periphery Networks
Consider the following process on a network: Each agent initially holds
either opinion blue or red; then, in each round, each agent looks at two random
neighbors and, if the two have the same opinion, the agent adopts it. This
process is known as the 2-Choices dynamics and is arguably the most basic
non-trivial opinion dynamics modeling voting behavior on social networks.
Despite its apparent simplicity, 2-Choices has been analytically characterized
only on networks with a strong expansion property -- under assumptions on the
initial configuration that establish it as a fast majority consensus protocol.
In this work, we aim at contributing to the understanding of the 2-Choices
dynamics by considering its behavior on a class of networks with core-periphery
structure, a well-known topological assumption in social networks. In a
nutshell, assume that a densely-connected subset of agents, the core, holds a
different opinion from the rest of the network, the periphery. Then, depending
on the strength of the cut between the core and the periphery, a
phase-transition phenomenon occurs: Either the core's opinion rapidly spreads
among the rest of the network, or a metastability phase takes place, in which
both opinions coexist in the network for superpolynomial time. The interest of
our result is twofold. On the one hand, by looking at the 2-Choices dynamics as
a simplistic model of competition among opinions in social networks, our
theorem sheds light on the influence of the core on the rest of the network, as
a function of the core's connectivity towards the latter. On the other hand, to
the best of our knowledge, we provide the first analytical result which shows a
heterogeneous behavior of a simple dynamics as a function of structural
parameters of the network. Finally, we validate our theoretical predictions
with extensive experiments on real networks
Unboundedness Problems for Machines with Reversal-Bounded Counters
We consider a general class of decision problems concerning formal languages, called (one-dimensional) unboundedness predicates, for automata that feature reversal-bounded counters (RBCA). We show that each problem in this class reduces-non-deterministically in polynomial time to the same problem for just nite automata. We also show an analogous reduction for automata that have access to both a push- down stack and reversal-bounded counters (PRBCA). This allows us to answer several open questions: For example, we settle the complexity of deciding whether a given (P)RBCA language L is bounded, meaning whether there exist words w1, . . . , wn with L ⊆ w1∗ · · · wn∗ . For PRBCA, even decidability was open. Our methods also show that there is no language of a (P)RBCA of intermediate growth. Part of our proof is likely of independent interest: We show that one can translate an RBCA into a machine with Z-counters in logarithmic space
Automated deduction with built-in theories: completeness results and constraint solving techniques
Postprint (published version
The PACE 2017 Parameterized Algorithms and Computational Experiments Challenge: The Second Iteration
In this article, the Program Committee of the Second Parameterized Algorithms and Computational Experiments challenge (PACE 2017) reports on the second iteration of the PACE challenge. Track A featured the Treewidth problem and Track B the Minimum Fill-In problem. Over 44 participants on 17 teams from 11 countries submitted their implementations to the competition
A Constant-Factor Approximation for Weighted Bond Cover
The Weighted ?-Vertex Deletion for a class ? of graphs asks, weighted graph G, for a minimum weight vertex set S such that G-S ? ?. The case when ? is minor-closed and excludes some graph as a minor has received particular attention but a constant-factor approximation remained elusive for Weighted ?-Vertex Deletion. Only three cases of minor-closed ? are known to admit constant-factor approximations, namely Vertex Cover, Feedback Vertex Set and Diamond Hitting Set. We study the problem for the class ? of ?_c-minor-free graphs, under the equivalent setting of the Weighted c-Bond Cover problem, and present a constant-factor approximation algorithm using the primal-dual method. For this, we leverage a structure theorem implicit in [Joret et al., SIDMA\u2714] which states the following: any graph G containing a ?_c-minor-model either contains a large two-terminal protrusion, or contains a constant-size ?_c-minor-model, or a collection of pairwise disjoint constant-sized connected sets that can be contracted simultaneously to yield a dense graph. In the first case, we tame the graph by replacing the protrusion with a special-purpose weighted gadget. For the second and third case, we provide a weighting scheme which guarantees a local approximation ratio. Besides making an important step in the quest of (dis)proving a constant-factor approximation for Weighted ?-Vertex Deletion, our result may be useful as a template for algorithms for other minor-closed families
Profinite trees, through monads and the lambda-calculus
In its simplest form, the theory of regular languages is the study of sets of
finite words recognized by finite monoids. The finiteness condition on monoids
gives rise to a topological space whose points, called profinite words, encode
the limiting behavior of words with respect to finite monoids. Yet, some
aspects of the theory of regular languages are not particular to monoids and
can be described in a general setting. On the one hand, Boja\'{n}czyk has shown
how to use monads to generalize the theory of regular languages and has given
an abstract definition of the free profinite structure, defined by codensity,
given a fixed monad and a notion of finite structure. On the other hand,
Salvati has introduced the notion of language of -terms, using
denotational semantics, which generalizes the case of words and trees through
the Church encoding. In recent work, the author and collaborators defined the
notion of profinite -term using semantics in finite sets and
functions, which extend the Church encoding to profinite words.
In this article, we prove that these two generalizations, based on monads and
denotational semantics, coincide in the case of trees. To do so, we consider
the monad of abstract clones which, when applied to a ranked alphabet, gives
the associated clone of ranked trees. This induces a notion of free profinite
clone, and hence of profinite trees. The main contribution is a categorical
proof that the free profinite clone on a ranked alphabet is isomorphic, as a
Stone-enriched clone, to the clone of profinite -terms of Church type.
Moreover, we also prove a parametricity theorem on families of semantic
elements which provides another equivalent formulation of profinite trees in
terms of Reynolds parametricity
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