492 research outputs found
Biased landscapes for random Constraint Satisfaction Problems
The typical complexity of Constraint Satisfaction Problems (CSPs) can be
investigated by means of random ensembles of instances. The latter exhibit many
threshold phenomena besides their satisfiability phase transition, in
particular a clustering or dynamic phase transition (related to the tree
reconstruction problem) at which their typical solutions shatter into
disconnected components. In this paper we study the evolution of this
phenomenon under a bias that breaks the uniformity among solutions of one CSP
instance, concentrating on the bicoloring of k-uniform random hypergraphs. We
show that for small k the clustering transition can be delayed in this way to
higher density of constraints, and that this strategy has a positive impact on
the performances of Simulated Annealing algorithms. We characterize the modest
gain that can be expected in the large k limit from the simple implementation
of the biasing idea studied here. This paper contains also a contribution of a
more methodological nature, made of a review and extension of the methods to
determine numerically the discontinuous dynamic transition threshold.Comment: 32 pages, 16 figure
Spectral Detection on Sparse Hypergraphs
We consider the problem of the assignment of nodes into communities from a
set of hyperedges, where every hyperedge is a noisy observation of the
community assignment of the adjacent nodes. We focus in particular on the
sparse regime where the number of edges is of the same order as the number of
vertices. We propose a spectral method based on a generalization of the
non-backtracking Hashimoto matrix into hypergraphs. We analyze its performance
on a planted generative model and compare it with other spectral methods and
with Bayesian belief propagation (which was conjectured to be asymptotically
optimal for this model). We conclude that the proposed spectral method detects
communities whenever belief propagation does, while having the important
advantages to be simpler, entirely nonparametric, and to be able to learn the
rule according to which the hyperedges were generated without prior
information.Comment: 8 pages, 5 figure
LP-based Covering Games with Low Price of Anarchy
We present a new class of vertex cover and set cover games. The price of
anarchy bounds match the best known constant factor approximation guarantees
for the centralized optimization problems for linear and also for submodular
costs -- in contrast to all previously studied covering games, where the price
of anarchy cannot be bounded by a constant (e.g. [6, 7, 11, 5, 2]). In
particular, we describe a vertex cover game with a price of anarchy of 2. The
rules of the games capture the structure of the linear programming relaxations
of the underlying optimization problems, and our bounds are established by
analyzing these relaxations. Furthermore, for linear costs we exhibit linear
time best response dynamics that converge to these almost optimal Nash
equilibria. These dynamics mimic the classical greedy approximation algorithm
of Bar-Yehuda and Even [3]
Random multi-index matching problems
The multi-index matching problem (MIMP) generalizes the well known matching
problem by going from pairs to d-uplets. We use the cavity method from
statistical physics to analyze its properties when the costs of the d-uplets
are random. At low temperatures we find for d>2 a frozen glassy phase with
vanishing entropy. We also investigate some properties of small samples by
enumerating the lowest cost matchings to compare with our theoretical
predictions.Comment: 22 pages, 16 figure
Minimum-cost multicast over coded packet networks
We consider the problem of establishing minimum-cost multicast connections over coded packet networks, i.e., packet networks where the contents of outgoing packets are arbitrary, causal functions of the contents of received packets. We consider both wireline and wireless packet networks as well as both static multicast (where membership of the multicast group remains constant for the duration of the connection) and dynamic multicast (where membership of the multicast group changes in time, with nodes joining and leaving the group). For static multicast, we reduce the problem to a polynomial-time solvable optimization problem, and we present decentralized algorithms for solving it. These algorithms, when coupled with existing decentralized schemes for constructing network codes, yield a fully decentralized approach for achieving minimum-cost multicast. By contrast, establishing minimum-cost static multicast connections over routed packet networks is a very difficult problem even using centralized computation, except in the special cases of unicast and broadcast connections. For dynamic multicast, we reduce the problem to a dynamic programming problem and apply the theory of dynamic programming to suggest how it may be solved
Dynamical systems on hypergraphs
We present a general framework that enables one to model high-order
interaction among entangled dynamical systems, via hypergraphs. Several
relevant processes can be ideally traced back to the proposed scheme. We shall
here solely elaborate on the conditions that seed the spontaneous emergence of
patterns, spatially heterogeneous solutions resulting from the many-body
interaction between fundamental units. In particular we will focus, on two
relevant settings. First, we will assume long-ranged mean field interactions
between populations, and then turn to considering diffusive-like couplings. Two
applications are presented, respectively to a generalised Volterra system and
the Brusselator model
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