4,173 research outputs found
Tractable Optimization Problems through Hypergraph-Based Structural Restrictions
Several variants of the Constraint Satisfaction Problem have been proposed
and investigated in the literature for modelling those scenarios where
solutions are associated with some given costs. Within these frameworks
computing an optimal solution is an NP-hard problem in general; yet, when
restricted over classes of instances whose constraint interactions can be
modelled via (nearly-)acyclic graphs, this problem is known to be solvable in
polynomial time. In this paper, larger classes of tractable instances are
singled out, by discussing solution approaches based on exploiting hypergraph
acyclicity and, more generally, structural decomposition methods, such as
(hyper)tree decompositions
Structural Decompositions for Problems with Global Constraints
A wide range of problems can be modelled as constraint satisfaction problems
(CSPs), that is, a set of constraints that must be satisfied simultaneously.
Constraints can either be represented extensionally, by explicitly listing
allowed combinations of values, or implicitly, by special-purpose algorithms
provided by a solver.
Such implicitly represented constraints, known as global constraints, are
widely used; indeed, they are one of the key reasons for the success of
constraint programming in solving real-world problems. In recent years, a
variety of restrictions on the structure of CSP instances have been shown to
yield tractable classes of CSPs. However, most such restrictions fail to
guarantee tractability for CSPs with global constraints. We therefore study the
applicability of structural restrictions to instances with such constraints.
We show that when the number of solutions to a CSP instance is bounded in key
parts of the problem, structural restrictions can be used to derive new
tractable classes. Furthermore, we show that this result extends to
combinations of instances drawn from known tractable classes, as well as to CSP
instances where constraints assign costs to satisfying assignments.Comment: The final publication is available at Springer via
http://dx.doi.org/10.1007/s10601-015-9181-
Tensor decomposition with generalized lasso penalties
We present an approach for penalized tensor decomposition (PTD) that
estimates smoothly varying latent factors in multi-way data. This generalizes
existing work on sparse tensor decomposition and penalized matrix
decompositions, in a manner parallel to the generalized lasso for regression
and smoothing problems. Our approach presents many nontrivial challenges at the
intersection of modeling and computation, which are studied in detail. An
efficient coordinate-wise optimization algorithm for (PTD) is presented, and
its convergence properties are characterized. The method is applied both to
simulated data and real data on flu hospitalizations in Texas. These results
show that our penalized tensor decomposition can offer major improvements on
existing methods for analyzing multi-way data that exhibit smooth spatial or
temporal features
Improved Optimal and Approximate Power Graph Compression for Clearer Visualisation of Dense Graphs
Drawings of highly connected (dense) graphs can be very difficult to read.
Power Graph Analysis offers an alternate way to draw a graph in which sets of
nodes with common neighbours are shown grouped into modules. An edge connected
to the module then implies a connection to each member of the module. Thus, the
entire graph may be represented with much less clutter and without loss of
detail. A recent experimental study has shown that such lossless compression of
dense graphs makes it easier to follow paths. However, computing optimal power
graphs is difficult. In this paper, we show that computing the optimal
power-graph with only one module is NP-hard and therefore likely NP-hard in the
general case. We give an ILP model for power graph computation and discuss why
ILP and CP techniques are poorly suited to the problem. Instead, we are able to
find optimal solutions much more quickly using a custom search method. We also
show how to restrict this type of search to allow only limited back-tracking to
provide a heuristic that has better speed and better results than previously
known heuristics.Comment: Extended technical report accompanying the PacificVis 2013 paper of
the same nam
Pure Nash Equilibria: Hard and Easy Games
We investigate complexity issues related to pure Nash equilibria of strategic
games. We show that, even in very restrictive settings, determining whether a
game has a pure Nash Equilibrium is NP-hard, while deciding whether a game has
a strong Nash equilibrium is SigmaP2-complete. We then study practically
relevant restrictions that lower the complexity. In particular, we are
interested in quantitative and qualitative restrictions of the way each players
payoff depends on moves of other players. We say that a game has small
neighborhood if the utility function for each player depends only on (the
actions of) a logarithmically small number of other players. The dependency
structure of a game G can be expressed by a graph DG(G) or by a hypergraph
H(G). By relating Nash equilibrium problems to constraint satisfaction problems
(CSPs), we show that if G has small neighborhood and if H(G) has bounded
hypertree width (or if DG(G) has bounded treewidth), then finding pure Nash and
Pareto equilibria is feasible in polynomial time. If the game is graphical,
then these problems are LOGCFL-complete and thus in the class NC2 of highly
parallelizable problems
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