161 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-
Computing hypergraph width measures exactly
Hypergraph width measures are a class of hypergraph invariants important in
studying the complexity of constraint satisfaction problems (CSPs). We present
a general exact exponential algorithm for a large variety of these measures. A
connection between these and tree decompositions is established. This enables
us to almost seamlessly adapt the combinatorial and algorithmic results known
for tree decompositions of graphs to the case of hypergraphs and obtain fast
exact algorithms.
As a consequence, we provide algorithms which, given a hypergraph H on n
vertices and m hyperedges, compute the generalized hypertree-width of H in time
O*(2^n) and compute the fractional hypertree-width of H in time
O(m*1.734601^n).Comment: 12 pages, 1 figur
On The Power of Tree Projections: Structural Tractability of Enumerating CSP Solutions
The problem of deciding whether CSP instances admit solutions has been deeply
studied in the literature, and several structural tractability results have
been derived so far. However, constraint satisfaction comes in practice as a
computation problem where the focus is either on finding one solution, or on
enumerating all solutions, possibly projected to some given set of output
variables. The paper investigates the structural tractability of the problem of
enumerating (possibly projected) solutions, where tractability means here
computable with polynomial delay (WPD), since in general exponentially many
solutions may be computed. A general framework based on the notion of tree
projection of hypergraphs is considered, which generalizes all known
decomposition methods. Tractability results have been obtained both for classes
of structures where output variables are part of their specification, and for
classes of structures where computability WPD must be ensured for any possible
set of output variables. These results are shown to be tight, by exhibiting
dichotomies for classes of structures having bounded arity and where the tree
decomposition method is considered
Beyond Hypertree Width: Decomposition Methods Without Decompositions
The general intractability of the constraint satisfaction problem has
motivated the study of restrictions on this problem that permit polynomial-time
solvability. One major line of work has focused on structural restrictions,
which arise from restricting the interaction among constraint scopes. In this
paper, we engage in a mathematical investigation of generalized hypertree
width, a structural measure that has up to recently eluded study. We obtain a
number of computational results, including a simple proof of the tractability
of CSP instances having bounded generalized hypertree width
Tree Projections and Constraint Optimization Problems: Fixed-Parameter Tractability and Parallel Algorithms
Tree projections provide a unifying framework to deal with most structural
decomposition methods of constraint satisfaction problems (CSPs). Within this
framework, a CSP instance is decomposed into a number of sub-problems, called
views, whose solutions are either already available or can be computed
efficiently. The goal is to arrange portions of these views in a tree-like
structure, called tree projection, which determines an efficiently solvable CSP
instance equivalent to the original one. Deciding whether a tree projection
exists is NP-hard. Solution methods have therefore been proposed in the
literature that do not require a tree projection to be given, and that either
correctly decide whether the given CSP instance is satisfiable, or return that
a tree projection actually does not exist. These approaches had not been
generalized so far on CSP extensions for optimization problems, where the goal
is to compute a solution of maximum value/minimum cost. The paper fills the
gap, by exhibiting a fixed-parameter polynomial-time algorithm that either
disproves the existence of tree projections or computes an optimal solution,
with the parameter being the size of the expression of the objective function
to be optimized over all possible solutions (and not the size of the whole
constraint formula, used in related works). Tractability results are also
established for the problem of returning the best K solutions. Finally,
parallel algorithms for such optimization problems are proposed and analyzed.
Given that the classes of acyclic hypergraphs, hypergraphs of bounded
treewidth, and hypergraphs of bounded generalized hypertree width are all
covered as special cases of the tree projection framework, the results in this
paper directly apply to these classes. These classes are extensively considered
in the CSP setting, as well as in conjunctive database query evaluation and
optimization
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
Tree Projections and Structural Decomposition Methods: Minimality and Game-Theoretic Characterization
Tree projections provide a mathematical framework that encompasses all the
various (purely) structural decomposition methods that have been proposed in
the literature to single out classes of nearly-acyclic (hyper)graphs, such as
the tree decomposition method, which is the most powerful decomposition method
on graphs, and the (generalized) hypertree decomposition method, which is its
natural counterpart on arbitrary hypergraphs. The paper analyzes this
framework, by focusing in particular on "minimal" tree projections, that is, on
tree projections without useless redundancies. First, it is shown that minimal
tree projections enjoy a number of properties that are usually required for
normal form decompositions in various structural decomposition methods. In
particular, they enjoy the same kind of connection properties as (minimal) tree
decompositions of graphs, with the result being tight in the light of the
negative answer that is provided to the open question about whether they enjoy
a slightly stronger notion of connection property, defined to speed-up the
computation of hypertree decompositions. Second, it is shown that tree
projections admit a natural game-theoretic characterization in terms of the
Captain and Robber game. In this game, as for the Robber and Cops game
characterizing tree decompositions, the existence of winning strategies implies
the existence of monotone ones. As a special case, the Captain and Robber game
can be used to characterize the generalized hypertree decomposition method,
where such a game-theoretic characterization was missing and asked for. Besides
their theoretical interest, these results have immediate algorithmic
applications both for the general setting and for structural decomposition
methods that can be recast in terms of tree projections
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