979 research outputs found
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
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*(2n) and compute the fractional hypertree-width of H in time O(1.734601n.m).
Hypergraph Acyclicity and Propositional Model Counting
We show that the propositional model counting problem #SAT for CNF- formulas
with hypergraphs that allow a disjoint branches decomposition can be solved in
polynomial time. We show that this class of hypergraphs is incomparable to
hypergraphs of bounded incidence cliquewidth which were the biggest class of
hypergraphs for which #SAT was known to be solvable in polynomial time so far.
Furthermore, we present a polynomial time algorithm that computes a disjoint
branches decomposition of a given hypergraph if it exists and rejects
otherwise. Finally, we show that some slight extensions of the class of
hypergraphs with disjoint branches decompositions lead to intractable #SAT,
leaving open how to generalize the counting result of this paper
Approximating Hereditary Discrepancy via Small Width Ellipsoids
The Discrepancy of a hypergraph is the minimum attainable value, over
two-colorings of its vertices, of the maximum absolute imbalance of any
hyperedge. The Hereditary Discrepancy of a hypergraph, defined as the maximum
discrepancy of a restriction of the hypergraph to a subset of its vertices, is
a measure of its complexity. Lovasz, Spencer and Vesztergombi (1986) related
the natural extension of this quantity to matrices to rounding algorithms for
linear programs, and gave a determinant based lower bound on the hereditary
discrepancy. Matousek (2011) showed that this bound is tight up to a
polylogarithmic factor, leaving open the question of actually computing this
bound. Recent work by Nikolov, Talwar and Zhang (2013) showed a polynomial time
-approximation to hereditary discrepancy, as a by-product
of their work in differential privacy. In this paper, we give a direct simple
-approximation algorithm for this problem. We show that up to
this approximation factor, the hereditary discrepancy of a matrix is
characterized by the optimal value of simple geometric convex program that
seeks to minimize the largest norm of any point in a ellipsoid
containing the columns of . This characterization promises to be a useful
tool in discrepancy theory
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
Instance and Output Optimal Parallel Algorithms for Acyclic Joins
Massively parallel join algorithms have received much attention in recent
years, while most prior work has focused on worst-optimal algorithms. However,
the worst-case optimality of these join algorithms relies on hard instances
having very large output sizes, which rarely appear in practice. A stronger
notion of optimality is {\em output-optimal}, which requires an algorithm to be
optimal within the class of all instances sharing the same input and output
size. An even stronger optimality is {\em instance-optimal}, i.e., the
algorithm is optimal on every single instance, but this may not always be
achievable.
In the traditional RAM model of computation, the classical Yannakakis
algorithm is instance-optimal on any acyclic join. But in the massively
parallel computation (MPC) model, the situation becomes much more complicated.
We first show that for the class of r-hierarchical joins, instance-optimality
can still be achieved in the MPC model. Then, we give a new MPC algorithm for
an arbitrary acyclic join with load O ({\IN \over p} + {\sqrt{\IN \cdot \OUT}
\over p}), where \IN,\OUT are the input and output sizes of the join, and
is the number of servers in the MPC model. This improves the MPC version of
the Yannakakis algorithm by an O (\sqrt{\OUT \over \IN} ) factor.
Furthermore, we show that this is output-optimal when \OUT = O(p \cdot \IN),
for every acyclic but non-r-hierarchical join. Finally, we give the first
output-sensitive lower bound for the triangle join in the MPC model, showing
that it is inherently more difficult than acyclic joins
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-
Gaussian width bounds with applications to arithmetic progressions in random settings
Motivated by problems on random differences in Szemer\'{e}di's theorem and on
large deviations for arithmetic progressions in random sets, we prove upper
bounds on the Gaussian width of point sets that are formed by the image of the
-dimensional Boolean hypercube under a mapping
, where each coordinate is a constant-degree
multilinear polynomial with 0-1 coefficients. We show the following
applications of our bounds. Let be the random
subset of containing each element independently with
probability .
A set is -intersective if
any dense subset of contains a proper -term
arithmetic progression with common difference in . Our main result implies
that is -intersective with probability provided for . This gives a polynomial improvement for all
of a previous bound due to Frantzikinakis, Lesigne and Wierdl, and
reproves more directly the same improvement shown recently by the authors and
Dvir.
Let be the number of -term arithmetic progressions in
and consider the large deviation rate
. We give quadratic
improvements of the best-known range of for which a highly precise estimate
of due to Bhattacharya, Ganguly, Shao and Zhao is valid for
all odd .
We also discuss connections with error correcting codes (locally decodable
codes) and the Banach-space notion of type for injective tensor products of
-spaces.Comment: 18 pages, some typos fixe
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