1,691 research outputs found
Dynamic Programming on Nominal Graphs
Many optimization problems can be naturally represented as (hyper) graphs,
where vertices correspond to variables and edges to tasks, whose cost depends
on the values of the adjacent variables. Capitalizing on the structure of the
graph, suitable dynamic programming strategies can select certain orders of
evaluation of the variables which guarantee to reach both an optimal solution
and a minimal size of the tables computed in the optimization process. In this
paper we introduce a simple algebraic specification with parallel composition
and restriction whose terms up to structural axioms are the graphs mentioned
above. In addition, free (unrestricted) vertices are labelled with variables,
and the specification includes operations of name permutation with finite
support. We show a correspondence between the well-known tree decompositions of
graphs and our terms. If an axiom of scope extension is dropped, several
(hierarchical) terms actually correspond to the same graph. A suitable
graphical structure can be found, corresponding to every hierarchical term.
Evaluating such a graphical structure in some target algebra yields a dynamic
programming strategy. If the target algebra satisfies the scope extension
axiom, then the result does not depend on the particular structure, but only on
the original graph. We apply our approach to the parking optimization problem
developed in the ASCENS e-mobility case study, in collaboration with
Volkswagen. Dynamic programming evaluations are particularly interesting for
autonomic systems, where actual behavior often consists of propagating local
knowledge to obtain global knowledge and getting it back for local decisions.Comment: In Proceedings GaM 2015, arXiv:1504.0244
A Backtracking-Based Algorithm for Computing Hypertree-Decompositions
Hypertree decompositions of hypergraphs are a generalization of tree
decompositions of graphs. The corresponding hypertree-width is a measure for
the cyclicity and therefore tractability of the encoded computation problem.
Many NP-hard decision and computation problems are known to be tractable on
instances whose structure corresponds to hypergraphs of bounded
hypertree-width. Intuitively, the smaller the hypertree-width, the faster the
computation problem can be solved. In this paper, we present the new
backtracking-based algorithm det-k-decomp for computing hypertree
decompositions of small width. Our benchmark evaluations have shown that
det-k-decomp significantly outperforms opt-k-decomp, the only exact hypertree
decomposition algorithm so far. Even compared to the best heuristic algorithm,
we obtained competitive results as long as the hypergraphs are not too large.Comment: 19 pages, 6 figures, 3 table
Counting hypergraph matchings up to uniqueness threshold
We study the problem of approximately counting matchings in hypergraphs of
bounded maximum degree and maximum size of hyperedges. With an activity
parameter , each matching is assigned a weight .
The counting problem is formulated as computing a partition function that gives
the sum of the weights of all matchings in a hypergraph. This problem unifies
two extensively studied statistical physics models in approximate counting: the
hardcore model (graph independent sets) and the monomer-dimer model (graph
matchings).
For this model, the critical activity
is the threshold for the uniqueness of Gibbs measures on the infinite
-uniform -regular hypertree. Consider hypergraphs of maximum
degree at most and maximum size of hyperedges at most . We show that
when , there is an FPTAS for computing the partition
function; and when , there is a PTAS for computing the
log-partition function. These algorithms are based on the decay of correlation
(strong spatial mixing) property of Gibbs distributions. When , there is no PRAS for the partition function or the log-partition
function unless NPRP.
Towards obtaining a sharp transition of computational complexity of
approximate counting, we study the local convergence from a sequence of finite
hypergraphs to the infinite lattice with specified symmetry. We show a
surprising connection between the local convergence and the reversibility of a
natural random walk. This leads us to a barrier for the hardness result: The
non-uniqueness of infinite Gibbs measure is not realizable by any finite
gadgets
Understanding the complexity of #SAT using knowledge compilation
Two main techniques have been used so far to solve the #P-hard problem #SAT.
The first one, used in practice, is based on an extension of DPLL for model
counting called exhaustive DPLL. The second approach, more theoretical,
exploits the structure of the input to compute the number of satisfying
assignments by usually using a dynamic programming scheme on a decomposition of
the formula. In this paper, we make a first step toward the separation of these
two techniques by exhibiting a family of formulas that can be solved in
polynomial time with the first technique but needs an exponential time with the
second one. We show this by observing that both techniques implicitely
construct a very specific boolean circuit equivalent to the input formula. We
then show that every beta-acyclic formula can be represented by a polynomial
size circuit corresponding to the first method and exhibit a family of
beta-acyclic formulas which cannot be represented by polynomial size circuits
corresponding to the second method. This result shed a new light on the
complexity of #SAT and related problems on beta-acyclic formulas. As a
byproduct, we give new handy tools to design algorithms on beta-acyclic
hypergraphs
Conjunctions of Among Constraints
Many existing global constraints can be encoded as a conjunction of among
constraints. An among constraint holds if the number of the variables in its
scope whose value belongs to a prespecified set, which we call its range, is
within some given bounds. It is known that domain filtering algorithms can
benefit from reasoning about the interaction of among constraints so that
values can be filtered out taking into consideration several among constraints
simultaneously. The present pa- per embarks into a systematic investigation on
the circumstances under which it is possible to obtain efficient and complete
domain filtering algorithms for conjunctions of among constraints. We start by
observing that restrictions on both the scope and the range of the among
constraints are necessary to obtain meaningful results. Then, we derive a
domain flow-based filtering algorithm and present several applications. In
particular, it is shown that the algorithm unifies and generalizes several
previous existing results.Comment: 15 pages plus appendi
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