586 research outputs found
Breaking Instance-Independent Symmetries In Exact Graph Coloring
Code optimization and high level synthesis can be posed as constraint
satisfaction and optimization problems, such as graph coloring used in register
allocation. Graph coloring is also used to model more traditional CSPs relevant
to AI, such as planning, time-tabling and scheduling. Provably optimal
solutions may be desirable for commercial and defense applications.
Additionally, for applications such as register allocation and code
optimization, naturally-occurring instances of graph coloring are often small
and can be solved optimally. A recent wave of improvements in algorithms for
Boolean satisfiability (SAT) and 0-1 Integer Linear Programming (ILP) suggests
generic problem-reduction methods, rather than problem-specific heuristics,
because (1) heuristics may be upset by new constraints, (2) heuristics tend to
ignore structure, and (3) many relevant problems are provably inapproximable.
Problem reductions often lead to highly symmetric SAT instances, and
symmetries are known to slow down SAT solvers. In this work, we compare several
avenues for symmetry breaking, in particular when certain kinds of symmetry are
present in all generated instances. Our focus on reducing CSPs to SAT allows us
to leverage recent dramatic improvement in SAT solvers and automatically
benefit from future progress. We can use a variety of black-box SAT solvers
without modifying their source code because our symmetry-breaking techniques
are static, i.e., we detect symmetries and add symmetry breaking predicates
(SBPs) during pre-processing.
An important result of our work is that among the types of
instance-independent SBPs we studied and their combinations, the simplest and
least complete constructions are the most effective. Our experiments also
clearly indicate that instance-independent symmetries should mostly be
processed together with instance-specific symmetries rather than at the
specification level, contrary to what has been suggested in the literature
Graph Symmetry Detection and Canonical Labeling: Differences and Synergies
Symmetries of combinatorial objects are known to complicate search
algorithms, but such obstacles can often be removed by detecting symmetries
early and discarding symmetric subproblems. Canonical labeling of combinatorial
objects facilitates easy equivalence checking through quick matching. All
existing canonical labeling software also finds symmetries, but the fastest
symmetry-finding software does not perform canonical labeling. In this work, we
contrast the two problems and dissect typical algorithms to identify their
similarities and differences. We then develop a novel approach to canonical
labeling where symmetries are found first and then used to speed up the
canonical labeling algorithms. Empirical results show that this approach
outperforms state-of-the-art canonical labelers.Comment: 15 pages, 10 figures, 1 table, Turing-10
Symmetry Breaking for Answer Set Programming
In the context of answer set programming, this work investigates symmetry
detection and symmetry breaking to eliminate symmetric parts of the search
space and, thereby, simplify the solution process. We contribute a reduction of
symmetry detection to a graph automorphism problem which allows to extract
symmetries of a logic program from the symmetries of the constructed coloured
graph. We also propose an encoding of symmetry-breaking constraints in terms of
permutation cycles and use only generators in this process which implicitly
represent symmetries and always with exponential compression. These ideas are
formulated as preprocessing and implemented in a completely automated flow that
first detects symmetries from a given answer set program, adds
symmetry-breaking constraints, and can be applied to any existing answer set
solver. We demonstrate computational impact on benchmarks versus direct
application of the solver.
Furthermore, we explore symmetry breaking for answer set programming in two
domains: first, constraint answer set programming as a novel approach to
represent and solve constraint satisfaction problems, and second, distributed
nonmonotonic multi-context systems. In particular, we formulate a
translation-based approach to constraint answer set solving which allows for
the application of our symmetry detection and symmetry breaking methods. To
compare their performance with a-priori symmetry breaking techniques, we also
contribute a decomposition of the global value precedence constraint that
enforces domain consistency on the original constraint via the unit-propagation
of an answer set solver. We evaluate both options in an empirical analysis. In
the context of distributed nonmonotonic multi-context system, we develop an
algorithm for distributed symmetry detection and also carry over
symmetry-breaking constraints for distributed answer set programming.Comment: Diploma thesis. Vienna University of Technology, August 201
Trivalent Graph isomorphism in polynomial time
It's important to design polynomial time algorithms to test if two graphs are
isomorphic at least for some special classes of graphs.
An approach to this was presented by Eugene M. Luks(1981) in the work
\textit{Isomorphism of Graphs of Bounded Valence Can Be Tested in Polynomial
Time}. Unfortunately, it was a theoretical algorithm and was very difficult to
put into practice. On the other hand, there is no known implementation of the
algorithm, although Galil, Hoffman and Luks(1983) shows an improvement of this
algorithm running in .
The two main goals of this master thesis are to explain more carefully the
algorithm of Luks(1981), including a detailed study of the complexity and, then
to provide an efficient implementation in SAGE system. It is divided into four
chapters plus an appendix.Comment: 48 pages. It is a Master Thesi
Efficient Linear Programming Decoding of HDPC Codes
We propose several improvements for Linear Programming (LP) decoding
algorithms for High Density Parity Check (HDPC) codes. First, we use the
automorphism groups of a code to create parity check matrix diversity and to
generate valid cuts from redundant parity checks. Second, we propose an
efficient mixed integer decoder utilizing the branch and bound method. We
further enhance the proposed decoders by removing inactive constraints and by
adapting the parity check matrix prior to decoding according to the channel
observations. Based on simulation results the proposed decoders achieve near-ML
performance with reasonable complexity.Comment: Submitted to the IEEE Transactions on Communications, November 200
Pseudo-Deterministic Proofs
We introduce pseudo-deterministic interactive proofs (psdIP): interactive proof systems for search problems where the verifier is guaranteed with high probability to output the same output on different executions. As in the case with classical interactive proofs, the verifier is a probabilistic polynomial time algorithm interacting with an untrusted powerful prover.
We view pseudo-deterministic interactive proofs as an extension of the study of pseudo-deterministic randomized polynomial time algorithms: the goal of the latter is to find canonical solutions to search problems whereas the goal of the former is to prove that a solution to a search problem is canonical to a probabilistic polynomial time verifier.
Alternatively, one may think of the powerful prover as aiding the probabilistic polynomial time verifier to find canonical solutions to search problems, with high probability over the randomness of the verifier. The challenge is that pseudo-determinism should hold not only with respect to the randomness, but also with respect to the prover: a malicious prover should not be able to cause the verifier to output a solution other than the unique canonical one.
The IP=PSPACE characterization implies that psdIP = IP. The challenge is to find constant round pseudo-deterministic interactive proofs for hard search problems. We show a constant round pseudo-deterministic interactive proof for the graph isomorphism problem: on any input pair of isomorphic graphs (G_0,G_1), there exist a unique isomorphism phi from G_0 to G_1 (although many isomorphism many exist) which will be output by the verifier with high probability, regardless of any dishonest prover strategy.
In contrast, we show that it is unlikely that psdIP proofs with constant rounds exist for NP-complete problems by showing that if any NP-complete problem has a constant round psdIP protocol, then the polynomial hierarchy collapses
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