1,052 research outputs found
Pebble Games and Linear Equations
We give a new, simplified and detailed account of the correspondence
between levels of the Sherali-Adams relaxation of graph isomorphism
and levels of pebble-game equivalence with counting (higher-dimensional Weisfeiler-Lehman colour refinement). The correspondence between basic colour refinement and fractional isomorphism, due to Ramana, Scheinerman and Ullman, is re-interpreted as the base level of Sherali-Adams and generalised to higher levels in this sense by Atserias and Maneva, who prove that the two resulting hierarchies interleave.
In carrying this analysis further, we here give (a) a precise characterisation of the level-k Sherali-Adams relaxation in terms of a modified counting pebble game; (b) a variant of the Sherali-Adams levels that precisely match the k-pebble counting game; (c) a proof that the interleaving between these two hierarchies is strict.
We also investigate the variation based on boolean arithmetic instead
of real/rational arithmetic and obtain analogous correspondences and
separations for plain k-pebble equivalence (without counting). Our
results are driven by considerably simplified accounts of the
underlying combinatorics and linear algebra
On Characterizing the Data Movement Complexity of Computational DAGs for Parallel Execution
Technology trends are making the cost of data movement increasingly dominant,
both in terms of energy and time, over the cost of performing arithmetic
operations in computer systems. The fundamental ratio of aggregate data
movement bandwidth to the total computational power (also referred to the
machine balance parameter) in parallel computer systems is decreasing. It is
there- fore of considerable importance to characterize the inherent data
movement requirements of parallel algorithms, so that the minimal architectural
balance parameters required to support it on future systems can be well
understood. In this paper, we develop an extension of the well-known red-blue
pebble game to develop lower bounds on the data movement complexity for the
parallel execution of computational directed acyclic graphs (CDAGs) on parallel
systems. We model multi-node multi-core parallel systems, with the total
physical memory distributed across the nodes (that are connected through some
interconnection network) and in a multi-level shared cache hierarchy for
processors within a node. We also develop new techniques for lower bound
characterization of non-homogeneous CDAGs. We demonstrate the use of the
methodology by analyzing the CDAGs of several numerical algorithms, to develop
lower bounds on data movement for their parallel execution
Relation algebras with n-dimensional relational bases
Accepted versio
Algorithms for detecting dependencies and rigid subsystems for CAD
Geometric constraint systems underly popular Computer Aided Design soft-
ware. Automated approaches for detecting dependencies in a design are critical
for developing robust solvers and providing informative user feedback, and we
provide algorithms for two types of dependencies. First, we give a pebble game
algorithm for detecting generic dependencies. Then, we focus on identifying the
"special positions" of a design in which generically independent constraints
become dependent. We present combinatorial algorithms for identifying subgraphs
associated to factors of a particular polynomial, whose vanishing indicates a
special position and resulting dependency. Further factoring in the Grassmann-
Cayley algebra may allow a geometric interpretation giving conditions (e.g.,
"these two lines being parallel cause a dependency") determining the special
position.Comment: 37 pages, 14 figures (v2 is an expanded version of an AGD'14 abstract
based on v1
Limitations of Algebraic Approaches to Graph Isomorphism Testing
We investigate the power of graph isomorphism algorithms based on algebraic
reasoning techniques like Gr\"obner basis computation. The idea of these
algorithms is to encode two graphs into a system of equations that are
satisfiable if and only if if the graphs are isomorphic, and then to (try to)
decide satisfiability of the system using, for example, the Gr\"obner basis
algorithm. In some cases this can be done in polynomial time, in particular, if
the equations admit a bounded degree refutation in an algebraic proof systems
such as Nullstellensatz or polynomial calculus. We prove linear lower bounds on
the polynomial calculus degree over all fields of characteristic different from
2 and also linear lower bounds for the degree of Positivstellensatz calculus
derivations.
We compare this approach to recently studied linear and semidefinite
programming approaches to isomorphism testing, which are known to be related to
the combinatorial Weisfeiler-Lehman algorithm. We exactly characterise the
power of the Weisfeiler-Lehman algorithm in terms of an algebraic proof system
that lies between degree-k Nullstellensatz and degree-k polynomial calculus
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