48,474 research outputs found
A feasible interpolation for random resolution
Random resolution, defined by Buss, Kolodziejczyk and Thapen (JSL, 2014), is
a sound propositional proof system that extends the resolution proof system by
the possibility to augment any set of initial clauses by a set of randomly
chosen clauses (modulo a technical condition). We show how to apply the general
feasible interpolation theorem for semantic derivations of Krajicek (JSL, 1997)
to random resolution. As a consequence we get a lower bound for random
resolution refutations of the clique-coloring formulas.Comment: Preprint April 2016, revised September and October 201
Sparser Random 3SAT Refutation Algorithms and the Interpolation Problem:Extended Abstract
We formalize a combinatorial principle, called the 3XOR principle, due to Feige, Kim and Ofek [12], as a family of unsatisfiable propositional formulas for which refutations of small size in any propo-sitional proof system that possesses the feasible interpolation property imply an efficient deterministic refutation algorithm for random 3SAT with n variables and Ω(n1.4) clauses. Such small size refutations would improve the state of the art (with respect to the clause density) efficient refutation algorithm, which works only for Ω(n1.5) many clauses [13]. We demonstrate polynomial-size refutations of the 3XOR principle in resolution operating with disjunctions of quadratic equations with small integer coefficients, denoted R(quad); this is a weak extension of cutting planes with small coefficients. We show that R(quad) is weakly autom-atizable iff R(lin) is weakly automatizable, where R(lin) is similar to R(quad) but with linear instead of quadratic equations (introduced in [25]). This reduces the problem of refuting random 3CNF with n vari-ables and Ω(n1.4) clauses to the interpolation problem of R(quad) and to the weak automatizability of R(lin)
Automating Resolution is NP-Hard
We show that the problem of finding a Resolution refutation that is at most
polynomially longer than a shortest one is NP-hard. In the parlance of proof
complexity, Resolution is not automatizable unless P = NP. Indeed, we show it
is NP-hard to distinguish between formulas that have Resolution refutations of
polynomial length and those that do not have subexponential length refutations.
This also implies that Resolution is not automatizable in subexponential time
or quasi-polynomial time unless NP is included in SUBEXP or QP, respectively
Absorbing Random Walks Interpolating Between Centrality Measures on Complex Networks
Centrality, which quantifies the "importance" of individual nodes, is among
the most essential concepts in modern network theory. As there are many ways in
which a node can be important, many different centrality measures are in use.
Here, we concentrate on versions of the common betweenness and it closeness
centralities. The former measures the fraction of paths between pairs of nodes
that go through a given node, while the latter measures an average inverse
distance between a particular node and all other nodes. Both centralities only
consider shortest paths (i.e., geodesics) between pairs of nodes. Here we
develop a method, based on absorbing Markov chains, that enables us to
continuously interpolate both of these centrality measures away from the
geodesic limit and toward a limit where no restriction is placed on the length
of the paths the walkers can explore. At this second limit, the interpolated
betweenness and closeness centralities reduce, respectively, to the well-known
it current betweenness and resistance closeness (information) centralities. The
method is tested numerically on four real networks, revealing complex changes
in node centrality rankings with respect to the value of the interpolation
parameter. Non-monotonic betweenness behaviors are found to characterize nodes
that lie close to inter-community boundaries in the studied networks
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Comparison of Current Gravity Estimation and Determination Models
This paper will discuss the history of gravity estimation and determination models while analyzing methods that are in development. Some fundamental methods for calculating the gravity field include spherical harmonics solutions, local weighted interpolation, and global point mascon modeling (PMC). Recently, high accuracy measurements have become more accessible, and the requirements for high order geopotential modeling have become more stringent. Interest in irregular bodies, accurate models of the hydrological system, and on-board processing has demanded a comprehensive model that can quickly and accurately compute the geopotential with low memory costs. This trade study of current geopotential modeling techniques will reveal that each modeling technique has a unique use case. It is notable that the spherical harmonics model is relatively accurate but poses a cumbersome inversion problem. PMC and interpolation models, on the other hand, are computationally efficient, but require more research to become robust models with high levels of accuracy. Considerations of the trade study will suggest further research for the point mascon model. The PMC model should be improved through mascon refinement, direct solutions that stem from geodetic measurements, and further validation of the gravity gradient. Finally, the potential for each model to be implemented with parallel computation will be shown to lead to large improvements in computing time while reducing the memory cost for each technique.Aerospace Engineering and Engineering Mechanic
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