1,932,131 research outputs found
Ranking the best instances
We formulate the local ranking problem in the framework of bipartite ranking
where the goal is to focus on the best instances. We propose a methodology
based on the construction of real-valued scoring functions. We study empirical
risk minimization of dedicated statistics which involve empirical quantiles of
the scores. We first state the problem of finding the best instances which can
be cast as a classification problem with mass constraint. Next, we develop
special performance measures for the local ranking problem which extend the
Area Under an ROC Curve (AUC/AROC) criterion and describe the optimal elements
of these new criteria. We also highlight the fact that the goal of ranking the
best instances cannot be achieved in a stage-wise manner where first, the best
instances would be tentatively identified and then a standard AUC criterion
could be applied. Eventually, we state preliminary statistical results for the
local ranking problem.Comment: 29 page
Physical Instances of Noncommuting Coordinates
Noncommuting spatial coordinates and fields can be realized in actual
physical situations. Plane wave solutions to noncommuting photodynamics exhibit
violaton of Lorentz invariance (special relativity).Comment: 13 pp., using sprocl and amsmath macros; Email correspondence to
[email protected]; talk given at Feza Gursey Institute, Istanbul, Turkey --
June 2001; "Symmetry Methods in Physics", Yerevan, Armenia -- July 2001; "CPT
and Lorentz Symmetry II", Bloomington, IN -- August 2001; "Particles and
Strings", Trento, Italy -- September 2001; "VIII Adriatic Meeting",
Dubrovnik, Croatia -- September 200
Edge Elimination in TSP Instances
The Traveling Salesman Problem is one of the best studied NP-hard problems in
combinatorial optimization. Powerful methods have been developed over the last
60 years to find optimum solutions to large TSP instances. The largest TSP
instance so far that has been solved optimally has 85,900 vertices. Its
solution required more than 136 years of total CPU time using the
branch-and-cut based Concorde TSP code [1]. In this paper we present graph
theoretic results that allow to prove that some edges of a TSP instance cannot
occur in any optimum TSP tour. Based on these results we propose a
combinatorial algorithm to identify such edges. The runtime of the main part of
our algorithm is for an n-vertex TSP instance. By combining our
approach with the Concorde TSP solver we are able to solve a large TSPLIB
instance more than 11 times faster than Concorde alone
Scale-Free Random SAT Instances
We focus on the random generation of SAT instances that have properties
similar to real-world instances. It is known that many industrial instances,
even with a great number of variables, can be solved by a clever solver in a
reasonable amount of time. This is not possible, in general, with classical
randomly generated instances. We provide a different generation model of SAT
instances, called \emph{scale-free random SAT instances}. It is based on the
use of a non-uniform probability distribution to select
variable , where is a parameter of the model. This results into
formulas where the number of occurrences of variables follows a power-law
distribution where . This property
has been observed in most real-world SAT instances. For , our model
extends classical random SAT instances.
We prove the existence of a SAT-UNSAT phase transition phenomenon for
scale-free random 2-SAT instances with when the clause/variable
ratio is . We also prove that scale-free
random k-SAT instances are unsatisfiable with high probability when the number
of clauses exceeds . %This implies that the SAT/UNSAT
phase transition phenomena vanishes when , and formulas are
unsatisfiable due to a small core of clauses. The proof of this result suggests
that, when , the unsatisfiability of most formulas may be due to
small cores of clauses. Finally, we show how this model will allow us to
generate random instances similar to industrial instances, of interest for
testing purposes
Community Structure in Industrial SAT Instances
Modern SAT solvers have experienced a remarkable progress on solving
industrial instances. Most of the techniques have been developed after an
intensive experimental process. It is believed that these techniques exploit
the underlying structure of industrial instances. However, there are few works
trying to exactly characterize the main features of this structure.
The research community on complex networks has developed techniques of
analysis and algorithms to study real-world graphs that can be used by the SAT
community. Recently, there have been some attempts to analyze the structure of
industrial SAT instances in terms of complex networks, with the aim of
explaining the success of SAT solving techniques, and possibly improving them.
In this paper, inspired by the results on complex networks, we study the
community structure, or modularity, of industrial SAT instances. In a graph
with clear community structure, or high modularity, we can find a partition of
its nodes into communities such that most edges connect variables of the same
community. In our analysis, we represent SAT instances as graphs, and we show
that most application benchmarks are characterized by a high modularity. On the
contrary, random SAT instances are closer to the classical Erd\"os-R\'enyi
random graph model, where no structure can be observed. We also analyze how
this structure evolves by the effects of the execution of a CDCL SAT solver. In
particular, we use the community structure to detect that new clauses learned
by the solver during the search contribute to destroy the original structure of
the formula. This is, learned clauses tend to contain variables of distinct
communities
OV Graphs Are (Probably) Hard Instances
© Josh Alman and Virginia Vassilevska Williams. A graph G on n nodes is an Orthogonal Vectors (OV) graph of dimension d if there are vectors v1, . . ., vn ∈ {0, 1}d such that nodes i and j are adjacent in G if and only if hvi, vji = 0 over Z. In this paper, we study a number of basic graph algorithm problems, except where one is given as input the vectors defining an OV graph instead of a general graph. We show that for each of the following problems, an algorithm solving it faster on such OV graphs G of dimension only d = O(log n) than in the general case would refute a plausible conjecture about the time required to solve sparse MAX-k-SAT instances: Determining whether G contains a triangle. More generally, determining whether G contains a directed k-cycle for any k ≥ 3. Computing the square of the adjacency matrix of G over Z or F2. Maintaining the shortest distance between two fixed nodes of G, or whether G has a perfect matching, when G is a dynamically updating OV graph. We also prove some complementary results about OV graphs. We show that any problem which is NP-hard on constant-degree graphs is also NP-hard on OV graphs of dimension O(log n), and we give two problems which can be solved faster on OV graphs than in general: Maximum Clique, and Online Matrix-Vector Multiplication
Community structure in industrial SAT instances
Modern SAT solvers have experienced a remarkable progress on solving industrial instances. It is believed that most of these successful techniques exploit the underlying structure of industrial instances. Recently, there have been some attempts to analyze the structure of industrial SAT instances in terms of complex networks, with the aim of explaining the success of SAT solving techniques, and possibly improving them.
In this paper, we study the community structure, or modularity, of industrial SAT instances. In a graph with clear community structure, or high modularity, we can find a partition of its nodes into communities such that most edges connect variables of the same community. Representing SAT instances as graphs, we show that most application benchmarks are characterized by a high modularity. On the contrary, random SAT instances are closer to the classical Erdös-Rényi random graph model, where no structure can be observed. We also analyze how this structure evolves by the effects of the execution of a CDCL SAT solver, and observe that new clauses learned by the solver during the search contribute to destroy the original structure of the formula. Motivated by this observation, we finally present an application that exploits the community structure to detect relevant learned clauses, and we show that detecting these clauses results in an improvement on the performance of the SAT solver. Empirically, we observe that this improves the performance of several SAT solvers on industrial SAT formulas, especially on satisfiable instances.Peer ReviewedPostprint (published version
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