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Some Dichotomy Theorems for Neural Learning Problems

Michael Schmitt
2004 Journal of machine learning research  
The results are established as dichotomy theorems showing that each problem is either NP-complete or solvable in polynomial time.  ...  In particular, we consider consistency and maximum consistency problems for neurons with binary weights, and maximum consistency problems for neurons with arbitrary weights.  ...  Acknowledgments I thank the anonymous reviewers for their helpful comments and suggestions.  ... 
dblp:journals/jmlr/Schmitt04 fatcat:bl3cwzg7xbbr7lzgl4qk37edam

An Upper Bound on the Minimum Number of Monomials Required to Separate Dichotomies of {−1, 1}n

Erhan Oztop
2006 Neural Computation  
In general, less than 2 n monomials are sufficient to solve a given dichotomy.  ...  The main theorem here states that for any binary classification problem in {−1, 1} n (n > 1), one can always find a polynomial function solution with 2 n − 2 n /4 or fewer monomials.  ...  I was introduced to the problem of establishing a bound on the maximum density by Marifi Guler. I thank Junmei Zhu and Jun Nakanishi for their comments on an earlier version of the manuscript.  ... 
doi:10.1162/neco.2006.18.12.3119 pmid:17052161 fatcat:hfrp3zzuh5gxjoqtoprgrw2bmu

Complexity of Homogeneous Co-Boolean Constraint Satisfaction Problems [article]

Florian Richoux
2010 arXiv   pre-print
In this short note we show a dichotomy theorem for every finite domain D of CSP built upon graphs of homogeneous co-Boolean functions, i.e., unary functions sharing the Boolean range 0, 1.  ...  It has been proved that for each CSP problem over a given set of relations there exists a corresponding CSP problem over graphs of unary functions belonging to the same complexity class.  ...  Similarly to our dichotomy theorem, which differs from Schaefer's Dichotomy Theorem in [7] , we conjecture that a dichotomy theorem for the latter problem will also be different from Bulatov's Dichotomy  ... 
arXiv:1011.4744v1 fatcat:5ptx3rduizfplfin4keg34iypm

Dichotomy Theorems for Alternation-Bounded Quantified Boolean Formulas [article]

Edith Hemaspaandra
2004 arXiv   pre-print
In 1978, Schaefer proved his famous dichotomy theorem for generalized satisfiability problems.  ...  In this paper, we prove dichotomy theorems for alternation-bounded quantified generalized Boolean formulas, by showing that these problems are either Σ_i^p-complete or in P, and we give a simple criterion  ...  Introduction In 1978, Schaefer proved his famous dichotomy theorem for generalized satisfiability problems.  ... 
arXiv:cs/0406006v1 fatcat:scmdburmvvc6dbljacs6ozfufy

Minimum cost and list homomorphisms to semicomplete digraphs

Gregory Gutin, Arash Rafiey, Anders Yeo
2006 Discrete Applied Mathematics  
The list homomorphism problem for H is a generalization of the homomorphism problem for H, where every vertex x ∈ V (D) is assigned a set L x of possible colors (vertices of H).  ...  Our dichotomy for the list homomorphism problem coincides with the one obtained by Bang-Jensen, Hell and MacGillivray in 1988 for the homomorphism problem when H is a semicomplete digraph: both problems  ...  We are very thankful to Dave Cohen, Martin Green, Pavol Hell and Andrei Krokhin for useful discussions on the topic of the paper.  ... 
doi:10.1016/j.dam.2005.11.006 fatcat:w7ym6atkvzf4djesr4hhrd7age

Inverse Circumscription

Hubie Chen
2003 International Joint Conference on Artificial Intelligence  
Inverse (or identification) problems involve decid ing whether or not an explicitly given set of data points have an implicit description, for instance, in the form of a constraint network.  ...  This pa per formalizes and studies the inverse circumscription problem, which (roughly speaking) is to de cide, given a set of models, if there exists a formula whose circumscription describes the input  ...  The author wishes to thank Bart Selman for useful discussions and suggestions, and Joe Halpern for his advice on the preparation of the final version of this paper.  ... 
dblp:conf/ijcai/Chen03 fatcat:asla3oerx5civdmbz3tivwj2ye

Steady-state and periodic exponential turnpike property for optimal control problems in Hilbert spaces [article]

Emmanuel Trelat, Can Zhang, Enrique Zuazua
2016 arXiv   pre-print
A similar statement holds true as well when replacing an optimal steady-state by an optimal periodic trajectory.  ...  In this work, we study the steady-state (or periodic) exponential turnpike property of optimal control problems in Hilbert spaces.  ...  maximum principle for optimal control problems is valid.  ... 
arXiv:1610.01912v2 fatcat:fwvpsxtgonfcrowcjio3lklrny

A Free Boundary Problem for a Leslie-Gower Predator-Prey Model in Higher Dimensions and Heterogeneous Environment

Shiwen Niu, Hongmei Cheng
2020 American Journal of Applied Mathematics  
We establish that a spreading-vanishing dichotomy is held for this model. We use the comparison principle. we will give the existence, uniqueness and some estimates of the solution to the problem.  ...  This paper is mainly concerned with some free boundary problems for a modified Leslie-Gower predator-prey model in higher dimensional and heterogeneous environment.  ...  We have proved that for both problems, a spreading-vanishing dichotomy holds (Theorems 3.1, 3.3 and 3.4), and when spreading occurs the spreading fronts expand at a nearly constant speed for large time  ... 
doi:10.11648/j.ajam.20200805.17 fatcat:grijqs6wxnc6tiwk6y27zd3egi

Steady-State and Periodic Exponential Turnpike Property for Optimal Control Problems in Hilbert Spaces

Emmanuel Trélat, Can Zhang, Enrique Zuazua
2018 SIAM Journal of Control and Optimization  
A similar statement holds true as well when replacing an optimal steady-state by an optimal periodic trajectory.  ...  In this work, we study the steady-state (or periodic) exponential turnpike property of optimal control problems in Hilbert spaces.  ...  maximum principle for optimal control problems is valid.  ... 
doi:10.1137/16m1097638 fatcat:hbelwgevhvbojm6hnmjujqu7be

On First-Species Counterpoint Theory

Juan Sebastián Arias-Valero, Octavio Alberto Agustín-Aquino, Emilio Lluis-Puebla
2021 MusMat|: Brazilian Journal of Music and Mathematics  
We generalize first-species counterpoint theory to arbitrary rings and obtain some new counting and maximization results that enrich the theory of admitted successors, pointing to a structural approach  ...  The original motivations of the theory, as well as all technical passages, are carefully reviewed so as to provide a complete exposition.  ...  Thus, for a general strong dichotomy, we locally characterize contrapuntal dissonances and consonances with 1.  ... 
doi:10.46926/musmat.2021v5n2.1-40 fatcat:idtvifgcgvc7zko2spetgeo6re

The Complexity of Computing Optimal Assignments of Generalized Propositional Formulae [article]

Steffen Reith, Heribert Vollmer
1998 arXiv   pre-print
We consider the problems of finding the lexicographically minimal (or maximal) satisfying assignment of propositional formulae for different restricted formula classes.  ...  It turns out that for each class from our framework, the above problem is either polynomial time solvable or complete for OptP.  ...  We are extremely grateful to Nadia Creignou, Caen, for a lot of helpful hints.  ... 
arXiv:cs/9809116v1 fatcat:p4raoiwtc5gndl5flxx74ficfe

The Complexity of Homomorphisms of Signed Graphs and Signed Constraint Satisfaction [chapter]

Florent Foucaud, Reza Naserasr
2014 Lecture Notes in Computer Science  
We also introduce the more general concept of signed constraint satisfaction problems and show that a dichotomy for such problems is equivalent to the statement of the Feder-Vardi Dichotomy Conjecture.  ...  We prove a dichotomy result for the class of all (C k , Σ)-Coloring problems (where C k is a cycle of length k ≥ 3): (C k , Σ)-Coloring is NP-complete, unless both k and the size of Σ are even.  ...  While it is a difficult problem to prove a dichotomy for S-CSP (equivalently, for CSP or for digraph homomorphism problems), it will be of interest to prove a dichotomy for signed graph homomorphism problems  ... 
doi:10.1007/978-3-642-54423-1_46 fatcat:cudazm72vre5pjgoq3c24revui

Constraint satisfaction problem and universal algebra

Libor Barto
2014 ACM SIGLOG News  
This column gives a brief survey of current research on the complexity of the constraint satisfaction problem (CSP) over fixed constraint languages.  ...  -The dichotomy theorem of Schaefer for CSPs over a two-element domain was generalized to a three-element domain [Bulatov 2011] .  ...  ., one that satisfies the maximum possible number of constraints.  ... 
doi:10.1145/2677161.2677165 fatcat:k5dstcfax5fujky2h5ekhogk2q

A combinatorial constraint satisfaction problem dichotomy classification conjecture

Jaroslav Nešetřil, Mark H. Siggers, László Zádori
2010 European journal of combinatorics (Print)  
The extension in this paper gives a polynomial-time reduction of CSP(H) for any relational system H to CSP(P ) for any relational system P that meets a certain technical partition condition, that of being  ...  This follows immediately from the theorem by the fact, shown in [12] , that CSP(K 3 ) is N P -complete (even for undirected instances).  ...  A substantial generalization of this dichotomy result was obtained in [1] for the H-colouring problems over digraphs with no sources and no sinks.  ... 
doi:10.1016/j.ejc.2009.02.007 fatcat:s3s5i7nbmvhxtnfrefgt2y7u6y

Fine-Grained Dichotomies for the Tutte Plane and Boolean #CSP

Cornelius Brand, Holger Dell, Marc Roth
2018 Algorithmica  
Another dichotomy theorem we strengthen is the one of Creignou and Hermann [6] for counting the number of satisfying assignments to a constraint satisfaction problem instance over the Boolean domain.  ...  We complete the dichotomy theorem for the Tutte polynomial under #ETH by proving that the number of all acyclic subgraphs of a given n-vertex graph cannot be determined in time exp o(n) unless #ETH fails  ...  dichotomies" work group at the Simons Institute in the spring of 2016.  ... 
doi:10.1007/s00453-018-0472-z fatcat:i2ys5fwwkngxbmy557eybmtbti
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