logical analysis of numerical data logical analysis of data lad is a methodology developed since the late eighties aimed at discovering hidden structural information in data sets lad was originally developed for analyzing binary data by using the theory of partially defined boolean functions an extension of lad for the analysis of numerical data sets is achieved through the process of binarization consisting in the replacement of each numerical variable by binary indicator variables each showing whether the value of the original variable is above or below a certain level binarization was successfully applied to the analysis of a variety of real life data sets this paper develops the theoretical foundations of the binarization process studying the combinatorial optimization problems related to the minimization of the number of binary variables to provide an algorithmic framework for the practical solution of such problems we construct compact linear integer programming formulations of them we develop polynomial time algorithms for some of these minimization problems and prove np hardness of others

lower bound for the online bin packing problem with restricted repacking in 1996 ivkovic and lloyd a fundamental restriction on fully dynamic maintenance of bin packing inform process lett 59 1996 pp 229 232 gave the lower bound frac 4 3 on the asymptotic worst case ratio for so called fully dynamic bin packing algorithms where the number of repackable items in each step is restricted by a constant in this paper we improve this result to about 1 3871 we present our proof for a semionline case of the classical bin packing but it works for fully dynamic bin packing as well we prove the lower bound by analyzing and solving a specific optimization problem the bound can be expressed exactly using the lambert w function

decision lists and related boolean functions we consider boolean functions represented by decision lists and study their relationships to other classes of boolean functions it turns out that the elementary class of 1 decision lists has interesting relationships to independently defined classes such as disguised horn functions read once functions nested differences of concepts threshold functions and 2 monotonic functions in particular 1 decision lists coincide with fragments of the mentioned classes we further investigate the recognition problem for this class as well as the extension problem in the context of partially defined boolean functions pdbfs we show that finding an extension of a given pdbf in the class of 1 decision lists is possible in linear time this improves on previous results moreover we present an algorithm for enumerating all such extensions with polynomial delay

learnability of quantified formulas we consider the following classes of quantified formulas fix a set of basic relations called a basis take conjunctions of these basic relations applied to variables and constants in arbitrary ways finally quantify existentially or universally some of the variables we introduce some conditions on the basis that guarantee efficient learnability furthermore we show that with certain restrictions on the basis the classification is complete we introduce as an intermediate tool a link between this class of quantified formulas and some well studied structures in universal algebra called clones more precisely we prove that the computational complexity of the learnability of these formulas is completely determined by a simple algebraic property of the basis of relations their clone of polymorphisms finally we use this technique to give a simpler proof of the already known dichotomy theorem over boolean domains

primal dual aggregation and disaggregation for stochastic linear programs a multistage stochastic linear program can be approximated by replacing the underlying information structure by a coarser structure given the ordinary lagrangian for the original problem this amounts to restricting both the primal and dual variables to be adapted to the coarser information structure the resulting primal problem has the same form as the original but with aggregated constraints and decisions mutual upper and lower bounds on the optimal values of the original and aggregated problems are obtained via partially aggregated problems formed by restricting only the primal variables or only the dual variables to the coarser structure the introduction of the partially aggregated problems leads to a general disaggregation framework for developing heuristic approaches to solving the original problem

matrix sandwich problems abstract the matrix sandwich problem msp is introduced here as follows given a 0 1 valued matrix a where is interpreted as do not care does there exist a fill in of the asterisks with 0s and 1s such that the completed 0 1 valued matrix m satisfies property we study the computational complexity of this problem for several matrix properties including the ferrers property block decompositions and certain forbidden submatrices matrix sandwich problems are an important special case of matrix completion problems the latter being generally defined over the real numbers rather than simply 0 1

learning conditional preference networks conditional preference networks cp nets have recently emerged as a popular language capable of representing ordinal preference relations in a compact and structured manner in this paper we investigate the problem of learning cp nets in the well known model of exact identification with equivalence and membership queries the goal is to identify a target preference ordering with a binary valued cp net by interacting with the user through a small number of queries each example supplied by the user or the learner is a preference statement on a pair of outcomes in this model we show that acyclic cp nets are not learnable with equivalence queries alone even if the examples are restricted to swaps for which dominance testing takes linear time by contrast acyclic cp nets are what is called attribute efficiently learnable when both equivalence queries and membership queries are available we indeed provide a learning algorithm whose query complexity is linear in the description size of the target concept but only logarithmic in the total number of attributes interestingly similar properties are derived for tree structured cp nets in the presence of arbitrary examples our learning algorithms are shown to be quasi optimal by deriving lower bounds on the vc dimension of cp nets in a nutshell our results reveal that active queries are required for efficiently learning cp nets in large multi attribute domains

on learning read k satisfy j dnf we study the learnability of read k satisfy j rksj dnf formulas these are boolean formulas in disjunctive normal form dnf in which the maximum number of occurrences of a variable is bounded by k and the number of terms satisfied by any assignment is at most j after motivating the investigation of this class of dnf formulas we present an algorithm that for any unknown rksj dnf formula to be learned with high probability finds a logically equivalent dnf formula using the well studied protocol of equivalence and membership queries the algorithm runs in polynomial time for k cdot j o log n over log log n where n is the number of input variables

a gaussian upper bound for gaussian multi stage stochastic linear programs this paper deals with two stage and multi stage stochastic programs in which the right hand sides of the constraints are gaussian random variables such problems are of interest since the use of gaussian estimators of random variables is widespread we introduce algorithms to find upper bounds on the optimal value of two stage and multi stage stochastic minimization programs with gaussian right hand sides the upper bounds are obtained by solving deterministic mathematical programming problems with dimensions that do not depend on the sample space size the algorithm for the two stage problem involves the solution of a deterministic linear program and a simple semidefinite program the algorithm for the multi stage problem invovles the solution of a quadratically constrained convex programming problem

logical analysis of binary data with missing bits we model a given pair of sets of positive and negative examples each of which may contain missing components as a partially defined boolean function with missing bits pbmb t f where email protected 0 1 n and email protected 0 1 n and stands for a missing bit then we consider the problem of establishing a boolean function an extension f 0 1 n 0 1 belonging to a given function class c such that f is true respectively false for every vector in t respectively in f this is a fundamental problem encountered in many areas such as learning theory pattern recognition example based knowledge bases logical analysis of data knowledge discovery and data mining in this paper depending upon how to deal with missing bits we formulate three types of extensions called robust consistent and most robust extensions for various classes of boolean functions such as general positive horn threshold decomposable and k dnf the complexity of the associated problems are then clarified some of them are solvable in polynomial time while the others are np hard

fully dynamic algorithms for bin packing being mostly myopic helps the problem of maintaining an approximate solution for one dimensional bin packing when items may arrive and depart dynamically is studied in accordance with various work on fully dynamic algorithms and in contrast to prior work on bin packing it is assumed that the packing may be arbitrarily rearranged to accommodate arriving and departing items in this context our main result is a fully dynamic approximation algorithm for bin packing mmp that is frac 5 4 competitive and requires theta log n time per operation i e for an insert or a delete of an item this competitive ratio of frac 5 4 is nearly as good as that of the best practical off line algorithms our algorithm utilizes the technique introduced here whereby the packing of an item is done with a total disregard for already packed items of a smaller size this myopic packing of an item may then cause several smaller items to be repacked in a similar fashion with a bit of additional sophistication to avoid certain bad cases the number of items either individual items or bundles of very small items treated as a whole that needs to be repacked is bounded by a constant

new horn revision algorithms a revision algorithm is a learning algorithm that identifies the target concept starting from an initial concept such an algorithm is considered efficient if its complexity in terms of the measured resource is polynomial in the syntactic distance between the initial and the target concept but only polylogarithmic in the number of variables in the universe we give efficient revision algorithms in the model of learning with equivalence and membership queries the algorithms work in a general revision model where both deletion and addition revision operators are allowed in this model one of the main open problems is the efficient revision of horn formulas two revision algorithms are presented for special cases of this problem for depth 1 acyclic horn formulas and for definite horn formulas with unique heads

aggregation and discretization in multistage stochastic programming multistage stochastic programs have applications in many areas and support policy makers in finding rational decisions that hedge against unforeseen negative events in order to ensure computational tractability continuous state stochastic programs are usually discretized and frequently the curse of dimensionality dictates that decision stages must be aggregated in this article we construct two discrete stage aggregated stochastic programs which provide upper and lower bounds on the optimal value of the original problem the approximate problems involve finitely many decisions and constraints thus principally allowing for numerical solution

an adaptive partition based approach for solving two stage stochastic programs with fixed recourse we study an adaptive partition based approach for solving two stage stochastic programs with fixed recourse a partition based formulation is a relaxation of the original stochastic program and we study a finitely converging algorithm in which the partition is adaptively adjusted until it yields an optimal solution a solution guided refinement strategy is developed to refine the partition by exploiting the relaxation solution obtained from a partition in addition to refinement we show that in the case of stochastic linear programs it is possible to merge some components in a partition without weakening the corresponding relaxation bound thus allowing the partition size to be kept small we also show that for stochastic linear programs with simple recourse there exists a small partition that yields an optimal solution the size of this partition is independent of the number of scenarios used in the model our computational results show that the proposed adaptive partition based approach converges v

algorithms for the relaxed online bin packing model the typical online bin packing problem requires the fitting of a sequence of rationals in 0 1 into a minimum number of bins of unit capacity by packing the ith input element without any knowledge of the sizes or the number of input elements that follow moreover unlike typical online problems this one issue does not admit any data reorganization i e no element can be moved from one bin to another in this paper first of all the relaxed online bin packing model will be formalized this model allows a constant number of elements to move from one bin to another as a consequence of the arrival of a new input element then in the context of this new model two online algorithms will be described the first presents linear time and space complexities with a 1 5 approximation ratio and moves at most once only small elements the second instead is an o n log n time and linear space algorithm with a 1 33 approximation ratio and moves each element a constant number of times in the worst case as a result of the arrival of a new input element the first algorithm moves no more than three elements while the second moves as many as seven elements please note that the number of movements performed is explicitly considered in the complexity analysis both algorithms are below the theoretical 1 536 lower bound effective for the online bin packing algorithms without the movement of elements moreover our algorithms are more online than any other linear space online bin packing algorithm because unlike the algorithms already known they allow the return of a possibly relevant fraction of bins before the work is carried out

decision tree approximations of boolean functions decision trees are popular representations of boolean functions we show that given an alternative representation of a boolean function f say as a read once branching program one can find a decision tree t which approximates f to any desired amount of accuracy moreover the size of the decision tree is at most that of the smallest decision tree which can represent f and this construction can be obtained in quasi polynomial time we also extend this result to the case where one has access only to a source of random evaluations of the boolean function f instead of a complete representation in this case we show that a similar approximation can be obtained with any specified amount of confidence as opposed to the absolute certainty of the former case this latter result implies proper pac learnability of decision trees under the uniform distribution without using membership queries

dynamic bin packing motivated by potential applications to computer storage allocation we generalize the classical one dimensional bin packing model to include dynamic arrivals and departures of items over time within this setting we prove close upper and lower bounds on the worst case performance of the commonly used first fit packing algorithm and using adversary type arguments we show that no on line packing algorithm can satisfy a substantially better performance bound than that for first fit

error free and best fit extensions of partially defined boolean functions abstract in this paper we address a fundamental problem related to the induction of boolean logic given a set of data represented as a set of binary truen vectors or positive examples and a set of falsen vectors or negative examples we establish a boolean function or an extension f so thatfis true resp false in every given true resp false vector we shall further require that such an extension belongs to a certain specified class of functions e g class of positive functions class of horn functions and so on the class of functions represents our a priori knowledge or hypothesis about the extensionf which may be obtained from experience or from the analysis of mechanisms that may or may not cause the phenomena under consideration the real world data may contain errors e g measurement and classification errors might come in when obtaining data or there may be some other influential factors not represented as variables in the vectors in such situations we have to give up the goal of establishing an extension that is perfectly consistent with the given data and we are satisfied with an extensionfhaving the minimum number of misclassifications both problems i e the problem of finding an extension within a specified class of boolean functions and the problem of finding a minimum error extension in that class will be extensively studied in this paper for certain classes we shall provide polynomial algorithms and for other cases we prove their np hardness

reasoning with models we develop a model based approach to reasoning in which the knowledge base is represented as a set of models satisfying assignments rather than a logical formula and the set of queries is restricted we show that for every propositional knowledge base kb there exists a set of characteristic models with the property that a query is true in kb if and only if it is satisfied by the models in this set we characterize a set of functions for which the model based representation is compact and provides efficient reasoning these include cases where the formula based representation does not support efficient reasoning in addition we consider the model based approach to abductive reasoning and show that for any propositional kb reasoning with its model based representation yields an abductive explanation in time that is polynomial in its size some of our technical results make use of the monotone theory a new characterization of boolean functions recently introduced the notion of restricted queries is inherent in our approach this is a wide class of queries for which reasoning is efficient and exact even when the model based representation kb provides only an approximate representation of the domain in question moreover we show that the theory developed here generalizes the model based approach to reasoning with horn expressions and captures even the notion of reasoning with horn approximations

on learning monotone dnf under product distributions we show that the class of monotone 2o logn term dnf formulae can be pac learned in polynomial time under the uniform distribution from random examples only this is an exponential improvement over the best previous polynomial time algorithms in this model which could learn monotone o log2 n term dnf we also show that various classes of small constant depth circuits which compute monotone functions are pac learnable in polynomial time under the uniform distribution all of our results extend to learning under any constant bounded product distribution

on the proper learning of axis parallel concepts we study the proper learnability of axis parallel concept classes in the pac learning and exact learning models these classes include union of boxes dnf decision trees and multivariate polynomials for constant dimensional axis parallel concepts c we show that the following problems have time complexities that are within a polynomial factor of each other c is properly exactly learnable with hypotheses of size at most times the target size from membership and equivalence queries c is properly pac learnable without membership queries under any product distribution there is an approximation algorithm for the minequic problem given a g c find a minimal size f c that is logically equivalent to g in particular if one has polynomial time complexity they all do using this we give the first proper learning algorithm of constant dimensional decision trees and the first negative results in proper learning from membership and equivalence queries for many classes for axis parallel concepts over a nonconstant dimension we show that with the equivalence oracle 1 3 we use this to show that binary decision trees are not properly learnable in polynomial time assuming p np and dnf is not se properly learnable e 1 in polynomial time even with an np oracle assuming 2p pnp

aggregation bounds in stochastic linear programming stochastic linear programs become extremely large and complex as additional uncertainties and possible future outcomes are included in their formulation row and column aggregation can significantly reduce this complexity but the solutions of the aggregated problem only provide an approximation of the true solution in this paper error bounds on the value of the optimal solution of the original problem are obtained from the solution of the aggregated problem these bounds apply for aggregation of both random variables and time periods

on learning ring sum expansions the problem of learning ring sum expansions from examples is studied ring sum expansions rse are representations of boolean functions over the base wedge oplus 1 which reflect arithmetic operations in gf 2 k rse is the class of ring sum expansions containing only monomials of length at most k k term is the class of ring sum expansions having at most k monomials it is shown that k rse k geq 1 is learnable while k term rse k geq 2 is not learnable if rp ne np without using a complexity theoretical hypothesis it is proven that k rse k geq 1 and k term rse k geq 2 cannot be learned from positive negative examples alone however if the restriction that the hypothesis which is output by the learning algorithm is also a k rse is suspended then k rse is learnable from positive negative examples only moreover it is proved that 2 term is learnable by a conjunction of a 2 cnf and a 1 dnf finally the paper presents learning on line prediction algorithms for k

the scenario generation algorithm for multistage stochastic linear programming a multistage stochastic linear program mslp is a model of sequential stochastic optimization where the objective and constraints are linear when any of the random variables used in the mslp are continuous the problem is infinite dimensional to numerically tackle such a problem we usually replace it with a finite dimensional approximation even when all the random variables have finite support the problem is often computationally intractable and must be approximated by a problem of smaller dimension one of the primary challenges in the field of stochastic programming deals with discovering effective ways to evaluate the importance of scenarios and to use that information to trim the scenario tree in such a way that the solution to the smaller optimization problem is not much different from the problem stated with the original tree the scenario generation sg algorithm proposed in this paper is a finite element method that addresses this problem for the class of mslp with random right hand sides

conjunctions of unate dnf formulas learning and structure abstract a central topic in query learning is to determine which classes of boolean formulas are efficiently learnable with membership and equivalence queries we consider the class r kconsisting of conjunctions ofkunate dnf formulas this class generalizes the class ofk clause cnf formulas and the class of unate dnf formulas both of which are known to be learnable in polynomial time with membership and equivalence queries we prove that r 2can be properly learned with a polynomial number of polynomial size membership and equivalence queries but can be properly learned in polynomial time with such queries if and only if p np thus the barrier to properly learning r 2with membership and equivalence queries is computational rather than informational few results of this type are known in our proofs we use recent results of hellersteinet al 1997 j assoc comput mach 43 5 840 862 characterizing the classes that are polynomial query learnable together with work of bshouty on the monotone dimension of boolean functions we extend some of our results to r kand pose open questions on learning dnf formulas of small monotone dimension we also prove structural results for r k we construct for any fixedk 2 a class of functionsfthat cannot be represented by any formula in r k but which cannot be easily shown to have this property more precisely for any functionfonnvariables in the class the value offon any polynomial size set of points in its domain is not a witness thatfcannot be represented by a formula in r k our construction is based on bch codes

an information based approximation scheme for stochastic optimization problems in continuous time dynamic stochastic optimization problems with a large possibly infinite number of decision stages and high dimensional state vectors are inherently difficult to solve in fact scenario tree based algorithms are unsuitable for problems with many stages while dynamic programming type techniques are unsuitable for problems with many state variables this paper proposes a stage aggregation scheme for stochastic optimization problems in continuous time thus having an extremely large i e uncountable number of decision stages by perturbing the underlying data and information processes we construct two approximate problems that provide bounds on the optimal value of the original problem moreover we prove that the gap between the bounds converges to zero as the stage aggregation is refined if massive aggregation of stages is possible without sacrificing too much accuracy the aggregate approximate problems can be addressed by means of scenario tree based methods the suggested approach applies to problems that exhibit randomness in the objective and the constraints while the constraint functions are required to be additively separable in the decision variables and random parameters

on the online bin packing problem a new framework for analyzing online bin packing algorithms is presented this framework presents a unified way of explaining the performance of algorithms based on the harmonic approach within this framework it is shown that a new algorithm harmonic has asymptotic performance ratio at most 1 58889 it is also shown that the analysis of harmonic 1 presented in richey 1991 is incorrect this is a fundamental logical flaw not an error in calculation or an omitted case the asymptotic performance ratio of harmonic 1 is at least 1 59217 thus harmonic provides the best upper bound for the online bin packing problem to date

exact learning boolean functions via the monotone theory we study the learnability of boolean functions from membership and equivalence queries we develop the monotone theory that proves 1 any boolean function is learnable in polynomial time in its minimal disjunctive normal form size its minimal conjunctive normal form size and the number of variables n in particular 2 decision trees are learnable our algorithms are in the model of exact learning with membership queries and unrestricted equivalence queries the hypotheses to the equivalence queries and the output hypotheses are depth 3 formulas

learning random log depth decision trees under uniform distribution we consider three natural models of random logarithmic depth decision trees over boolean variables we give an efficient algorithm that for each of these models learns all but an inverse polynomial fraction of such trees using only uniformly distributed random examples from 0 1 n the learning algorithm constructs a decision tree as its hypothesis

theory revision with queries horn read once and parity formulas a theory in this context is a boolean formula it is used to classify instances or truth assignments theories can model real world phenomena and can do so more or less correctly the theory revision or concept revision problem is to correct a given roughly correct concept this problem is considered here in the model of learning with equivalence and membership queries a revision algorithm is considered efficient if the number of queries it makes is polynomial in the revision distance between the initial theory and the target theory and polylogarithmic in the number of variables and the size of the initial theory the revision distance is the minimal number of syntactic revision operations such as the deletion or addition of literals needed to obtain the target theory from the initial theory efficient revision algorithms are given for horn formulas and read once formulas where revision operators are restricted to deletions of variables or clauses and for parity formulas where revision operators include both deletions and additions of variables we also show that the query complexity of the read once revision algorithm is near optimal

constraint aggregation in infinite dimensional spaces and applications an aggregation technique for constraints with values in hilbert spaces is suggested the technique allows us to replace the original optimization problem by a sequence of subproblems having scalar or finite dimensional constraints applications to optimal control games and stochastic programming are discussed in detail

variations on extending partially defined boolean functions with missing bits in this paper we consider four possible definitions for extending a partially defined boolean function in which the input contains some missing bits we show somewhat surprisingly that for many general and frequently used families of function classes three of these notions of an extension are mathematically equivalent though such an equivalence does not hold universally as demonstrated by several examples

on computing all abductive explanations from a propositional horn theory abduction is a fundamental mode of reasoning with applications in many areas of ai and computer science the computation of abductive explanations is an important computational problem which is at the core of early systems such as the atms and clause management systems and is intimately related to prime implicate generation in propositional logic many algorithms have been devised for computing some abductive explanation and the complexity of the problem has been well studied however little attention has been paid to the problem of computing multiple explanations and in particular all explanations for an abductive query we fill this gap and consider the computation of all explanations of an abductive query from a propositional horn theory or of a polynomial subset of them our study pays particular attention to the form of the query ranging from a literal to a compound formula to whether explanations are based on a set of abducible literals and to the representation of the horn theory either by a horn conjunctive normal form cnf or model based in terms of its characteristic models for these combinations we present either tractability results in terms of polynomial total time algorithms intractability results in terms of nonexistence of such algorithms unless p e np or semi tractability results in terms of solvability in quasi polynomial time established by polynomial time equivalence to the problem of dualizing a monotone cnf expression our results complement previous results in the literature and refute a longstanding conjecture by selman and levesque they elucidate the complexity of generating all abductive explanations and shed light on related problems such as generating sets of restricted prime implicates of a horn theory the algorithms for tractable cases can be readily applied for generating a polynomial subset of explanations in polynomial time

fully dynamic bin packing revisited we consider the fully dynamic bin packing problem where items arrive and depart in an online fashion and repacking of previously packed items is allowed the goal is of course to minimize both the number of bins used as well as the amount of repacking a recently introduced way of measuring the repacking costs at each timestep is the migration factor defined as the total size of repacked items divided by the size of an arriving or departing item concerning the trade off between number of bins and migration factor if we wish to achieve an asymptotic competitive ratio of 1 epsilon for the number of bins a relatively simple argument proves a lower bound of omega 1 epsilon for the migration factor we establish a nearly matching upper bound of o 1 epsilon 4 log 1 epsilon using a new dynamic rounding technique and new ideas to handle small items in a dynamic setting such that no amortization is needed the running time of our algorithm is polynomial in the number of items nand in 1 epsilon the previous best trade off was for an asymptotic competitive ratio of 5 4 for the bins rather than 1 epsilon and needed an amortized number of o log n repackings while in our scheme the number of repackings is independent of n and non amortized

learning from examples with unspecified attribute values a challenging problem within machine learning is how to make good inferences from data sets in which pieces of information are missing while it is valuable to have algorithms that perform well for specific domains to gain a fundamental understanding of the problem one needs a theory about how to learn with incomplete data the important contribution of such a theory is not so much the specific algorithmic results but rather that it provides good ways of thinking about the problem formally in this paper we introduce the unspecified attribute value uav learning model as a first step towards a theoretical framework for studying the problem of learning from incomplete data in the exact learning framework in the uav learning model an example x is classified positive resp negative if all possible assignments for the unspecified attributes result in a positive resp negative classification otherwise the classification given to x is for unknown given an example x in which some attributes are unspecified the oracle uav mq responds with the classification of x given a hypothesis h the oracle uav eq returns an example x that could have unspecified attributes for which h x is incorrect we show that any class of functions learnable in angluin s exact model using the mq and eq oracles is also learnable in the uav model using the mq and uav eq oracles as long as the counterexamples provided by the uav eq oracle have a logarithmic number of unspecified attributes we also show that any class learnable in the exact model using the mq and eq oracles is also learnable in the uav model using the uav mq and uav eq oracles as well as an oracle to evaluate a given boolean formula on an example with unspecified attributes for some hypothesis classes such as decision trees and unate formulas the evaluation can be done in polynomial time without an oracle we also study the learnability of a universal class of decision trees under the uav model and of dnf formulas under a representation dependent variation of the uav model

structure identification in relational data this paper presents several investigations into the prospects for identifying meaningful structures in empirical data namely structures permitting effective organization of the data to meet requirements of future queries we propose a general framework whereby the notion of identifiability is given a precise formal definition similar to that of learnability using this framework we then explore if a tractable procedure exists for deciding whether a given relation is decomposable into a constraint network or a cnf theory with desirable topology and if the answer is positive identifying the desired decomposition finally we address the problem of expressing a given relation as a horn theory and if this is impossible finding the best k horn approximation to the given relation we show that both problems can be solved in time polynomial in the length of the data

the complexity and approximability of finding maximum feasible subsystems of linear relations we study the combinatorial problem which consists given a system of linear relations of finding a maximum feasible subsystem that is a solution satisfying as many relations as possible the computational complexity of this general problem named max fls is investigated for the four types of relations and various constrained versions of max fls where a subset of relations must be satisfied or where the variables take bounded discrete values are also considered we establish the complexity of solving these problems optimally and whenever they are intractable we determine their degree of approximability max fls with or relations is np hard even when restricted to homogeneous systems with bipolar coefficients whereas it can be solved in polynomial time for relations with real coefficients the various np hard versions of max fls belong to different approximability classes depending on the type of relations and the additional constraints we show that the range of approximability stretches from apx complete problems which can be approximated within a constant but not within every constant unless p np to npo pb complete ones that are as hard to approximate as all np optimization problems with polynomially bounded objective functions while max fls with equations and integer coefficients cannot be approximated within pe for some e 0 where p is the number of relations the same problem over gf q for a prime q can be approximated within q but not within qe for some e 0 max fls with strict or nonstrict inequalities can be approximated within 2 but not within every constant factor our results also provide strong bounds on the approximability of two variants of max fls with and relations that arise when training perceptrons which are the building blocks of artificial neural networks and when designing linear classifiers

horn extensions of a partially defined boolean function given a partially defined boolean function pdbf t f we investigate in this paper how to find a horn extension f 0 1 n mapsto 0 1 which is consistent with t f where t subseteq 0 1 n denotes a set of true boolean vectors or positive examples and f subseteq 0 1 n denotes a set of false boolean vectors or negative examples given a pdbf t f it is known that the existence of a horn extension can be checked in polynomial time as there are many horn extensions however we consider those extensions f which have maximal and minimal sets t f of the true vectors of f respectively for a pdbf t f there always exists the unique maximal i e maximum horn extension but there are in general many minimal horn extensions we first show that a polynomial time membership oracle can be constructed for the maximum extension even if its disjunctive normal form dnf can be very long our main contribution is to show that checking if a given horn dnf represents a minimal extension and generating a horn dnf of a minimal horn extension can both be done in polynomial time we also can check in polynomial time if a pdbf t f has the unique minimal horn extension however the problems of finding a horn extension f with the smallest t f and of obtaining a horn dnf whose number of literals is smallest are both np hard

aggregation and disaggregation techniques and methodology in optimization a fundamental issue in the use of optimization models is the tradeoff between the level of detail and the ease of using and solving the model aggregation and disaggregation techniques have proven to be valuable tools for manipulating data and determining the appropriate policies to employ for this tradeoff furthermore aggregation and disaggregation techniques offer promise for solving large scale optimization models supply a set of promising methodologies for studying the underlying structure of both univariate and multivariate data sets and provide a set of tools for manipulating data for different levels of decision makers in this paper we develop a general framework for aggregation and disaggregation methodology survey previous work regarding aggregation and disaggregation techniques for optimization problems illuminate the appropriate role of aggregation and disaggregation methodology for optimization applications and propose future research directions

the monotone theory for the pac model in this paper we extend the monotone theory to the pac learning model with membership queries using this extention we show that a dnf formula that has at least one 1 poly heavy clause in one of its cnf representation a clause that is not satisfied with probability 1 poly n s where n is the number of variables and s is the number of terms in f with respect to a distribution d is weakly learnable under this distribution so dnf that are not weakly learnable under the distribution d do not have any 1 poly heavy clauses in any of their cnf representations a dnf f is called cdnf if there is and a cnf representation of f that contains poly n s clauses that approximates f according to a distribution d we show that the class of all cdnf is weakly e pac learnable with membership queries under the distribution d we then show how to change our algorithm to a parallel algorithm that runs in polylogarithmic time with a polynomial number of processors in particular decision trees are strongly pac learnable with membership queries under any distribution in parallel in polylogarithmic time with a polynomial number of processors finally we show that no efficient parallel exact learning algorithm exists for decision trees

robust approximation schemes for cube packing square bin packing where square items are to be packed into a minimum number of unit size squares and its multidimensional generalization cube packing where d dimensional cubic items are to be packed into a minimum number of unit size cubes of the same dimension are well studied generalizations of the classical one dimensional bin packing problem these problems are known to have asymptotic polynomial time approximation schemes aptas in this paper we design robust approximation schemes for these problems at each step a robust algorithm receives a single input item to be added to the packing the new item must be packed and the packing can be modified but the total volume of items which may migrate between bins or change their positions inside bins must be bounded by a constant factor times the volume of the new item that is the solution is created by slightly adjusting the solution upon the arrival of each new item previous approximation schemes partition items into three types according

learning functions represented as multiplicity automata we study the learnability of multiplicity automata in angluin s exact learning model and we investigate its applications our starting point is a known theorem from automata theory relating the number of states in a minimal multiplicity automaton for a function to the rank of its hankel matrix with this theorem in hand we present a new simple algorithm for learning multiplicity automata with improved time and query complexity and we prove the learnability of various concept classes these include among others the class of disjoint dnf and more generally satisfy o 1 dnf the class of polynomials over finite fields the class of bounded degree polynomials over infinite fields the class of xor of terms certain classes of boxes in high dimensions in addition we obtain the best query complexity for several classes known to be learnable by other methods such as decision trees and polynomials over gf 2 while multiplicity automata are shown to be useful to prove the learnability of some subclasses of dnf formulae and various other classes we study the limitations of this method we prove that this method cannot be used to resolve the learnability of some other open problems such as the learnability of general dnf formulas or even k term dnf for k log n or satisfy s dnf formulas for s 1 these results are proven by exhibiting functions in the above classes that require multiplicity automata with super polynomial number of states

primal and dual linear decision rules in stochastic and robust optimization linear stochastic programming provides a flexible toolbox for analyzing real life decision situations but it can become computationally cumbersome when recourse decisions are involved the latter are usually modeled as decision rules i e functions of the uncertain problem data it has recently been argued that stochastic programs can quite generally be made tractable by restricting the space of decision rules to those that exhibit a linear data dependence in this paper we propose an efficient method to estimate the approximation error introduced by this rather drastic means of complexity reduction we apply the linear decision rule restriction not only to the primal but also to a dual version of the stochastic program by employing techniques that are commonly used in modern robust optimization we show that both arising approximate problems are equivalent to tractable linear or semidefinite programs of moderate sizes the gap between their optimal values estimates the loss of optimality incurred by the linear decision rule approximation our method remains applicable if the stochastic program has random recourse and multiple decision stages it also extends to cases involving ambiguous probability distributions

complexity of identification and dualization of positive boolean functions abstract we consider in this paper the problem of identifying min t and max f of a positive i e monotone boolean function by using membership queries only where min t max f denotes the set of minimal true vectors maximal false vectors of it is shown that the existence of an incrementally polynomial algorithm for this problem is equivalent to the existence of the following algorithms where and g are positive boolean functions an incrementally polynomial algorithm to dualize an incrementally polynomial algorithm to self dualize a polynomial algorithm to decide if and are mutually dual a polynomial algorithm to decide if is self dual a polynomial algorithm to decide if is saturated a polynomial algorithm in min max to identify min only some of these are already well known open problems in the respective fields other related topics including various equivalent problems encountered in hypergraph theory and theory of coteries used in distributed systems are also discussed

computational limitations on learning from examples the computational complexity of learning boolean concepts from examples is investigated it is shown for various classes of concept representations that these cannot be learned feasibly in a distribution free sense unless r np these classes include a disjunctions of two monomials b boolean threshold functions and c boolean formulas in which each variable occurs at most once relationships between learning of heuristics and finding approximate solutions to np hard optimization problems are given