A Model for Managing Activity Constraints

We present a method for constructing optimized pseudopotentials for general polyatomic systems which yield the “smoothest” orbitals localized in particular regions of space. (“Smoothest” orbitals are those for which the expectation value of the kinetic energy operator is a minimum.) We specifically consider the general case in which one can have several occupied orbitals localized in a particular region, rather than a single orbital. Recently, Anderson presented a pseudopotential formalism for constructing localized orbitals for a single isolated band in a polyatomic system. We have extended his formalism to the general multiorbital case, and show that for a specific choice of pseudopotential the Anderson equation yields optimal orbitals. We also show that the Hückel form of the Anderson pseudopotential equation yields identical eigenvalues (neglecting terms of order S2) to the optimal form, and orbitals differing from optimal orbitals by terms linear in S. The correction terms to the Hückel orbitals are obtained by first-order perturbation theory. Finally we examine the Adams–Gilbert equation, which is valid only for systems representable by a single Slater determinant, and indicate how one may solve this equation for the localized orbitals using perturbation theory with Anderson orbitals as the basis set.

New potential energy surfaces are calculated for the hydronium ion using high-order coupled cluster ab initio methods. Large basis sets are used especially for the inversion part of the full surface. Electronic energies obtained with different correlation consistent basis sets are extrapolated to the infinite basis set limit. Core-valence and first order relativistic effects are also included. The influence of these two contributions and basis set sizes on both the inversion barrier height and equilibrium geometry are investigated thoroughly. The same methods are also adopted for ammonia in order to further improve a recently published surface [J. Chem. Phys. 118, 6358 (2003)]. The vibrational eigenvalues are calculated variationally both for the symmetric and asymmetric isotopomers using exact six-dimensional kinetic energy operators and successive basis set contractions. With the new surfaces, the mean absolute deviations obtained for all experimentally observed inversion splittings for different isotopomers of H3O+ (8 states) and NH314 (17 states) are 0.78 and 0.25 cm−1, respectively. Inversion levels are calculated with almost similar accuracy. These results indicate that the calculated inversion barrier heights for H3O+ and NH3, 650 and 1792 cm−1, respectively, are close to the real values. The value for ammonia is also close to the height determined from published experimental data in our previous work. The lowest energies for the high-frequency modes are computed with the mean absolute deviation being less than 2 cm−1 for isotopomers of H3O+ and less than 4.5 cm−1 for NH314 with respect to experimental energies.

This paper presents a new probabilistie primality test. Upon termination the test outputs "composite" or "prime", along with a short proof of correctness, which can be verified in deterministic polynomial time. The test is different from the tests of Miller [M], Solovay-Strassen [SSI, and Rabin [R] in that its assertions of primality are certain, rather than being correct with high probability or dependent on an unproven assumption.

Chapter 10 Complexity theoretic model theory and algebra

We study stable matching problems in networks where players are embedded in a social context, and may incorporate friendship relations or altruism into their decisions. Each player is a node in a social network and strives to form a good match with a neighboring player. We consider the existence, computation, and inefficiency of stable matchings from which no pair of players wants to deviate. When the benefits from a match are the same for both players, we show that incorporating the well-being of other players into their matching decisions significantly decreases the price of stability, while the price of anarchy remains unaffected. Furthermore, a good stable matching achieving the price of stability bound always exists and can be reached in polynomial time. We extend these results to more general matching rewards, when players matched to each other may receive different utilities from the match. For this more general case, we show that incorporating social context (i.e., “caring about your friends”) can make an even larger difference, and greatly reduce the price of anarchy. We show a variety of existence results, and present upper and lower bounds on the prices of anarchy and stability for various matching utility structures.

It is well known that a search engine can significantly benefit from an auxiliary database, which can suggest interpretations of the search query by means of the involved concepts and their interrelationship. The difficulty is to translate abstract notions like concept and interpretation into a concrete search algorithm that operates over the auxiliary database. To surpass existing heuristics, there is a need for a formal basis, which is realized in this paper through the framework of a search database system, where an interpretation is identified as a parse. It is shown that the parses of a query can be generated in polynomial time in the combined size of the input and the output, even if parses are restricted to those having a nonempty evaluation. Identifying that one parse is more specific than another is important for ranking answers, and this framework captures the precise semantics of being more specific; moreover, performing this comparison between parses is tractable. Lastly, the paper studies the problem of finding the most specific parses. Unfortunately, this problem turns out to be intractable in the general case. However, under reasonable assumptions, the parses can be enumerated in an order of decreasing specificity, with polynomial delay and polynomial space.

In this paper, we show that checking some properties of Boolean functions which are given by the lists of monomials in their polynomial representations can be implemented in polynomial time. Multi-linear polynomials over GF(2) are often a convenient way to represent Boolean functions. There is a single polynomial for each Boolean function, and the length of the polynomial (i.e. the number of its monomials) which represents a function of n variables can be far less than 2n. Therefore, in some cases, polynomials are a compressed description of Boolean functions. Besides, polynomial representations of Boolean functions have applications in circuit lower bounds, computational learning, error-correcting codes, cryptography. We construct polynomial-time algorithms for checking some properties of Boolean functions which are given by the lists of monomials in their polynomial representations. The considered properties are self-anti-duality (evenness), self-duality, periodicity, 1-invariance (Mobius transform invariance, coincidence). Note that checking each of these properties directly by definition gives, in the general case, only exponential-time algorithms. The approach to construct our algorithms is the following. Firstly, we prove that if a Boolean function has a certain property then its polynomial has a special structure. And secondly, we check the property by its characterization, cutting negative cases by proven facts. More precisely, we analyze an exponential time algorithm and prove that if the number of steps of the algorithm exceeds a bound, which is polynomial in the input size, and may be computed in advance, then the input will be necessarily rejected.

The complexity of manipulative attacks in nearly single-peaked electorates

LP-Based Approaches to Stationary-Constrained Markov Decision Problems

Probabilistic circuits are a class of probabilistic generative models that allow us to compute different types of probabilistic queries in polynomial time. Unlike many of the mainstream approaches for generative modeling, they can compute exact likelihoods, marginals, and expectations. Yet, assessing the reliability of their inferences is not straightforward. Credal probabilistic circuits are the imprecise counterpart of probabilistic circuits allowing, among other queries, computations of cautious inferences and sensitivity analyses. In this work, we propose an efficient algorithm to compute the lower and upper expectations for factorizing functions using a credal probabilistic circuit. We discuss under what structural assumptions and types of factorizing functions the algorithm works. We prove that such algorithm has polynomial time complexity in the input size. In the general case, we prove that computing cautious inferences using credal probabilistic circuits is an NP-hard problem, yet the proposed algorithm can be used as an approximation. Some experiments show how the approximation degrades with the complexity of the model structure.

Optimal schedule for works running on machines, in a general case, is a hard problem and there is no complete optimal deterministic algorithm in polynomial time. Optimal and approximated solutions were issued for some specific cases with constraints. One can find the solutions for the cases 1 and 2 machines [4,16] as the initial algorithms. In [16], author solved late works problem using algorithm with setup times included. [4] solve the due works problem on two machines, and can be extended for the case of 3 machines with some conditions on works. Other authors looked at schedule problems in specific cases like [5,8,17]. In this paper, we apply duration automata [1,2] to solve the schedule problem dynamically for the works with uncertain processing time in a cluster computer, which is a system consisting of many computers (computing nodes) co-working together. We solve the schedule problem in a cluster with $m$ machines for two cases: all machines are the same and different in configurations (machine's resources are formally considered as a configuration information, represented by the time need to finish the works). Because of uncertainly processing time, tranditional algorithms can not be used effectively. By using DA model with DA's order criterion, we issue schedule algorithms and practically prove to be better in time consuming compare to FIFO (natural order), the fastest first (greedy) and the longest first (safety) methods without synchronized points.

This paper addresses a one-machine scheduling problem in which the efficiency of the machine is not constant, that is the duration of a task is longer in badly efficient time periods. Each task has an irregular completion cost. Under the assumption that the efficiency constraints are time-periodic, we show that the special case where the sequence is fixed can be solved in polynomial time. The general case is NP-complete so that we propose a two-phase heuristic to find good solutions. Our approach is tested on problems with earliness-tardiness costs.

We study the decision problem of a Proposer who has a set of applications to submit for approval to an Authority and can choose an order of submission. The Proposer’s utility depends on the Authority’s rulings. The Authority has to be consistent with its past decisions, which we model using the nearest-neighbor criterion. If the Proposer’s utility increases with the set of approved applications, then any greedy strategy is optimal for her: She should submit any application that, given the current history, would be approved. However, if her utility increases with some approvals but decreases with others, the Proposer’s problem becomes significantly more complex. In the single-dimensional case, an optimal strategy can be computed in polynomial time. In the general case, however, finding an optimal strategy is NP-hard. Thus, even in the absence of uncertainty or strategic behavior on the part of the Authority, evaluating the impact of current submissions on future outcomes can be computationally intractable.

We introduce an exponential neighborhood for the Vehicle Routing Problem (vrp) with unit customers’ demands, and we show that it can be explored efficiently in polynomial time by reducing its exploration to a particular case of the Restricted Complete Matching (rcm) problem that we prove to be polynomial time solvable using flow techniques. Furthermore, we show that in the general case with non-unit customers’ demands the exploration of the neighborhood becomes an \(\mathcal{NP}\) -hard problem.

The various robust linear programming models investigated so far in the literature essentially appear to be based either on what is referred to as rowwise uncertainty models or on columnwise uncertainty models (these basically assume that the rows, or the columns, of the constraint matrix are subject to changes within a well-specified uncertainty set). In this chapter, we discuss a special case of columnwise uncertainty, namely the subclass of robust linear programming (LP) models with uncertainty limited to the right hand side (RHS) only (this subclass does not appear to have been significantly investigated so far). In this context we introduce the concept of a ‘two-stage robust LP model’ as opposed to the standard case (which might be referred to as a ‘single-stage robust LP model’) and we address the question of whether LP duality can be used to convert a LP problem with RHS uncertainty into a robust LP problem with uncertainty on the objective function. We show how to derive both statements of (a) the dual to the robust model and (b) the robust version of the dual. The resulting expressions of the objective function to be optimized in both cases, appear to be clearly distinct. Moreover, from a complexity point of view, one appears to be efficiently solvable (it reduces to a convex optimization problem), whereas the other, as a nonconvex optimization problem, is expected to be computationally difficult in the general case. As an application of the two-stage robust LP model introduced here, we next investigate the PERT (program evaluation and review technique) scheduling problem, considering two possible natural ways of specifying the uncertainty set for the task durations: the case where the uncertainty set is a scaled ball with respect to the L ∞ norm; the case where the uncertainty set is a scaled Hamming ball of bounded radius (which, though leading to a quite different model, bears some resemblance to the well-known Bertsimas–Sim approach to robustness). We show that in both cases the resulting robust optimization problem can be efficiently solved in polynomial time.