Solvable Class Research Articles

Branch‐and‐bound approach for optima localization in scheduling multiprocessor jobs

AbstractWe consider multiprocessor task scheduling problems with dedicated processors. We determine the tight optima localization intervals for different subproblems of the basic problem. Based on the ideas of a computer‐aided technique developed by Sevastianov and Tchernykh for shop scheduling problems, we elaborate a similar method for the multiprocessor task scheduling problem. Our method allows us to find an upper bound for the length of the optimal schedule in terms of natural lower bound. As a byproduct of our results, a family of linear‐time approximation algorithms with a constant ratio performance guarantee is designed for the NP‐hard subproblems of the basic problem, and new polynomially solvable classes of problems are found.

Task scheduling in distributed real-time systems

An approach to the flow shop scheduling of the computation process in distributed realtime systems is considered. This approach is based on the concept of a solvable class of systems for which simple optimal scheduling algorithms exist.

(Co)homology of poset Lie algebras

We investigate the (co)homological properties of two classes of Lie algebras that are constructed from any finite poset: the solvable class $\frak{gl}^\preceq$ and the nilpotent class $\frak{gl}^\prec$. We confirm the conjecture of Jollenbeck that says: every prime power $p^r\!\leq\!n\!-\!2$ appears as torsion in $H_\ast(\frak{nil}_n;\mathbb{Z})$, and every prime power $p^r\!\leq\!n\!-\!1$ appears as torsion in $H_\ast(\frak{sol}_n;\mathbb{Z})$. If $\preceq$ is a bounded poset, then the (co)homology of $\frak{gl}^\preceq$ is \emph{torsion-convex}, i.e. if it contains $p$-torsion, then it also contains $p'$-torsion for every prime $p'\!<\!p$. \par We obtain new explicit formulas for the (co)homology of some families over arbitrary fields. Among them are the solvable non-nilpotent analogs of the Heisenberg Lie algebras from the Cairns & Jambor article, the 2-step Lie algebras from Armstrong & Cairns & Jessup article, strictly block-triangular Lie algebras, etc. The resulting generating functions and the combinatorics of how they are obtained are interesting in their own right. \par All this is done by using AMT (algebraic Morse theory). This article serves as a source of examples of how to construct useful acyclic matchings, each of which in turn induces compelling combinatorial problems and solutions. It also enables graph theory to be used in homological algebra.

On a solvable class of product-type systems of difference equations

It is shown that the following class of systems of difference equations zn+1 = ?zanwbn, wn+1 = ?wcnzdn-2, n ? N0, where a,b,c,d ? Z, ?, ?, z-2, z-1, z0,w0 ? C \ {0}, is solvable, continuing our investigation of classification of solvable product-type systems with two dependent variables. We present closed form formulas for solutions to the systems in all the cases. In the main case, when bd ? 0, a detailed investigation of the form of the solutions is presented in terms of the zeros of an associated polynomial whose coefficients depend on some of the parameters of the system.

Solvability of a class of product-type systems of difference equations

A solvable class of product-type systems of difference equations with two dependent variables on the complex domain is presented. The main results complement some recent ones in the literature, while their proofs contain some refined methodological details. We provide closed form formulas for general solutions to the system or give procedures for how to get them.

Isotropic Brownian motions over complex fields as a solvable model for May–Wigner stability analysis

We consider matrix-valued stochastic processes known as isotropic Brownian motions, and show that these can be solved exactly over complex fields. While these processes appear in a variety of questions in mathematical physics, our main motivation is their relation to a May–Wigner-like stability analysis, for which we obtain a stability phase diagram. The exact results establish the full joint probability distribution of the finite-time Lyapunov exponents, and may be used as a starting point for a more detailed analysis of the stability-instability phase transition. Our derivations rest on an explicit formulation of a Fokker–Planck equation for the Lyapunov exponents. This formulation happens to coincide with an exactly solvable class of models of the Calgero–Sutherland type, originally encountered for a model of phase-coherent transport. The exact solution over complex fields describes a determinantal point process of biorthogonal type similar to recent results for products of random matrices, and is also closely related to Hermitian matrix models with an external source.

Using solvable classes in flowshop scheduling

A periodic processes scheduling problem appears for flexible manufacturing systems, computer systems, and other applications. The paper considers an approach to flow shop scheduling in terms of real-time distributed computing systems. The approach is based on the concept of solvable class of systems, for which simple optimal scheduling algorithms exist. The proposed approach belongs to the class of fast approximate algorithms; it slightly falls behind the known heuristic NEH algorithm regarding the optimality criterion, but it is much faster.

New solvable class of product-type systems of difference equations on the complex domain and a new method for proving the solvability

New solvable class of product-type systems of difference equations on the complex domain and a new method for proving the solvability

On speed scaling via integer programming

We consider a class of convex mixed-integer nonlinear programs motivated by speed scaling of heterogeneous parallel processors with sleep states and convex power consumption curves. We show that the problem is NP-hard and identify some polynomially solvable classes. Furthermore, a dynamic programming and a greedy approximation algorithms are proposed to obtain a fully polynomial-time approximation scheme for a special case. For the general case, we implement an outer approximation algorithm.

New quasi-exactly solvable class of generalized isotonic oscillators

We introduce a new family of quasi-exactly solvable (QES) generalized isotonic oscillators which are based on the Hermite exceptional orthogonal polynomials. We obtain exact closed-form expressions for the energies and wavefunctions as well as the allowed potential parameters for the first two members of the family using the Bethe ansatz method. Numerical calculations of the energies reveal that member potentials have multiple QES eigenstates and the number of states for higher members are parameter dependent.

Constraint Solving via Fractional Edge Covers

Many important combinatorial problems can be modeled as constraint satisfaction problems. Hence, identifying polynomial-time solvable classes of constraint satisfaction problems has received a lot of attention. In this article, we are interested in structural properties that can make the problem tractable. So far, the largest structural class that is known to be polynomial-time solvable is the class of bounded hypertree width instances introduced by Gottlob et al. [2002]. Here we identify a new class of polynomial-time solvable instances: those having bounded fractional edge cover number. Combining hypertree width and fractional edge cover number, we then introduce the notion of fractional hypertree width. We prove that constraint satisfaction problems with bounded fractional hypertree width can be solved in polynomial time (provided that the tree decomposition is given in the input). Together with a recent approximation algorithm for finding such decompositions [Marx 2010], it follows that bounded fractional hypertree width is now the most generally known structural property that guarantees polynomial-time solvability.

Properties of bigravity solutions in a solvable class

We consider the properties of solutions in the bigravity theory for general models, which are parametrized by two parameters $\alpha_{3}$ and $\alpha_{4}$. Assuming that two metric tensors $g_{\mu \nu}$ and $f_{\mu \nu}$ satisfy the condition $f_{\mu \nu}=C^{2}g_{\mu \nu}$ where $C$ is a constant, we investigate the conditions for the parameters so that the solutions with $C\neq1$ could exist. We also discuss the magnitude and the sign of corresponding cosmological constants. For the black hole solution, we consider the black hole entropy to which the massive spin-$2$ field contributes. In order to obtain the black hole entropy, we take an approach which uses the Virasoro algebra and the central charge corresponding to the surface term in the action.

Problem With Inhomogeneous Integral Time Condition for a Partial Differential Equation of the First Order With Respect to Time and of Infinite Order With Respect to the Space Variables

We specify a class of quasipolynomials as a class of unique solvability of the problem with inhomogeneous integral time condition for a homogeneous partial differential equation of the first order with respect to time and, in the general case, of infinite order with respect to the space variables with constant coefficients. In this class, the solution of the problem is represented in the form of action of a differential expression whose symbol is the right-hand side of the integral condition upon a meromorphic function of parameters with subsequent setting these parameters equal to zero. In a wider class of quasipolynomials, namely, in the class of existence of a nonunique solution of the problem, we propose a formula for the construction of its partial solutions.

Matching Problems with Delta-Matroid Constraints

Given an undirected graph $G=(V, E)$ and a delta-matroid $(V,{\cal F})$, the delta-matroid matching problem is to find a maximum cardinality matching $M$ such that the set of the end vertices of $M$ belongs to ${\cal F}$. This problem is a natural generalization of the matroid matching problem to delta-matroids, and thus it cannot be solved in polynomial time in general. This paper introduces a class of the delta-matroid matching problem, where the given delta-matroid is a projection of a linear delta-matroid. We first show that it can be solved in polynomial time if the given linear delta-matroid is generic. This result enlarges a polynomially solvable class of matching problems with precedence constraints on vertices such as the 2-master/slave matching. In addition, we design a polynomial-time algorithm when the graph is bipartite and the delta-matroid is defined on one vertex side. This result is extended to the case where a linear matroid constraint is additionally imposed on the other vertex side.

Finding hypernetworks in directed hypergraphs

The term “hypernetwork” (more precisely, s-hypernetwork and (s, d)-hypernetwork) has been recently adopted to denote some logical structures contained in a directed hypergraph. A hypernetwork identifies the core of a hypergraph model, obtained by filtering off redundant components. Therefore, finding hypernetworks has a notable relevance both from a theoretical and from a computational point of view.In this paper we provide a simple and fast algorithm for finding s-hypernetworks, which substantially improves on a method previously proposed in the literature. We also point out two linearly solvable particular cases.Finding an (s, d)-hypernetwork is known to be a hard problem, and only one polynomially solvable class has been found so far. Here we point out that this particular case is solvable in linear time.

The complexity of mixed multi-unit combinatorial auctions: Tractability under structural and qualitative restrictions

Mixed multi-unit combinatorial auctions (MMUCAs) are extensions of classical combinatorial auctions (CAs) where bidders trade transformations of goods rather than just sets of goods. Solving MMUCAs, i.e., determining the sequences of bids to be accepted by the auctioneer, is computationally intractable in general. However, differently from classical combinatorial auctions, little was known about whether polynomial-time solvable classes of MMUCAs can be singled out on the basis of their characteristics. The paper fills this gap, by studying the computational complexity of MMUCA instances under structural and qualitative restrictions, which characterize interactions among bidders and types of bids involved in the various transformations, respectively.

Approximating solutions of generalized pseudocontractive variational inequalities by admissible perturbation type iterative methods

In this paper we give the solvability class of generalized strongly nonlinear variational inequalities modified by the use of the new concept of admissible perturbation operator on nonempty closed convex sets in Hilbert spaces.

The critical manifolds of inhomogeneous bond percolation on bow-tie and checkerboard lattices

We give a conditional derivation of the inhomogeneous critical percolation manifold of the bow-tie lattice with five different probabilities, a problem that does not appear at first to fall into any known solvable class. Although our argument is mathematically rigorous only on a region of the manifold, we conjecture that the formula is correct over its entire domain, and we provide a non-rigorous argument for this that employs the negative probability regime of the triangular lattice critical surface. We discuss how the rigorous portion of our result substantially broadens the range of lattices in the solvable class to include certain inhomogeneous and asymmetric bow-tie lattices, and that, if it could be put on a firm foundation, the negative probability portion of our method would extend this class to many further systems, including F Y Wu’s checkerboard formula for the square lattice. We conclude by showing that this latter problem can in fact be proved using a recent result of Grimmett and Manolescu for isoradial graphs, lending strong evidence in favor of our other conjectured results.This article is part of ‘Lattice models and integrability’, a special issue of Journal of Physics A: Mathematical and Theoretical in honour of F Y Wu's 80th birthday.

On the squeezed limit of the bispectrum in general single field inflation

We investigate the consistency relation relating the squeezed limit of the bispectrum to the scalar spectral index in single field models of inflation. We give a simple integral formula for the bispectrum in the squeezed limit in terms of the free mode functions of the primordial curvature perturbation, in any Lorentz invariant single field model of inflation and without resorting to any approximation, generalizing a recent result obtained by Ganc and Komatsu in the case of canonical kinetic terms. We use our result to verify the consistency relation in an exactly solvable class of models with a non-trivial speed of sound. We then verify the consistency relation at the first non-trivial order in the slow-varying approximation in general single field inflation (a known result) and at second order in this approximation in canonical single field inflation.

Problem with nonlocal two-point condition in time for a homogeneous partial differential equation of infinite order with respect to space variables

We specify a class of unique solvability of a problem with nonlocal boundary condition for a homogeneous partial differential equation of the first order with respect to time and of infinite order with respect to space variables with constant complex coefficients. In the class of quasipolynomials of special form, we indicate formulas for the construction of a solution of the problem that require a finite number of differentiation operations of analytically given functions. For the case where there exists a nonunique solution of the problem, we present an algorithm of the construction of its partial solution.