Elementary Closure Research Articles

The Chvátal–Gomory procedure for integer SDPs with applications in combinatorial optimization

AbstractIn this paper we study the well-known Chvátal–Gomory (CG) procedure for the class of integer semidefinite programs (ISDPs). We prove several results regarding the hierarchy of relaxations obtained by iterating this procedure. We also study different formulations of the elementary closure of spectrahedra. A polyhedral description of the elementary closure for a specific type of spectrahedra is derived by exploiting total dual integrality for SDPs. Moreover, we show how to exploit (strengthened) CG cuts in a branch-and-cut framework for ISDPs. Different from existing algorithms in the literature, the separation routine in our approach exploits both the semidefinite and the integrality constraints. We provide separation routines for several common classes of binary SDPs resulting from combinatorial optimization problems. In the second part of the paper we present a comprehensive application of our approach to the quadratic traveling salesman problem (QTSP). Based on the algebraic connectivity of the directed Hamiltonian cycle, two ISDPs that model the QTSP are introduced. We show that the CG cuts resulting from these formulations contain several well-known families of cutting planes. Numerical results illustrate the practical strength of the CG cuts in our branch-and-cut algorithm, which outperforms alternative ISDP solvers and is able to solve large QTSP instances to optimality.

Complete Representations and Neat Embeddings

Let \(2<n<\omega\). Then \({\sf CA}_n\) denotes the class of cylindric algebras of dimension \(n\), \({\sf RCA}_n\) denotes the class of representable \(\sf CA_n\)s, \({\sf CRCA}_n\) denotes the class of completely representable \({\sf CA}_n\)s, and \({\sf Nr}_n{\sf CA}_{\omega}(\subseteq {\sf CA}_n\)) denotes the class of \(n\)-neat reducts of \({\sf CA}_{\omega}\)s. The elementary closure of the class \({\sf CRCA}_n\)s (\(\mathbf{K_n}\)) and the non-elementary class \({\sf At}({\sf Nr}_n{\sf CA}_{\omega})\) are characterized using two-player zero-sum games, where \({\sf At}\) is the operator of forming atom structures. It is shown that \(\mathbf{K_n}\) is not finitely axiomatizable and that it coincides with the class of atomic algebras in the elementary closure of \(\mathbf{S_c}{\sf Nr}_n{\sf CA}_{\omega}\) where \(\mathbf{S_c}\) is the operation of forming complete subalgebras. For any class \(\mathbf{L}\) such that \({\sf At}{\sf Nr}_n{\sf CA}_{\omega}\subseteq \mathbf{L}\subseteq {\sf At}\mathbf{K_n}\), it is proved that \({\bf SP}\mathfrak{Cm}\mathbf{L}={\sf RCA}_n\), where \({\sf Cm}\) is the dual operator to \(\sf At\); that of forming complex algebras. It is also shown that any class \(\mathbf{K}\) between \({\sf CRCA}_n\cap \mathbf{S_d}{\sf Nr}_n{\sf CA}_{\omega}\) and \(\mathbf{S_c}{\sf Nr}_n{\sf CA}_{n+3}\) is not first order definable, where \(\mathbf{S_d}\) is the operation of forming dense subalgebras, and that for any \(2<n<m\), any \(l\geq n+3\) any any class \(\mathbf{K}\) (such that \({\sf At}({\sf Nr}_n{\sf CA}_{m})\cap {\sf CRCA}_n\subseteq \mathbf{K}\subseteq {\sf At}\mathbf{S_c}{\sf Nr}_n{\sf CA}_{l}\), \(\mathbf{K}\) is not not first order definable either.

Relationship between screen time among children and lower economic status during elementary school closures due to the coronavirus disease 2019 pandemic

BackgroundThis study aimed to examine whether the extended use of a variety of digital screen devices was associated with lower economic status and other environmental factors among Korean elementary school children and their caregivers during school closures precipitated by the coronavirus disease 2019 (COVID-19) pandemic.MethodsA total of 217 caregivers of children 7–12 years of age from Suwon, Korea, were recruited and asked to respond to a self-administered questionnaire in June 2020. The questionnaire addressed demographic information and children’s use of digital media, in addition to their caregivers. The t-test was used for continuous variables, and the Kruskal-Wallis test was used for variables measured on an interval scale. A multiple regression analyses were performed to examine the effects of significant correlative factors on screen time in children as predictors.ResultsChildren with lower household incomes demonstrated a higher frequency and longer duration of smartphone and tablet personal computer use compared to those from higher income households. Children of households in which incomes decreased after COVID-19 used smartphones and tablet PCs more often and for longer durations. Children from households that experienced decreased income(s) after COVID-19 used personal computers more often and for a longer duration, and children from low-income families engaged in longer screen time on smartphones. A change in primary caregiver(s) may have increased children’s screen time on smartphones.ConclusionLower household income was associated with longer screen time among children, and poor mental health among caregivers during school closures precipitated the COVID-19 pandemic.

Various notions of represetability for cylindric and polyadic algebras

Abstract For β an ordinal, let PEAβ (SetPEAβ) denote the class of polyadic equality (set) algebras of dimension β. We show that for any infinite ordinal α, if is atomic, then for any n < ω, the n-neat reduct of , in symbols , is a completely representable PEAn (regardless of the representability of ). That is to say, for all non-zero , there is a and a homomorphism such that fa(a) ≠ 0 and for any for which exists. We give new proofs that various classes consisting solely of completely representable algebras of relations are not elementary; we further show that the class of completely representable relation algebras is not closed under ≡∞,ω. Various notions of representability (such as ‘satisfying the Lyndon conditions’, weak and strong) are lifted from the level of atom structures to that of atomic algebras and are further characterized via special neat embeddings. As a sample, we show that the class of atomic CAns satisfying the Lyndon conditions coincides with the class of atomic algebras in ElScNrnCAω, where El denotes ‘elementary closure’ and Sc is the operation of forming complete subalgebras.

Self-consistency conditions for elementary Reynolds stress closures

ABSTRACTThis paper summarizes the analytical representation of the correlation tensor in homogeneous anisotropic turbulence and sketches an application to Reynolds stress transport turbulence models. It is shown that the analytical approach can address some of the limitations of conventional modeling.

On the relative strength of different generalizations of split cuts

Split cuts are among the most important and well-understood cuts for general mixed-integer programs. In this paper we consider some recent generalizations of split cuts and compare their relative strength. More precisely, we compare the elementary closures of split, cross, crooked cross and general multi-branch split cuts as well as cuts obtained from multi-row and basic relaxations.We present a complete containment relationship between the closures of split, rank-2 split, cross, crooked cross and general multi-branch split cuts. More specifically, we show that 3-branch split cuts strictly dominate crooked cross cuts, which in turn strictly dominate cross cuts. We also show that multi-branch split cuts are incomparable to rank-2 split cuts. In addition, we also show that cross cuts, and hence crooked cross cuts, cannot always be obtained from 2-row relaxations or from basic relaxations. Together, these results settle some open questions raised in earlier papers.

RLT insights into lift-and-project closures

This paper examines relationships between the two methods for constructing polyhedral outer-approximations of mixed 0-1 linear programs known in the literature as the level-1 Reformulation–Linearization-Technique (RLT) and the elementary closure of lift-and-project (L&P) cuts based on the level-1 L&P. The latter can be obtained, via a transformation of variables, as a relaxed version of the former through the removal of constraints. Whereas both approaches derive formulations in lifted spaces by scaling problem constraints with suitable multipliers, a key difference is that the multipliers used by the RLT are defined in terms of the binary variables of the given problem. As a consequence, these multipliers, in contrast to those used in the L&P, portend important advantages, including the ability to handle nonlinear expressions, to recognize redundant inequalities, and to exploit special structures. This paper compares the two methods, and uses RLT constructs to provide insights into the L&P approach. Some particular insights are illustrated on the quadratic assignment and MAX-2SAT problems, with level-2 RLT and level-2 L&P comparisons also given for the latter problem.

Constant Flux States in Anisotropic Turbulence

An elementary closure theory is used to compute the scaling of anisotropic contributions to the correlation function in homogeneous turbulence. These contributions prove to decay with wavenumber more rapidly than the energy spectrum; this property is sometimes called the “recovery of isotropy” at small scales and is a key hypothesis of the Kolmogorov theory. Although comparisons with a more comprehensive theory suggest that the present theory is too crude, its elementary character makes the scaling analysis straightforward. The analysis reveals some characteristic features of anisotropic turbulence, including “angular” energy transfer in wavevector space.

Reassessment of the classical closures for scalar turbulence

In deducing the consequences of the Direct Interaction Approximation, Kraichnan was sometimes led to consider the properties of special classes of nonlinear interactions in degenerate triads in which one wavevector is very small. Such interactions can be described by simplified models closely related to elementary closures for homogeneous isotropic turbulence such as the Heisenberg and Leith models. These connections can be exploited to derive considerably improved versions of the Heisenberg and Leith models that are only slightly more complicated analytically. This paper applies this approach to derive some new simplified closure models for passive scalar advection and investigates the consistency of these models with fundamental properties of scalar turbulence. Whereas some properties, such as the existence of the Kolmogorov–Obukhov range and the existence of thermal equilibrium ensembles, follow the velocity case closely, phenomena special to the scalar case arise when the diffusive and viscous effects become important at different scales of motion. These include the Batchelor and Batchelor–Howells–Townsend ranges pertaining, respectively, to high and low molecular Schmidt number. We also consider the spectrum in the diffusive range that follows the Batchelor range. We conclude that improved elementary models can be made consistent with many nontrivial properties of scalar turbulence, but that such models have unavoidable limitations.

A simple construction of representable relation algebras with non‐representable completions

AbstractWe give a simple new construction of representable relation algebras with non‐representable completions. Using variations on our construction, we show that the elementary closure of the class of completely representable relation algebras is not finitely axiomatizable (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Using rank-1 lift-and-project closures to generate cuts for 0–1 MIPs, a computational investigation

Various techniques for building relaxations and generating valid inequalities for pure or mixed integer programming problems without special structure are reviewed and compared computationally. Besides classical techniques such as Gomory cuts, Mixed Integer Rounding cuts, lift-and-project and reformulation–linearization techniques, a new variant is also investigated: the use of the relaxation corresponding to the intersection of simple disjunction polyhedra (i.e. the so-called elementary closure of lift-and-project cuts). Systematic comparative computational results are reported on series of test problems including multidimensional knapsack problems (MKP) and MIPLIB test problems. From the results obtained, the relaxation based on the elementary closure of lift-and-project cuts appears to be one of the most promising.

Exact MAX-2SAT solution via lift-and-project closure

We present a new approach for exact solution of MAX-2SAT problems based on a strong reformulation deduced from an optimal continuous solution over the elementary closure of lift-and-project cuts. Computational results show that this formulation leads to a reduced number of nodes in the branch-and-bound tree and short computing times.

A precise correspondence between lift-and-project cuts, simple disjunctive cuts, and mixed integer gomory cuts for 0-1 programming

We establish a precise correspondence between lift-and-project cuts for mixed 0-1 programs, simple disjunctive cuts (intersection cuts) and mixed-integer Gomory cuts. The correspondence maps members of one family onto members of the others. It also maps bases of the higher-dimensional cut generating linear program (CGLP) into bases of the linear programming relaxation. It provides new bounds on the number of facets of the elementary closure, and on the rank, of the standard linear programming relaxation of the mixed 0-1 polyhedron with respect to the above families of cutting planes.

Cutting Planes and the Elementary Closure in Fixed Dimension

The elementary closure P′ of a polyhedron P is the intersection of P with all its Gomory-Chvátal cutting planes. P′ is a rational polyhedron provided that P is rational. The known bounds for the number of inequalities defining P′ are exponential, even in fixed dimension. We show that the number of inequalities needed to describe the elementary closure of a rational polyhedron is polynomially bounded in fixed dimension. If P is a simplicial cone, we construct a polytope Q, whose integral elements correspond to cutting planes of P. The vertices of the integer hull QI include the facets of P′. A polynomial upper bound on their number can be obtained by applying a result of Cook et al. (1992). Finally, we present a polynomial algorithm in varying dimension, which computes cutting planes for a simplicial cone that correspond to vertices of QI.

Elementary closures for integer programs

In integer programming, the elementary closure associated with a family of cuts is the convex set defined by the intersection of all the cuts in the family. In this paper, we compare the elementary closures arising from several classical families of cuts: three versions of Gomory's fractional cuts, three versions of Gomory's mixed integer cuts, two versions of intersection cuts and their strengthened forms, Chvátal cuts, MIR cuts, lift-and-project cuts without and with strengthening, two versions of disjunctive cuts, Sherali–Adams cuts and Lovász–Schrijver cuts with positive semi-definiteness constraints.

NOTE - On the Membership Problem for the Elementary Closure of a Polyhedron

, an integral vector b and some rational vector , decide whether is outside the elementary closure , is NP-complete. This result is achieved by an extension of a result by Caprara and Fischetti.

Induction rules, reflection principles, and provably recursive functions

A well-known result (Leivant, 1983) states that, over basic Kalmar elementary arithmetic EA, the induction schema for ∑ n formulas is equivalent to the uniform reflection principle for ∑ n + 1 formulas ( n ⩾ 1). We show that fragments of arithmetic axiomatized by various forms of induction rules admit a precise axiomatization in terms of reflection principles as well. Thus, the closure of EA under the induction rule for ∑ n (or Π n + 1 ) formulas is equivalent to ω times iterated ∑ n reflection principle. Moreover, for k < ω, k times iterated ∑ n reflection principle over EA precisely corresponds to the extension of EA by ⩽ k nested applications of ∑ n induction rule. The above relationship holds in greater generality than just stated. In fact, we give general formulas characterizing in terms of iterated reflection principles the extension of any given theory (containing EA) by ⩽ k nested applications of ∑ n or Π n induction rules. In particular, the closure of a theory T under just one application of ∑ 1 induction rule is equivalent to T together with ∑ 1 reflection principle for each finite Π 2 axiomatized subtheory of T. These results have closely parallel ones in the theory of subrecursive function classes. The rules under study correspond, in a canonical way, to natural closure operators on the classes of provably recursive functions. Thus, ∑ 1 induction rule precisely corresponds to the primitive recursive closure operator, and ∑ 1 collection rule, introduced below, corresponds to the elementary closure operator.

Relationships among $PL$, $\#L$, and the determinant

Recent results by Toda, Vinay, Damm, and Valiant have shown that the complexity of the determinant is characterized by the complexity ofcounting the number ofaccepting computations ofa nondeterministic logspace-bounded machine. (This class of functions is known as #L) By using that characterization and by establishing a few elementary closure properties, we give a very simple proof of theorem of Jung, showing that probabilistic logspace-bounded (PL) machines lose none of their computational power if they are restricted to run in polynomial time. We also present new results comparing and contrasting the classes of functions reducible to PL, #L, and the determinant, using various notions of reducibility.

Input Proofs and Rank One Cutting Planes

Input resolution and “unit support” resolution (a generalization of unit resolution) are complete inference methods for Horn clauses in propositional logic. We show that they have a close analog in cutting plane theory. Namely, a logical clause can be deduced using input or unit support resolution if and only if it belongs to the elementary closure of the premises and is therefore a rank one cut in Chvátal's sense. This connection leads to a cutting plane algorithm for solving non-Horn inference problems. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

On the set covering polytope: I. All the facets with coefficients in {0, 1, 2}

While the set packing polytope, through its connection with vertex packing, has lent itself to fruitful investigations, little is known about the set covering polytope. We characterize the class of valid inequalities for the set covering polytope with coefficients equal to 0, 1 or 2, and give necessary and sufficient conditions for such an inequality to be minimal and to be facet defining. We show that all inequalities in the above class are contained in the elementary closure of the constraint set, and that 2 is the largest value ofk such that all valid inequalities for the set covering polytope with integer coefficients no greater thank are contained in the elementary closure. We point out a connection between minimal inequalities in the class investigated and certain circulant submatrices of the coefficient matrix. Finally, we discuss conditions for an inequality to cut off a fractional solution to the linear programming relaxation of the set covering problem and to improve the lower bound given by a feasible solution to the dual of the linear programming relaxation.