Family Of Inequalities Research Articles

The Hamiltonian p-median problem: Polyhedral results and branch-and-cut algorithms

In this paper we study the Hamiltonian p-median problem, in which we are given an edge-weighted graph and we are asked to determine p vertex-disjoint cycles spanning all vertices of the graph and having minimum total weight. We introduce two new families of valid inequalities for a formulation of the problem in the space of edge variables. Each one of the families forbids solutions to the 2-factor relaxation of the problem that have less than p cycles. The inequalities in one of the families are associated with large cycles of the underlying graph and generalize known inequalities associated with Hamiltonian cycles. The other family involves inequalities for the case with p=n/3, associated with edge cuts and multi-cuts whose shores have specific cardinalities. We identify inequalities from both families that define facets of the polytope associated with the problem. We design branch-and-cut algorithms based on these families of inequalities and on inequalities associated with 2-opt moves removing sub-optimal solutions. Computational experiments on benchmark instances show that the proposed algorithms exhibit a comparable performance with respect to existing exact methods from the literature. Moreover the algorithms solve to optimality new instances with up to 400 vertices.

Theoretical and computational analysis of a new formulation for the Rural Postman Problem and the General Routing Problem

The Rural Postman Problem (RPP) is one of the most well-known problems in arc routing. Given an undirected graph, the RPP consists of finding a closed walk traversing and servicing a given subset of edges with minimum total cost. In the General Routing Problem (GRP), there is also a subset of vertices that must be visited. Both problems were introduced by Orloff and proved to be NP-hard. In this paper, we propose a new formulation for the RPP and the GRP using two sets of binary variables representing the first and second traversal, respectively, of each edge. We present several families of valid inequalities that induce facets of the polyhedron of solutions under mild conditions. Using this formulation and these families of inequalities, we propose a branch-and-cut algorithm, test it on a large set of benchmark instances, and compare its performance against the exact procedure that, as far as we know, produced the best results. The results obtained show that the proposed formulation is useful for solving undirected RPP and GRP instances of very large size.

Some Families of Jensen-like Inequalities with Application to Information Theory.

It is well known that the traditional Jensen inequality is proved by lower bounding the given convex function, f(x), by the tangential affine function that passes through the point (E{X},f(E{X})), where E{X} is the expectation of the random variable X. While this tangential affine function yields the tightest lower bound among all lower bounds induced by affine functions that are tangential to f, it turns out that when the function f is just part of a more complicated expression whose expectation is to be bounded, the tightest lower bound might belong to a tangential affine function that passes through a point different than (E{X},f(E{X})). In this paper, we take advantage of this observation by optimizing the point of tangency with regard to the specific given expression in a variety of cases and thereby derive several families of inequalities, henceforth referred to as "Jensen-like" inequalities, which are new to the best knowledge of the author. The degree of tightness and the potential usefulness of these inequalities is demonstrated in several application examples related to information theory.

The maximum 2D subarray polytope: Facet-inducing inequalities and polyhedral computations

Given a matrix with real-valued entries, the maximum 2D subarray problem consists in finding a rectangular submatrix with consecutive rows and columns maximizing the sum of its entries. In this work we start a polyhedral study of an integer programming formulation for this problem. We thus define the 2D subarray polytope , explore conditions ensuring the validity of linear inequalities, and provide several families of facet-inducing inequalities. We also report computational experiments assessing the reduction of the dual bound for the linear relaxation achieved by these families of inequalities.

A path forward: Tropicalization in extremal combinatorics

Many important problems in extremal combinatorics can be stated as proving a pure binomial inequality in graph homomorphism numbers, i.e., proving that hom(H1,G)a1⋯hom(Hk,G)ak≥hom(Hk+1,G)ak+1⋯hom(Hm,G)am holds for some fixed graphs H1,…,Hm and all graphs G. One prominent example is Sidorenko's conjecture. For a fixed collection of graphs U={H1,…,Hm}, the exponent vectors of valid pure binomial inequalities in graphs of U form a convex cone. We compute this cone for several families of graphs including complete graphs, even cycles, stars and paths; the latter is the most interesting and intricate case that we compute. In all of these cases, we observe a tantalizing polyhedrality phenomenon: the cone of valid pure binomial inequalities is actually rational polyhedral, and therefore all valid pure binomial inequalities can be generated from the finite collection of exponent vectors of the extreme rays. Using the work of Kopparty and Rossman ([17]), we show that the cone of valid inequalities is indeed rational polyhedral when all graphs Hi are series-parallel and chordal, and we conjecture that polyhedrality holds for any finite collection U. We demonstrate that the polyhedrality phenomenon also occurs in matroids and simplicial complexes. Our description of the inequalities for paths involves a generalization of the Erdős-Simonovits conjecture recently proved in its original form in [29] and a new family of inequalities not observed previously. We also solve an open problem of Kopparty and Rossman on the homomorphism domination exponent of paths. One of our main tools is tropicalization, a well-known technique in complex algebraic geometry, first applied in extremal combinatorics in [3]. We prove several results about tropicalizations which may be of independent interest.

Quantum Bell inequalities from Information Causality – tight for Macroscopic Locality

In a Bell test, the set of observed probability distributions complying with the principle of local realism is fully characterized by Bell inequalities. Quantum theory allows for a violation of these inequalities, which is famously regarded as Bell nonlocality. However, finding the maximal degree of this violation is, in general, an undecidable problem. Consequently, no algorithm can be used to derive quantum analogs of Bell inequalities, which would characterize the set of probability distributions allowed by quantum theory. Here we present a family of inequalities, which approximate the set of quantum correlations in Bell scenarios where the number of settings or outcomes can be arbitrary. We derive these inequalities from the principle of Information Causality, and thus, we do not assume the formalism of quantum mechanics. Moreover, we identify a subspace in the correlation space for which the derived inequalities give the necessary and sufficient conditions for the principle of Macroscopic Locality. As a result, we show that in this subspace, the principle of Information Causality is strictly stronger than the principle of Macroscopic Locality.

A branch‐and‐cut algorithm for the pickup‐and‐delivery traveling salesman problem with handling costs

AbstractIn the Pickup‐and‐Delivery Traveling Salesman Problem with Handling Costs (PDTSPH), a single vehicle has to satisfy multiple customer requests, each defined by a pickup location and a delivery location. Cargo handling is performed at the rear end of the vehicle, in a Last‐In‐First‐Out (LIFO) order for PDTSPH. However, additional handling operations are permitted with a penalty if other loads that block the access to the delivery have to be unloaded and reloaded. The objective of PDTSPH is to minimize the total transportation and handling cost. In this paper, we present a new Mixed Integer Programming (MIP) model and a branch‐and‐cut algorithm to solve PDTSPH. We also present new integral separation procedures to effectively handle the exponential number of constraints in our MIP model. A family of inequalities are introduced to enhance the scalability of our implementation. The performance of our approach is compared with a compact formulation from the literature (Veenstra et al. [21]) in instances ranging from 9 to 21 customer requests. Computational results show our algorithm outperforming the compact formulation in 69% of instances with an average runtime improvement of 57%.

Further generalizations of some Hecke-type identities

Since the study by Jacobi and Hecke, Hecke-type identities have received a lot of attention. In this paper, we obtain a general transformation formula for [Formula: see text]-series. Based on this formula, we derive a new [Formula: see text]-formula in this paper, which clearly includes infinitely many [Formula: see text]-identities. Other identities of Andrews and Liu and new identities for Hecke-type series are also discussed. As applications, we utilize some of these Hecke-type identities to establish families of inequalities for several partition functions.

Second law of thermodynamics for batteries with vacuum state

In stochastic thermodynamics work is a random variable whose average is bounded by the change in the free energy of the system. In most treatments, however, the work reservoir that absorbs this change is either tacitly assumed or modelled using unphysical systems with unbounded Hamiltonians (i.e. the ideal weight). In this work we describe the consequences of introducing the ground state of the battery and hence — of breaking its translational symmetry. The most striking consequence of this shift is the fact that the Jarzynski identity is replaced by a family of inequalities. Using these inequalities we obtain corrections to the second law of thermodynamics which vanish exponentially with the distance of the initial state of the battery to the bottom of its spectrum. Finally, we study an exemplary thermal operation which realizes the approximate Landauer erasure and demonstrate the consequences which arise when the ground state of the battery is explicitly introduced. In particular, we show that occupation of the vacuum state of any physical battery sets a lower bound on fluctuations of work, while batteries without vacuum state allow for fluctuation-free erasure.

Scanning integer points with lex-inequalities: a finite cutting plane algorithm for integer programming with linear objective

We consider the integer points in a unimodular cone K ordered by a lexicographic rule defined by a lattice basis. To each integer point x in K we associate a family of inequalities (lex-inequalities) that define the convex hull of the integer points in K that are not lexicographically smaller than x. The family of lex-inequalities contains the Chvátal–Gomory cuts, but does not contain and is not contained in the family of split cuts. This provides a finite cutting plane method to solve the integer program min {cx: xin Scap mathbb {Z}^n}, where Ssubset mathbb {R}^n is a compact set and cin mathbb {Z}^n. We analyze the number of iterations of our algorithm.

A mass transportation proof of the sharp one-dimensional Gagliardo–Nirenberg inequalities

The aim of this paper is to give a mass transportation proof for a full family of sharp Gagliardo–Nirenberg inequalities in dimension one. In fact, we shall establish a duality principle which derives this family of inequalities as a consequence. We also characterize all optimizers for these inequalities via the mass transportation method.

A polyhedral study of the diameter constrained minimum spanning tree problem

This paper provides a first polyhedral study of the diameter constrained minimum spanning tree problem (DMSTP). We introduce a new set of inequalities, the circular-jump inequalities which strengthen the well-known jump inequalities. These inequalities are further generalized in two ways: either by increasing the number of partitions defining a jump, or by combining jumps with cutsets. Most of the proposed new inequalities are shown to define facets of the DMSTP polytope under very mild conditions. Currently best known lower bounds for the DMSTP are obtained from an extended formulation on a layered graph using the concept of central nodes/edges. A subset of the new families of inequalities is shown to be not implied by this layered graph formulation.

New Valid Inequalities in Branch-and-Cut-and-Price for Multi-Agent Path Finding

BCP, a state-of-the-art algorithm for optimal Multi-agent Path Finding, uses the branch-and-cut-and-price framework to decompose the problem into (1) a master problem that selects a set of collision-free low-cost paths, (2) a pricing problem that adds lower-cost paths to the master problem, (3) separation problems that resolve various kinds of conflicts in the master problem, and (4) branching rules that split the nodes in the high-level branch-and-bound search tree. This paper focuses on the separation aspects of the decomposition by introducing five new classes of fractional conflicts and valid inequalities that remove these conflicts to tighten the linear programming relaxation in the master problem. Experimental results on 12820 instances across 16 maps indicate that including the five families of inequalities allows BCP to solve an additional 585 instances, optimize the same instances 41% faster, and solve 2068 more instances than CBSH-RM and 157 more than Lazy CBS.

Lutwak–Petty projection inequalities for Minkowski valuations and their duals

Lutwak's volume inequalities for polar projection bodies of all orders are generalized to polarizations of Minkowski valuations generated by even, zonal measures on the Euclidean unit sphere. This is based on analogues of mixed projection bodies for such Minkowski valuations and a generalization of the notion of centroid bodies. A new integral representation is used to single out Lutwak's inequalities as the strongest among these families of inequalities, which in turn are related to a conjecture on affine quermassintegrals. In the dual setting, a generalization of volume inequalities for intersection bodies of all orders by Leng and Lu is proved. These results are related to Grinberg's inequalities for dual affine quermassintegrals.

Functional Brunn-Minkowski inequalities induced by polarity

We prove a new family of inequalities, which compare the integral of a geometric convolution of non-negative functions with the integrals of the original functions. For classical inf-convolution, this type of inequality is called the Prékopa-Leindler inequality, which, restricted to indicators of convex bodies, gives the classical Brunn-Minkowski inequality. The convolution we consider is a different one, which arises from the study of the polarity transform for functions. While inf-convolution arises as the pull back of usual addition of convex functions under the Legendre transform, our geometric inf-convolution arises as the pull back of the second order reversing transform on geometric convex functions (called either polarity transform or A-transform). These are, up to linear terms, the only order reversing isomorphisms on the class of geometric convex functions. We prove that the integral of this new geometric convolution of two functions is bounded from below by the harmonic average of the individual integrals. Our inequality implies the Brunn-Minkowski inequality, as well as some other, new, inequalities for volumes of bodies. Our inequalities are intimately connected with Busemann's convexity theorem, a new variant of which we prove for 1-convex hulls and log-concave densities.

An Intrinsic Characterization of Five Points in a CAT(0) Space

Abstract Gromov (2001) and Sturm (2003) proved that any four points in a CAT(0) space satisfy a certain family of inequalities. We call those inequalities the ⊠-inequalities, following the notation used by Gromov. In this paper, we prove that a metric space X containing at most five points admits an isometric embedding into a CAT(0) space if and only if any four points in X satisfy the ⊠-inequalities. To prove this, we introduce a new family of necessary conditions for a metric space to admit an isometric embedding into a CAT(0) space by modifying and generalizing Gromov’s cycle conditions. Furthermore, we prove that if a metric space satisfies all those necessary conditions, then it admits an isometric embedding into a CAT(0) space. This work presents a new approach to characterizing those metric spaces that admit an isometric embedding into a CAT(0) space.

A cutting-plane algorithm for the Steiner team orienteering problem

The Team Orienteering Problem (TOP) is an NP-hard routing problem in which a fleet of identical vehicles aims at collecting rewards (prizes) available at given locations, while satisfying restrictions on the travel times. In TOP, each location can be visited by at most one vehicle, and the goal is to maximize the total sum of rewards collected by the vehicles within a given time limit. In this paper, we propose a generalization of TOP, namely the Steiner Team Orienteering Problem (STOP). In STOP, we provide, additionally, a subset of mandatory locations. In this sense, STOP also aims at maximizing the total sum of rewards collected within the time limit, but, now, every mandatory location must be visited. In this work, we propose a new commodity-based formulation for STOP and use it within a cutting-plane scheme. The algorithm benefits from the compactness and strength of the proposed formulation and works by separating three families of inequalities, which consist of some general connectivity constraints, classical lifted cover inequalities based on dual bounds and a class of conflict cuts. To our knowledge, the last class of inequalities is also introduced in this work. A state-of-the-art branch-and-cut algorithm from the literature of TOP is adapted to STOP and used as baseline to evaluate the performance of the cutting-plane. Extensive computational experiments show the competitiveness of the new algorithm while solving several STOP and TOP instances. In particular, it is able to solve, in total, 15 more TOP instances than any other previous exact algorithm and finds eight new optimality certificates. With respect to the new STOP instances introduced in this work, our algorithm solves 30 more instances than the baseline.

Discriminants of cyclic homogeneous inequalities of three variables

In this article, we study the structure of the cone of semidefinite forms. It is a closed semialgebraic set but usually is not basic closed semialgebraic set. A discriminant is a defining equation of an irreducible component of algebraic boundary of this cone. We calculate discriminants using new tools – characteristic variety and local cones. A characteristic variety is a semialgebraic subset of a real projective variety on which the family of inequalities is essentially defined as linear functions. Local cone is a subcone of the PSD cone which corresponds to a maximal ideal. This theory works well for a family of polynomials which are invariant under an action of a finite group. After we construct an abstract general theory, we apply it to a family of cyclic homogeneous polynomials of three real variables of degree d. We calculate some discriminants for d=3, 4, 5 and 6, and we show that this theory derives many new results.

On Some New Inequalities For Continuous Fusion Frames in Hilbert Spaces

Continuous frames and fusion frames were considered recently as generalizations of frames in Hilbert spaces. In this paper, for any continuous fusion frame, we obtain a new family of inequalities which are parametrized by a parameter $$\lambda \in \mathbb {R}$$ . By suitable choices of $$\lambda $$ , one obtains the previous results as special cases. Moreover, these new inequalities involve the expressions $$\langle S_Yh,h \rangle $$ , $$\Vert S_Yh\Vert $$ , etc., where $$S_Y$$ is a “truncated form” of the continuous fusion frame operator.

Concerning an infinite series of Ramanujan related to the natural logarithm

We consider an infinite series, due to Ramanujan, which converges to a simple expression involving the natural logarithm. We show that Ramanujan’s series represents a completely monotone function, and explore some of its consequences, including a non-trivial family of inequalities satisfied by the natural logarithm, some formulas for the Euler–Mascheroni constant, and a recurrence satisfied by the Bernoulli numbers. We also provide a one-parameter generalization of Ramanujan’s series, which includes as a special case another related infinite series evaluation due to Ramanujan.