Polyhedral Combinatorics Research Articles

A Large Deviations Approach to the Geometry of Random Polytopes

The aim of this article is to present a general “large deviations approach” to the geometry of polytopes spanned by random points with independent coordinates. The origin of our work is in the study of the structure of ±1-polytopes, the convex hulls of subsets of the combinatorial cube . Understanding the complexity of this class of polytopes is important for the “polyhedral combinatorics” approach to combinatorial optimization, and was put forward by Ziegler in [20]. Many natural questions regarding the behaviour of ±1-polytopes in high dimensions are open, since, for many important geometric parameters, low-dimensional intuition does not help to identify the extremal ±1-polytopes. The study of random ±1-polytopes sheds light to some of these questions, the main reason being that random behaviour is often the extremal one.

Polyhedral combinatorics of the cardinality constrained quadratic knapsack problem and the quadratic selective travelling salesman problem

This paper considers the Cardinality Constrained Quadratic Knapsack Problem (QKP) and the Quadratic Selective Travelling Salesman Problem (QSTSP). The QKP is a generalization of the Knapsack Problem and the QSTSP is a generalization of the Travelling Salesman Problem. Thus, both problems are NP hard. The QSTSP and the QKP can be solved using branch-and-cut methods. Good bounds can be obtained if strong constraints are used. Hence it is important to identify strong or even facet-defining constraints. This paper studies the polyhedral combinatorics of the QSTSP and the QKP, i.e. amongst others we identify facet-defining constraints for the QSTSP and the QKP, and provide mathematical proofs that they do indeed define facets.

Nondecomposable solutions to group equations and an application to polyhedral combinatorics

This paper is based on the study of the set of nondecomposable integer solutions in a Gomory corner polyhedron, which was recently used in a reformulation method for integer linear programs. In this paper, we present an algorithm for efficiently computing this set. We precompute a database of nondecomposable solutions for cyclic groups up to order 52. As a second application of this database, we introduce an algorithm for computing nontrivial simultaneous lifting coefficients. The lifting coefficients are exact for a discrete relaxation of the integer program that consists of a group relaxation plus bound constraints.

Two-party Bell inequalities derived from combinatorics via triangular elimination

We establish a relation between the two-party Bell inequalities for two-valued measurements and a high-dimensional convex polytope called the cut polytope in polyhedral combinatorics. Using this relation, we propose a method, triangular elimination, to derive tight Bell inequalities from facets of the cut polytope. This method gives two hundred million inequivalent tight Bell inequalities from currently known results on the cut polytope. In addition, this method gives general formulas which represent families of infinitely many Bell inequalities. These results can be used to examine general properties of Bell inequalities.

Another complete invariant for Steiner triple systems of order 15

AbstractIn this note, the 80 non‐isomorphic triple systems on 15 points are revisited from the viewpoint of the convex hull of the characteristic vectors of their blocks. The main observation is that the numbers, of facets of these 80 polyhedra are all different, thus producing a new proof of the non‐isomorphism of these triple systems. The space dimension of these polyhedra is also discussed. Finally, we observe the large number of facets of some of these polyhedra with few vertices, in relation with the upper bound problem for combinatorial polyhedra. © 2005 Wiley Periodicals, Inc. J Combin Designs.

Commutator-Based (A)[X]n(SU(2)×Sn) NMR Cluster Systems: Establishment of the Universality of [n](Sn) Salients and Constraints on ϕ±11(1.1) Polarisations to the [1n] Salient: Permutational Spin Symmetry (PSS) Within NMR Spin Dynamics - an Analytic View

Analytic SU(2) × Sn dual tensorial (DT) spin dynamics over uniform NMR spins is invoked in examining the modern quantum basis for the universal non-observability rule which governs dominant intracluster JXX' couplings of (A)[X]n NMR systems as a specific form of (abstract) permutational spin symmetry (PSS) with well defined properties on spin-alone space. This is shown to be linked to DT constraints that apply to the cross-product ϕ1±1(1.1) polarisation development i.e., as being confined to [1n](Sn) (Liouvillian) salient, with the existence of [n] (rotating frame) null subspaces. Both these arise within the spin dynamics of (A)[X]2 spin systems (or subsystems thereof) within (a hierarchy of) dominant JXX' governing the internal L(0); such spin systems provide analytic sequels to comparative spin dynamics studies of XX' PSS and AX broken-PSS systems in a Liouvillian coupled tensorial basis formalism, since both draw on (Sanctuary B. C.: Mol. Phys. 1985, 55, 1017), and on the realisation that proper PSS over a (uniform) spin-space L(0) = [H(0),.] zeroth-order Liouvillian and its internal (hierarchical subsets of) JXX' (Temme F. P.: J. Mol. Struct. (THEOCHEM) 2002, 547, 153) i.e., for abstract Sn ↓ G group embeddings. The present work also examines the general irrep-structure of DT spin symmetries for the extent of unit-character irreps and the role of Sn ⊃ Sn-1 ⊃ .. ⊃ [2](S2) group chains in defining the Sn multiple invariants under democratic recoupling of PSS of uniform spin systems. As group measures, these properties apply to both (A)[X]n and [AX]n PSS symmetries, with the invariant cardinality |SI|(n) being related to time-reversal invariance (TRI) and its inherent democratic recoupling (DR) over Weyl (I • I) pairs. For [X]2n uniform spin clusters, |SI|(2n) is best derived via n-fold polyhedral combinatorics of the underlying DR (Temme F. P.: Proc. R. Soc. London, Ser. A 2005, 461, 321) i.e., as an augmented post-Weyl view of the essential role of TRI in (group) invariant cardinality, with the Sn-invariants represented by certain Sn subduction properties.

Time-reversal-based [formula omitted] scalar invariants as (Lie Algebraic) group measures: a structured overview of generalised democratic-recoupled, uniform non-Abelian [ AX] n NMR spin systems, as abstract [formula omitted] chain networks

The physics of dual group scalar invariants (SIs) as (Lie algebraic) group measures (L-GMs) and its significance to non-Abelian NMR spin systems motivates this overview of uniform general-2 n [ AX] 2 n spin evolution, which represents an extensive addendum to Corio’s earlier (essentially restricted) view of Abelian spin system SU(2)-based SI-cardinalities. The |D 0( U )|((⊗SU(2)) (2n))|SI| values in [J. Magn. Reson., 134 (1998) 131] arise from strictly linear recoupled time-reversal invariance (TRI) models. In contrast, here we discuss the physical significance of an alternative polyhedral combinatorics approach to democratic recoupling (DR), a property inherent in both the TRI and statistical sampling. Recognition of spin ensemble SIs as being L-GMs over isomorphic algebras is invaluable in many DR-based NMR problems. Various [ AX] n model spin systems, including the [ AX] 3 bis odd-odd parity spin system, are examined as direct applications of these L-GM- and combinatorial-based SI ideas. Hence in place of | SI|=15 (implied by Corio’s | D 0|((⊗ SU(2)) 2 n ) approach), the bis 3-fold spin system cardinality is seen now as constrained to a single invariant on an isomorphic product algebra under L-GMs, in accord with the subspectral analysis of Jones et al. [Canad. J. Chem., 43 (1965) 683]. The group projective ideas cited here for DR (as cf. to graph theoretic views) apply to highly degenerate non-Abelian problems. Over dual tensorial bases, they define models of spin dynamical evolution whose (SR) quasiparticle superboson carrier (sub)spaces are characterised by SIs acting as explicit auxiliary labels [Physica, A198 (1993) 245; J. Math. Chem., 31 (2002) 281]. A deeper S 2n network-based view of spin-alone space developed in Balasubramanian’s work [J. Chem. Phys., 78 (1983) 6358] is especially important, (e.g.) in the study of spin waves [J. Math. Chem., 31 (2002) 363]. Beyond the specific NMR SIs derived here, there are DR applications where a sporadic, still higher, 2 n-fold regular uniform spin ensemble exhibits a topological FG duality to some known modest | SI| (2 i<2 n) cardinality—in principle providing for the (sparce) existence of other | SI| (2 n) DR-based values.

On Pedigree Polytopes and Hamiltonian Cycles

Abstract Polyhedral combinatorics deals with solving combinatorial optimisation problems using the polytope whose vertices are the characteristic vectors of the combinatorial objects of interest. Given the complete graph with n vertices, Kn, with rational edge weights, the symmetric travelling salesman problem (STSP) is to find a Hamiltonian cycle such that the total edge weight of the Hamiltonian cycle is as small as possible. Polyhedral approach to solve STSP, starts with the standard tour poly-tope, Qn, whose vertices are the characteristic vectors of Hamiltonian cycles, and given a point from a polytope, P containing Qn, it tries to find a separating hyperplane that separates the point and Qn or shows that the point is in Qn. In this paper we define a combinatorial object called pedigree, and study an alternative polytope, called Pedigree polytope. Pedigrees are in one-to-one correspondence with the Hamiltonian cycles on Kn. Interestingly the pedigree polytope seems to differ from Qn with respect to the complexity of testing whether two given vertices of the polytope are non-adjacent. A polynomial time algorithm is given for non-adjacency testing in pedigree polytope, whereas the corresponding problem is known to be NP-complete for Qn. In this paper we discuss some properties of the pedigree polytope and illustrate with examples.

SU(3)×𝒮20 algebras for uniform spin‐1 ensembles on [2H12C]20, or [14N]20, dodecahedrane‐type lattices and analogous isotopomeric [M2012C40] met‐carb subensembles: M‐based cardinalities and completeness of 𝒮20 spin irreps, via hierarchical {𝒞λ⊢(n=20):(M)} designs of polyhedral combinatorics*

AbstractThe M‐based hierarchy cardinalities of spin irreps for \documentclass{article}\pagestyle{empty}\begin{document}$[A]_{20}^{(I_{i}=1)}$\end{document} uniform nuclear magnetic resonance (NMR) /isotopomer spin ensembles are derived. Such ideas define the completeness of the number‐partition‐based (intermediate) combinatorial designs (on M) themselves inherent in specific λ⊢n digit assembly combinatorial properties. Illustrative M‐subspatial irrep subsets are derived via Schur decompositions from symbolic algorithms (Sagan, B. E. Symmetric Group: Representations, C‐Algorithms, Symmetric Functions; Wadsworth: Belmont, CA, 1991; SYMMETRICA package, as per Kerber, A.; Kohnert, A.; Lascoux, A. Symbolic Comput 1992, 14, 195). The results are discussed in the context of the independant cardinality of underlying system scalar invariants (SI) corresponding to the democratic auxiliary labels (Chem Phys 1998, 238, 245; J Math Chem in press), or projective recoupling, of 20‐fold dual tensorial sets. Landau‐like 𝒮n‐maps for fundamental terms plus statistical weighted subsidary maps yield the independent ∣SI∣s (see Europhys Lett in press). Geometric aspects of the dual group mappings imply that eventually large cage NMR ensembles must be governed by a local, rather than a global, symmetry, with the former related to established spectral deceptive NMR. This suggests a further role for dynamical networks in NMR, beyond that given by K. Balasubramanian (J Chem Phys 1983, 78, 6369) as implied by D. Watts (Small Worlds; Princeton Univ. Press: Princeton, NJ, 1999). © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002

Ideal clutters

The Operations Research model known as the Set Covering Problem has a wide range of applications. See for example the survey by Ceria, Nobili and Sassano and edited by Dell'Amico, Maffioli and Martello (Annotated Bibliographies in Combinatorial Optimization, Wiley, New York, 1997). Sometimes, due to the special structure of the constraint matrix, the natural linear programming relaxation yields an optimal solution that is integer, thus solving the problem. Under which conditions do such integrality properties hold? This question is of both theoretical and practical interest. On the theoretical side, polyhedral combinatorics and graph theory come together in this rich area of discrete mathematics. In this tutorial, we present the state of the art and open problems on this question.

The QAP-polytope and the star transformation

Polyhedral Combinatorics has been successfully applied to obtain considerable algorithmic progress towards the solution of many prominent hard combinatorial optimization problems. Until a few years ago, the quadratic assignment problem (QAP) was one of the exceptions. The work of Padberg and Rijal (Location, Scheduling, Design and Integer Programming, Kluwer Academic Publishers, Dordrecht, 1996; Rijal, Ph.D. Thesis, New York University, 1995) has on the one hand yielded some basic facts about the associated quadratic assignment polytope, but has on the other hand shown that investigations even of the very basic questions (like the dimension, the affine hull, and the trivial facets) soon become extremely complicated. In this paper, we propose an isomorphic transformation of the “natural” realization of the quadratic assignment polytope, which simplifies the polyhedral investigations enormously. We demonstrate this by giving short proofs of the basic results on the polytope that indicate that, exploiting the techniques developed in this paper, deeper polyhedral investigations of the QAP now become possible.

Branch-and-cut for combinatorial optimization problems without auxiliary binary variables

Many optimisation problems involve combinatorial constraints on continuous variables. An example of a combinatorial constraint is that at most one variable in a group of nonnegative variables may be positive. Traditionally, in the mathematical programming community, such problems have been modeled as mixed-integer programs by introducing auxiliary binary variables and additional constraints. Because the number of variables and constraints becomes larger and the combinatorial structure is not used to advantage, these mixed-integer programming models may not be solved satisfactorily, except for small instances. Traditionally, constraint programming approaches to such problems keep and use the combinatorial structure, but do not use linear programming bounds in the search for an optimal solution. Here we present a branch-and-cut approach that considers the combinatorial constraints without the introduction of binary variables. We review the development of this approach and show how strong constraints can be derived using ideas from polyhedral combinatorics. To illustrate the ideas, we present a production scheduling model that arises in the manufacture of fibre optic cables.

Probabilistic satisfiability with imprecise probabilities

Treatment of imprecise probabilities within the probabilistic satisfiability approach to uncertainty in knowledge-based systems is surveyed and discussed. Both probability intervals and qualitative probabilities are considered. Analytical and numerical methods to test coherence and bound the probability of a conclusion are reviewed. They use polyhedral combinatorics and advanced methods of linear programming.

Geometrical Techniques for Estimating Numbers of Linear Extensions

LetPbe a two-dimensional order, and __Pany complement ofP, i.e., any partial order whose comparability graph is the complement of the comparability graph ofP. Lete(Q) denote the number of linear extensions of the partial orderQ. Sidorenko showed thate(P)e(__P) ≥n!, for any two-dimensional partial orderP. In this note, we use results from polyhedral combinatorics, and from the geometry ofRn, to give a companion upper bound one(P)e(__P), as well as an alternative proof of the lower bound. We use these results to obtain bounds on the number of linear extensions of a random two-dimensional partial order.

Polyhedral aspects of partial orders and comparability graphs

Abstract Polyhedral combinatorics studies convex polytopes defined as convex hulls of certain combinatorial sets embedded in Rd. Important issues in the study of such polytopes are the determination of the facets, edges and automorphism group. We consider all partial orders on a given set of size n, represented in Rn(n−1) by means of their characteristic vectors. The associated convex hull is the partial order polytope. A remarkable feature is that axioms for partial orders define facets of the partial order polytope. Unfortunately, most facets have no such nice interpretation. Finding them all turns out to be quite hard. With the help of the computer program porta, we obtained all facets of the partial order polytope up to n = 4, and a partial list of facets for n = 5. This list contains a total of 43244 facets and we were able to prove it is incomplete thanks to a personal algorithm testing whether a list of facets of a polytope contains all facets. For all n, the automorphism group of the partial order polytope is entirely determined and some edges are characterized. The results on the edges give rise to a conjecture stating a strikingly simple criterion for adjacency. A simple, undirected graph is a comparability graph if its edges can be transitively oriented, thus producing a partial order. Using characteristic vectors in Rn(n−1)/2, we encode all comparability graphs on a given set of size n. The comparability graph polytope is the convex hull of these vectors. The latter appears as the image of the partial order polytope under a canonical projection which maps the characteristic vector of a partial order to the characteristic vector of its comparability graph. This projection can be exploited to infer new facets of the partial order polytope from facets of the comparability graph polytope. Some of these facets, that we call lifted odd cycle facets, are literal translations of the characterization of comparability graphs due to Ghouila-Houri in 1962 (independently obtained by Gilmore and Hoffman in 1964).

An exact solution for the segment-to-segment multiple sequence alignment problem.

In molecular biology, sequence alignment is a crucial tool in studying the structure and function of molecules, as well as the evolution of species. In the segment-to-segment variation of the multiple alignment problem, the input can be seen as a set of non-gapped segment pairs (diagonals). Given a weight function that assigns a weight score to every possible diagonal, the goal is to choose a consistent set of diagonals of maximum weight. We show that the segment-to-segment multiple alignment problem is equivalent to a novel formulation of the Maximum Trace problem: the Generalized Maximum Trace (GMT) problem. Solving this problem to optimality, therefore, may improve upon the previous greedy strategies that are used for solving the segment-to-segment multiple sequence alignment problem. We show that the GMT can be stated in terms of an integer linear program and then solve the integer linear program using methods from polyhedral combinatorics. This leads to a branch-and-cut algorithm for segment-to-segment multiple sequence alignment. We report on our first computational experiences with this novel method and show that the program is able to find optimal solutions for real-world test examples.

Annotated Bibliographies in Combinatorial Optimization.

Part I: General methodologies complexity and approximability polyhedral combinatorics branch-and-cut algorithms matroids and submodular functions advances in linear programming decomposition and column generation stochastic integer programming randomized algorithms local search graphs and matrices. Part II: Specific topics and applications sequencing and scheduling Travelling Salesman Problem max cut location problems network design flows and paths quadratic and 3-dimensional assignments linear assignment vehicle routing cutting and packing combinatorial topics in VLSI design applications in computational biology.

Construction of convex continuations for functions defined on a hypersphere

Construction of convex continuations for functions defined on the vertices of some combinatorial polyhedra, in particular the permutation polyhedron and the arrangement polyhedron, has been studied in [1, 2]. Subsequently this result has been generalized to functions defined at the extreme points of an arbitrary polyhedron [3]. For purposes of combinatorial optimization [4-6] it is relevant to consider the existence and construction of convex continuations from continua, in particular, when the function is defined on a hypersphere in thek-dimensional space. Unfortunately, passage to the limit from discrete sets to continua does not produce positive results in this case. We are thus forced to develop special approaches to investigating the existence of convex continuations of functions defined on continua.

A polyhedral approach to RNA sequence structure alignment.

Ribonucleic acid (RNA) is a polymer composed of four bases denoted A, C, G, and U. It generally is a single-stranded molecule where the bases form hydrogen bonds within the same molecule leading to structure formation. In comparing different homologous RNA molecules it is important to consider both the base sequence and the structure of the molecules. Traditional alignment algorithms can only account for the sequence of bases, but not for the base pairings. Considering the structure leads to significant computational problems because of the dependencies introduced by the base pairings. In this paper we address the problem of optimally aligning a given RNA sequence of unknown structure to one of known sequence and structure. We phrase the problem as an integer linear program and then solve it using methods from polyhedral combinatorics. In our computational experiments we could solve large problem instances--23S ribosomal RNA with more than 1400 bases--a size intractable for former algorithms.

Cuts, matrix completions and graph rigidity

This paper brings together several topics arising in distinct areas: polyhedral combinatorics, in particular, cut and metric polyhedra; matrix theory and semidefinite programming, in particular, completion problems for positive semidefinite matrices and Euclidean distance matrices; distance geometry and structural topology, in particular, graph realization and rigidity problems. Cuts and metrics provide the unifying theme. Indeed, cuts can be encoded as positive semidefinite matrices (this fact underlies the approximative algorithm for max-cut of Goemans and Williamson) and both positive semidefinite and Euclidean distance matrices yield points of the cut polytope or cone, after applying the functions 1/π arccos(.) or √. When fixing the dimension in the Euclidean distance matrix completion problem, we find the graph realization problem and the related question of unicity of realization, which leads to the question of graph rigidity. Our main objective here is to present in a unified setting a number of results and questions concerning matrix completion, graph realization and rigidity problems. These problems contain indeed very interesting questions relevant to mathematical programming and we believe that research in this area could yield to cross-fertilization between the various fields involved.