Adjacency on polymatroids

This paper characterizes adjacency for extreme points of a polymatroid. Two extreme points of a polymatroid are adjacent if and only if they either differ in exactly one component or differ in exactly two components with the two components satisfying a certain ordering relation. A polynomial algorithm generates and lists all extreme points adjacent to a given extreme point of a polymatroid. Similar results hold for the core of a convex game.     

Finding lower bounds on the complexity of secret sharing schemes by linear programming

Optimizing the maximum, or average, length of the shares in relation to the length of the secret for every given access structure is a difficult and long-standing open problem in cryptology. Most of the known lower bounds on these parameters have been obtained by implicitly or explicitly using that every secret sharing scheme defines a polymatroid related to the access structure. The best bounds that can be obtained by this combinatorial method can be determined by using linear programming, and this can be effectively done for access structures on a small number of participants.By applying this linear programming approach, we improve some of the known lower bounds for the access structures on five participants and the graph access structures on six participants for which these parameters were still undetermined. Nevertheless, the lower bounds that are obtained by this combinatorial method are not tight in general. For some access structures, they can be improved by adding to the linear program non-Shannon information inequalities as new constraints. We obtain in this way new separation results for some graph access structures on eight participants and for some ports of non-representable matroids. Finally, we prove that, for two access structures on five participants, the combinatorial lower bound cannot be attained by any linear secret sharing scheme.     

Polymatroid greedoids

This paper discusses polymatroid greedoids, a superclass of them, called local poset greedoids, and their relations to other subclasses of greedoids. Polymatroid greedoids combine in a certain sense the different relaxation concepts of matroids as polymatroids and as greedoids. Some characterization results are given especially for local poset greedoids via excluded minors. General construction principles for intersection of matroids and polymatroid greedoids with shelling structures are given. Furthermore, relations among many subclasses of greedoids which are known so far, are demonstrated.     

Optimal complexity of secret sharing schemes with four minimal qualified subsets

The complexity of a secret sharing scheme is defined as the ratio between the maximum length of the shares and the length of the secret. This paper deals with the open problem of optimizing this parameter for secret sharing schemes with general access structures. Specifically, our objective is to determine the optimal complexity of the access structures with exactly four minimal qualified subsets. Lower bounds on the optimal complexity are obtained by using the known polymatroid technique in combination with linear programming. Upper bounds are derived from decomposition constructions of linear secret sharing schemes. In this way, the exact value of the optimal complexity is determined for several access structures in that family. For the other ones, we present the best known lower and upper bounds.     

Monotonicity of polymatroids

Unlike matroids, it is observed that there is an upper bound for the numberk such that the polymatroid isk-monotone. We define this upper bound as the monotonicity of polymatroid. We aim to calculate the value of this quantity in terms of the ground rank function of the polymatroid.     

Matroid matching and some applications

The polymatroid matching problem, also known as the matchoid problem or the matroid parity problem, is polynomially unsolvable in general but solvable for linear matroids. The solution for linear matroids is analysed and results concerning arbitrary matroids are given from which the linear case follows immediately. The same general result is then applied to find a maximum circuitfree partial hypergraph of a 3-uniform hypergraph, to generalize a theorem of Mader on packing openly disjoint paths starting and ending in a given set, and to study a problem in structural rigidity.     

Worst-case and average-case analysis of an algorithm solving a generalized knapsack problem

The paper deals with a discontinuous linear knapsack problem of minimizing a linear objective function on a certain nonconnected subset of a polymatroid. The general problem is NP-hard but some practically interesting subclasses are solvable by a polynomial-time algorithm. Bounds of the expense of the solution process are given depending only on the subclass considered. The average-case behaviour is considered, too.     

An intersection theorem for supermatroids

We generalize the matroid intersection theorem to distributive supermatroids, a structure that extends the matroid to the partially ordered ground set. Distributive supermatroids are special cases of both supermatroids and greedoids, and they generalize polymatroids. This is the first good characterization proved for the intersection problem of an independence system where the ground set is partially ordered. The characterization given has a more complex structure than the matroid (or polymatroid) intersection theorem.     

Poly-symmetry in processor-sharing systems

We consider a system of processor-sharing queues with state-dependent service rates. These are allocated according to balanced fairness within a polymatroid capacity set. Balanced fairness is known to be both insensitive and Pareto-efficient in such systems, which ensures that the performance metrics, when computable, will provide robust insights into the real performance of the system considered. We first show that these performance metrics can be evaluated with a complexity that is polynomial in the system size if the system is partitioned into a finite number of parts, so that queues are exchangeable within each part and asymmetric across different parts. This in turn allows us to derive stochastic bounds for a larger class of systems which satisfy less restrictive symmetry assumptions. These results are applied to practical examples of tree data networks, such as backhaul networks of Internet service providers, and computer clusters.     

Performance Bounds with Curvature for Batched Greedy Optimization

The batched greedy strategy is an approximation algorithm to maximize a set function subject to a matroid constraint. Starting with the empty set, the batched greedy strategy iteratively adds to the current solution set a batch of elements that results in the largest gain in the objective function while satisfying the matroid constraints. In this paper, we develop bounds on the performance of the batched greedy strategy relative to the optimal strategy in terms of a parameter called the total batched curvature. We show that when the objective function is a polymatroid set function, the batched greedy strategy satisfies a harmonic bound for a general matroid constraint and an exponential bound for a uniform matroid constraint, both in terms of the total batched curvature. We also study the behavior of the bounds as functions of the batch size. Specifically, we prove that the harmonic bound for a general matroid is nondecreasing in the batch size and the exponential bound for a uniform matroid is nondecreasing in the batch size under the condition that the batch size divides the rank of the uniform matroid. Finally, we illustrate our results by considering a task scheduling problem and an adaptive sensing problem.     

The parity problem of polymatroids without double circuits

According to the present state of the theory of the matroid parity problem, the existence of a good characterization to the size of a maximum matching depends on the behavior of certain substructures, called double circuits. In this paper we prove that if a polymatroid has no double circuits then a partition type min-max formula characterizes the size of a maximum matching. Applications to parity constrained orientations and to a rigidity problem are given. Research is supported by OTKA grants K60802, TS049788 and by European MCRTN Adonet, Contract Grant No. 504438.     

Decomposition of binary matroids

We prove results relating to the decomposition of a binary matroid, including its uniqueness when the matroid is cosimple. We extend the idea of “freedom” of an element in a matroid to “freedom” of a set, and show that there is a unique maximal integer polymatroid inducing a given binary matroid.     

Characterizing and recognizing generalized polymatroids

Generalized polymatroids are a family of polyhedra with several nice properties and applications. One property of generalized polymatroids used widely in existing literature is “total dual laminarity;” we make this notion explicit and show that only generalized polymatroids have this property. Using this we give a polynomial-time algorithm to check whether a given linear program defines a generalized polymatroid, and whether it is integral if so. Additionally, whereas it is known that the intersection of two integral generalized polymatroids is integral, we show that no larger class of polyhedra satisfies this property.     

A geometric theory of hypergraph colouring

Summary Ak-colouring of a hypergraph is an assignment of no more thank colours to the vertices of the hypergraph in such a way that the coloured hypergraph has no monochromatic edges. In this paper a polymatroid is associated with a hypergraph. It is shown that the number ofk-colourings of the hypergraph is determined by an evaluation of the characteristic polynomial of this polymatroid. Hypergraph colouring is then related to an extension of the critical problem developed by the author. It is shown that, ifq is a prime power, then a hypergraph isq ^k -colourable if and only if the critical exponent overGF(q) of its associated polymatroid is less than or equal tok.     

A Lagrangian relax-and-cut approach for the sequential ordering problem with precedence relationships

The sequential ordering problem with precedence relationships was introduced in Escudero [7]. It has a broad range of applications, mainly in production planning for manufacturing systems. The problem consists of finding a minimum weight Hamiltonian path on a directed graph with weights on the arcs, subject to precedence relationships among nodes. Nodes represent jobs (to be processed on a single machine), arcs represent sequencing of the jobs, and the weights are sums of processing and setup times. We introduce a formulation for the constrained minimum weight Hamiltonian path problem. We also define Lagrangian relaxation for obtaining strong lower bounds on the makespan, and valid cuts for further tightening of the lower bounds. Computational experience is given for real-life cases already reported in the literature.     

A new exact algorithm for the vehicle routing problem based onq-paths andk-shortest paths relaxations

We consider the basic Vehicle Routing Problem (VRP) in which a fleet ofM identical vehicles stationed at a central depot is to be optimally routed to supply customers with known demands subject only to vehicle capacity constraints. In this paper, we present an exact algorithm for solving the VRP that uses lower bounds obtained from a combination of two relaxations of the original problem which are based on the computation ofq-paths andk-shortest paths. A set of reduction tests derived from the computation of these bounds is applied to reduce the size of the problem and to improve the quality of the bounds. The resulting lower bounds are then embedded into a tree-search procedure to solve the problem optimally. Computational results are presented for a number of problems taken from the literature. The results demonstrate the effectiveness of the proposed method in solving problems involving up to about 50 customers and in providing tight lower bounds for problems up to about 150 customers.     

Distributionally robust stochastic shortest path problem

This paper considers a stochastic version of the shortest path problem, the Distributionally Robust Stochastic Shortest Path Problem(DRSSPP) on directed graphs. In this model, the arc costs are deterministic, while each arc has a random delay. The mean vector and the second-moment matrix of the uncertain data are assumed known, but the exact information of the distribution is unknown. A penalty occurs when the given delay constraint is not satisfied. The objective is to minimize the sum of the path cost and the expected path delay penalty. As it is NP-hard, we approximate the DRSSPP with a semidefinite programming (SDP for short) problem, which is solvable in polynomial time and provides tight lower bounds.     

Adhesivity of polymatroids

Two polymatroids are adhesive if a polymatroid extends both in such a way that two ground sets become a modular pair. Motivated by entropy functions, the class of polymatroids with adhesive restrictions and a class of selfadhesive polymatroids are introduced and studied. Adhesivity is described by polyhedral cones of rank functions and defining inequalities of the cones are identified, among them known and new non-Shannon type information inequalities for entropy functions. The selfadhesive polymatroids on a four-element set are characterized by Zhang-Yeung inequalities.     

A Greedy Algorithm for Capacitated Lot-Sizing Problems

We show that the convex hull of the set of feasible solutions of single-item capacitated lot-sizing problem (CLSP) is a base polyhedron of a polymatroid. We present a greedy algorithm to solve CLSP with linear objective function. The proposed algorithm is an effective implementation of the classical Edmonds' algorithm for maximizing linear function over a polymatroid. We consider some special cases of CLSP with nonlinear objective function that can be solved by the proposed greedy algorithm in O ( n ) time.     

Fractal and smoothness properties of space-time Gaussian models

Spatio-temporal models are widely used for inference in statistics and many applied areas. In such contexts, interests are often in the fractal nature of the sample surfaces and in the rate of change of the spatial surface at a given location in a given direction. In this paper, we apply the theory of Yaglom (1957) to construct a large class of space-time Gaussian models with stationary increments, establish bounds on the prediction errors, and determine the smoothness properties and fractal properties of this class of Gaussian models. Our results can be applied directly to analyze the stationary spacetime models introduced by Cressie and Huang (1999), Gneiting (2002), and Stein (2005), respectively.