Scheduling Problems and Mixed Graph Colorings

Some scheduling problems induce a mixed graph coloring, i.e., an assignment of positive integers (colors) to vertices of a mixed graph such that, if two vertices are joined by an edge, then their colors have to be different, and if two vertices are joined by an arc, then the color of the startvertex has to be not greater than the color of the endvertex. We discuss some algorithms for coloring the vertices of a mixed graph with a small number t of colors and present computational results for calculating the chromatic number, i.e., the minimal possible value of such a t . We also study the chromatic polynomial of a mixed graph which may be used for calculating the number of feasible schedules.     

Stability of two player game structures

We have extended a two player game-theoretical model proposed by V. Gurvich [To theory of multi-step games, USSR Comput. Math and Math. Phys. 13 (1973)] and H. Moulin [The Strategy of Social Choice, North Holland, Amsterdam, 1983]: All the considered game situations are framed by the same game structure. The structure determines the families of potential decisions of the two players, as well as the subsets of possible outcomes allowed by pairs of such choices. To be a solution of a game, a pair of decisions has to determine a (pure) functional equilibrium of the situational pair of payoff mappings which transforms the realized outcome into real-valued rewards of the players. Accordingly we understand that a structure is stable, if it admits functional equilibria for all possible game situations; and that it is complete, if every situation that only partitions the potential outcomes, is dominated by one of the players. We have generalized and strengthened a theorem by V. Gurvich [Equilibrium in pure strategies, Soviet Math. Dokl. 38 (1989)], proving that a proper structure is stable iff it is complete. Additional results provide game-theoretical insight that focuses the inquiry on the complexity of the stability decision problem; in particular, for coherent structures.These results also have combinatorial importance because every structure is characterized by a pair of hypergraphs [C. Berge, Graphes et Hypergraphes, Dunod, 1970] over a common ground set. The structure is dual (complete/coherent) iff the clutter of one hypergraph equals (includes/is included in) the blocker of the other one. So, for non-void coherent structures, the stability decision problem is equivalent to the much studied subexponential [M.L. Fredman, L. Khachiyan, On the complexity of dualization of monotone disjunctive normal forms, J. Algorithms 21 (1996)] hypergraph duality decision problem.     

On realizations of point determining graphs, and obstructions to full homomorphisms

A graph is point determining if distinct vertices have distinct neighbourhoods. A realization of a point determining graph H is a point determining graph G such that each vertex-removed subgraph G-x which is point determining, is isomorphic to H. We study the fine structure of point determining graphs, and conclude that every point determining graph has at most two realizations.A full homomorphism of a graph G to a graph H is a vertex mapping f such that for distinct vertices u and v of G, we have uv an edge of G if and only if f(u)f(v) is an edge of H. For a fixed graph H, a full H-colouring of G is a full homomorphism of G to H. A minimal H-obstruction is a graph G which does not admit a full H-colouring, such that each proper induced subgraph of G admits a full H-colouring. We analyse minimal H-obstructions using our results on point determining graphs. We connect the two problems by proving that if H has k vertices, then a graph with k+1 vertices is a minimal H-obstruction if and only if it is a realization of H. We conclude that every minimal H-obstruction has at most k+1 vertices, and there are at most two minimal H-obstructions with k+1 vertices.We also consider full homomorphisms to graphs H in which loops are allowed. If H has ? loops and k vertices without loops, then every minimal H-obstruction has at most (k+1)(?+1) vertices, and, when both k and ? are positive, there is at most one minimal H-obstruction with (k+1)(?+1) vertices.In particular, this yields a finite forbidden subgraph characterization of full H-colourability, for any graph H with loops allowed.     

Exploiting equalities in polynomial programming

We propose a novel approach for solving polynomial programs over compact domains with equality constraints. By means of a generic transformation, we show that existing solution schemes for the, typically simpler, problem without equalities can be used to address the problem with equalities.     

Polynomial integrability of the Hamiltonian systems with homogeneous potential of degree − 3

In this paper, we study the polynomial integrability of natural Hamiltonian systems with two degrees of freedom having a homogeneous potential of degree k given either by a polynomial, or by an inverse of a polynomial. For k=−2,−1,…,3,4, their polynomial integrability has been characterized. Here, we have two main results. First, we characterize the polynomial integrability of those Hamiltonian systems with homogeneous potential of degree −3. Second, we extend a relation between the nontrivial eigenvalues of the Hessian of the potential calculated at a Darboux point to a family of Hamiltonian systems with potentials given by an inverse of a homogeneous polynomial. This relation was known for such Hamiltonian systems with homogeneous polynomial potentials. Finally, we present three open problems related with the polynomial integrability of Hamiltonian systems with a rational potential.     

Quasi-likelihood analysis for nonsynchronously observed diffusion processes

We consider nonsynchronous sampling of parameterized stochastic regression models, which contain stochastic differential equations. Constructing a quasi-likelihood function, we prove that the quasi-maximum likelihood estimator and the Bayes type estimator are consistent and asymptotically mixed normal when the sampling frequency of the nonsynchronous data becomes large.     

On the complexity of project scheduling to minimize exposed time

We consider project scheduling where the project manager’s objective is to minimize the time from when an adversary discovers the project until the completion of the project. We analyze the complexity of the problem identifying both polynomially solvable and NP-hard versions of the problem. The complexity of the problem is seen to be dependent on the nature of renewable resource constraints, precedence constraints, and the ability to crash activities in the project.     

Polynomial cases of the economic lot sizing problem with cost discounts

In this paper we study the economic lot sizing problem with cost discounts. In the economic lot sizing problem a facility faces known demands over a discrete finite horizon. At each period, the ordering cost function and the holding cost function are given and they can be different from period to period. There are no constraints on the quantity ordered in each period and backlogging is not allowed. The objective is to decide when and how much to order so as to minimize the total ordering and holding costs over the finite horizon without any shortages. We study two different cost discount functions. The modified all-unit discount cost function alternates increasing and flat sections, starting with a flat section that indicates a minimum charge for small quantities. While in general the economic lot sizing problem with modified all-unit discount cost function is known to be NP-hard, we assume that the cost functions do not vary from period to period and identify a polynomial case. Then we study the incremental discount cost function which is an increasing piecewise linear function with no flat sections. The efficiency of the solution algorithms follows from properties of the optimal solution. We computationally test the polynomial algorithms against the use of CPLEX.     

交换环上多项式结式的性质

本文主要讨论交换环上多项式结式的一些性质.首先,我们证明了交换环上一种乘积的结式等于结式的乘积的性质,然后,我们证明了交换环上一种和的结式具有的性质,并且给出了交换环上结式为零的一个充分条件.