Primal separation for 0/1 polytopes

 The 0/1 primal separation problem is: Given an extreme point xˉ of a 0/1 polytope P and some point x ^*, find an inequality which is tight at xˉ, violated by x ^* and valid for P or assert that no such inequality exists. It is known that this separation variant can be reduced to the standard separation problem for P. We show that 0/1 optimization and 0/1 primal separation are polynomial time equivalent. This implies that the problems 0/1 optimization, 0/1 standard separation, 0/1 augmentation, and 0/1 primal separation are polynomial time equivalent. Then we provide polynomial time primal separation procedures for matching, stable set, maximum cut, and maximum bipartite graph problems, giving evidence that these algorithms are conceptually simpler and easier to implement than their corresponding counterparts for standard separation. In particular, for perfect matching we present an algorithm for primal separation that rests only on simple max-flow computations. In contrast, the known standard separation method relies on an explicit minimum odd cut algorithm. Consequently, we obtain a very simple proof that a maximum weight perfect matching of a graph can be computed in polynomial time. Received: August 20, 2001 / Accepted: April 2002 Published online: December 9, 2002 RID="⋆" ID="⋆" This research was developed while the author was on leave at the Istituto di Analisi dei Sistemi ed Informatica, Viale Manzoni 30, 00185 Roma, supported by the project TMR-DONET nr. ERB FMRX-CT98-0202 of the European Union. Mathematics Subject Classification (2000): 90C10, 90C60, 90C57     

Computing low-capacity 0–1 knapsack polytopes

A procedure is proposed that, given a linear inequalityL having positive integral coefficients in 0–1 variables, finds all the facets of the convex hull of the solutions ofL. This is done for allL by doing once and for all the computations for a particular sequenceM ₁,M ₂,... of such linear inequalities, called master knapsack problems. To find the facets for a givenL, it is enough to know the facets for the master knapsack problemM _b, whereb is the free term inL (no matter how many variablesL might have). ThisM _b has no more than order ofb logb variables. The procedure is based on polarity. All the facets forM ₁,...,M ₇ are tabulated.

---

Zusammenfassung Es wird ein Verfahren vorgestellt, das für eine lineare UngleichungL in binären Variablen mit positiven ganzzahligen Koeffizienten alle Facetten der konvexen Hülle der Lösungen vonL bestimmt. Um diese Facetten für irgendeine UngleichungL mit rechter Seiteb zu erhalten, braucht man nur die Facetten des sogenannten Master-Knapsack-ProblemsM _b zu kennen, dasO (b log b) Variablen besitzt. Fürb=1,..., 7 werden alle Facetten vonM _b angegeben.

     

Separation algorithms for 0-1 knapsack polytopes

Valid inequalities for 0-1 knapsack polytopes often prove useful when tackling hard 0-1 Linear Programming problems. To generate such inequalities, one needs separation algorithms for them, i.e., routines for detecting when they are violated. We present new exact and heuristic separation algorithms for several classes of inequalities, namely lifted cover, extended cover, weight and lifted pack inequalities. Moreover, we show how to improve a recent separation algorithm for the 0-1 knapsack polytope itself. Extensive computational results, on MIPLIB and OR Library instances, show the strengths and limitations of the inequalities and algorithms considered.     

Integer-empty polytopes in the 0/1-cube with maximal Gomory–Chvátal rank

We provide a complete characterization of all polytopes P⊆[0,1]ⁿ with empty integer hulls, whose Gomory–Chvátal rank is n (and, therefore, maximal). In particular, we show that the first Gomory–Chvátal closure of all these polytopes is identical.     

Approximate formulations for 0-1 knapsack sets

We show that for each 0<?≤1 there exists an extended formulation for the knapsack problem, of size polynomial in the number of variables, whose value is at most (1+?) times the value of the integer program.     

A G/M/1-queue with exponential retrial

Summary We consider a G/M/1 retrial model in which the delays between retrials are i.i.d. exponentially distributed random variables. We investigate the steady-state distribution of the embedded Markov chain at completion service epochs, the stationary distribution at anytime and the virtual waiting time.     

The GI/M/1 queue with exponential vacations

In this paper, we give a detailed analysis of the GI/M/1 queue with exhaustive service and multiple exponential vacation. We express the transition matrix of the imbedded Markov chain as a block-Jacobi form and give a matrix-geometric solution. The probability distribution of the queue length at arrival epochs is derived and is shown to decompose into the distribution of the sum of two independent random variables. In addition, we discuss the limiting behavior of the continuous time queue length processes and obtain the probability distributions for the waiting time and the busy period.     

Revlex-initial 0/1-polytopes

We introduce revlex-initial 0/1-polytopes as the convex hulls of reverse-lexicographically initial subsets of 0/1-vectors. These polytopes are special knapsack-polytopes. It turns out that they have remarkable extremal properties. In particular, we use these polytopes in order to prove that the minimum numbers g_nfac(d,n) of facets and the minimum average degree g_avdeg(d,n) of the graph of a d-dimensional 0/1-polytope with n vertices satisfy g_nfac(d,n)?3d and g_avdeg(d,n)?d+4. We furthermore show that, despite the sparsity of their graphs, revlex-initial 0/1-polytopes satisfy a conjecture due to Mihail and Vazirani, claiming that the graphs of 0/1-polytopes have edge-expansion at least one.     

Improved 0/1-interchange scheduling

Consideration is given to the problem of nonpreemptively scheduling a set ofN independent tasks to a system ofM identical processors, with the objective to minimize the overall finish time. It is proved that the 0/1-INTERCHANGE scheduling heuristic can be modified, without increasing its time complexity fromO(N logM), so that its worst-case performance bound is reduced from 2 to 4/3 times optimal.     

Busy period analysis for the GI/M/1 queue with exponential vacations

We consider the busy period in the GI/M/1 queue with multiple exponential vacations. We derive the transform of the joint distribution of the length of a busy period, the number of customers served during the busy period, and the residual interarrival time at the instant the busy period ends.     

Some properties of cliques in 0–1 mixed integer programs

Summary In this note we present new properties of cliques induced constraints straintsX(C _r ⁺ )-X(C _r ^- ) ≤ 1 - |C _r ^- | for λ εS, whereS is the set of cliques that are implied by 0–1 mixed integer programs. These properties allow to further fixing of 0–1 variables, to detect instance's infeasibility and to imply new cliques.     

Extremal Properties of 0/1-Polytopes

We provide lower and upper bounds for the maximal number of facets of a d-dimensional 0/1-polytope, and for the maximal number of vertices that can appear in a two-dimensional projection (``shadow') of such a polytope. Received June 14, 1996, and in revised form September 30, 1996.     

On the 0/1 knapsack polytope

This paper deals with the 0/1 knapsack polytope. In particular, we introduce the class ofweight inequalities. This class of inequalities is needed to describe the knapsack polyhedron when the weights of the items lie in certain intervals. A generalization of weight inequalities yields the so-called “weight-reduction principle” and the class of extended weight inequalities. The latter class of inequalities includes minimal cover and (l,k)-configuration inequalities. The properties of lifted minimal cover inequalities are extended to this general class of inequalities.     

A generalized extended rational expansion method and its application to (1 + 1)-dimensional dispersive long wave equation

In this Letter, a generalized extended rational expansion method is used to construct exact solutions of the (1 + 1)-dimensional dispersive long wave equation. As a result, many new and more general exact solutions are obtained, the solutions obtained in this Letter include rational triangular periodic wave solutions, rational solitary wave solutions.     

Approximation of 1/x by exponential sums in [1, {infty})

^** Email: braess{at}num.rub.de^*** Email: wh{at}mis.mpg.de Approximations of 1/x by sums of exponentials are well studiedfor finite intervals. Here the error decreases like (exp(–ck))with the order k of the exponential sum. In this paper we investigateapproximations of 1/x in the interval [1, ). We prove estimatesof the error by and confirm this asymptotic estimate by numerical results. Numericalresults lead to the conjecture that the constant in the exponentequals .     

On stochastic decomposition in the GI/M/1 queue with single exponential vacation

We consider a GI/M/1 queueing system in which the server takes exactly one exponential vacation each time the system empties. We derive the PGF of the stationary queue length and the LST of the stationary FIFO sojourn time. We show that both the queue length and the sojourn time can be stochastically decomposed into meaningful quantities.     

Fluid model driven by an M/G/1 queue with multiple exponential vacations

This paper studies a fluid model driven by an M/G/1 queue with multiple exponential vacations. By introducing various vacation strategies to the fluid model, we can provide greater flexibility for the design and control of input rate and output rate. The Laplace transform of the steady-state distribution of the buffer content is expressed through the minimal positive solution to a crucial equation. Then the performance measure-mean buffer content, which is independent of the vacation parameter, is obtained. Finally, with some numerical examples, the parameter effect on the mean buffer content is presented.