Primitive Zonotopes

We introduce and study a family of polytopes which can be seen as a generalization of the permutahedron of type $B_d$. We highlight connections with the largest possible diameter of the convex hull of a set of points in dimension $d$ whose coordinates are integers between $0$ and $k$, and with the computational complexity of multicriteria matroid optimization.Comment: The title was slightly modified, and the determination of the computational complexity of multicriteria matroid optimization was adde

Convex Integer Optimization by Constantly Many Linear Counterparts

In this article we study convex integer maximization problems with composite objective functions of the form $f(Wx)$, where $f$ is a convex function on $\R^d$ and $W$ is a $d\times n$ matrix with small or binary entries, over finite sets $S\subset \Z^n$ of integer points presented by an oracle or by linear inequalities. Continuing the line of research advanced by Uri Rothblum and his colleagues on edge-directions, we introduce here the notion of {\em edge complexity} of $S$, and use it to establish polynomial and constant upper bounds on the number of vertices of the projection \conv(WS) and on the number of linear optimization counterparts needed to solve the above convex problem. Two typical consequences are the following. First, for any $d$, there is a constant $m(d)$ such that the maximum number of vertices of the projection of any matroid $S\subset\{0,1\}^n$ by any binary $d\times n$ matrix $W$ is $m(d)$ regardless of $n$ and $S$; and the convex matroid problem reduces to $m(d)$ greedily solvable linear counterparts. In particular, $m(2)=8$. Second, for any $d,l,m$, there is a constant $t(d;l,m)$ such that the maximum number of vertices of the projection of any three-index $l\times m\times n$ transportation polytope for any $n$ by any binary $d\times(l\times m\times n)$ matrix $W$ is $t(d;l,m)$; and the convex three-index transportation problem reduces to $t(d;l,m)$ linear counterparts solvable in polynomial time

Approximate Tradeoffs on Matroids

International audienceWe consider problems where a solution is evaluated with a couple. Each coordinate of this couple represents an agent’s utility. Due to the possible conflicts, it is unlikely that one feasible solution is optimal for both agents. Then, a natural aim is to find tradeoffs. We investigate tradeoff solutions with guarantees for the agents.The focus is on discrete problems having a matroid structure. We provide polynomial-time deterministic algorithms which achieve several guarantees and we prove that some guarantees are not possible to reach

Euler Polytopes and Convex Matroid Optimization

Del Pia and Michini recently improved the upper bound of kd due to Kleinschmidt and Onn for the largest possible diameter of the convex hull of a set of points in dimension d whose coordinates are integers between 0 and k. We introduce Euler polytopes which include a family of lattice polytopes with diameter (k + 1)d/2, and thus reduce the gap between the lower and upper bounds. In addition, we highlight connections between Euler polytopes and a parameter studied in convex matroid optimization and strengthen the lower and upper bounds for this parameter

Combinatorial optimization under ellipsoidal uncertainty

We study combinatorial problems with ellipsoidal uncertainty in the objective function concerning their theoretical and practical solvability. Ellipsoidal uncertainty is a natural model when the coefficients are normally distributed random variables. Robust versions of typical combinatorial problems can be very hard to solve compared to their linear versions. Complexity and approaches differ fundamentally depending on whether uncorrelated or correlated uncertainty occurs. We distinguish between these two cases and consider first the unconstrained binary optimization under uncorrelated ellipsoidal uncertainty. For this we develop an algorithm which computes an optimal solution by merely sorting the variables and, correspondingly, has a running time of O(n log n). The algorithm is based on the diminishing returns-property, which is characteristic for submodular functions. We introduce a new and a more general p-norm-uncertainty and show that with only slight modifications the sorting algorithm can be easily applied. We also extend the algorithm to general integer variables, which in this case only leads to a pseudo-polynomial time. The next step to the general case is investigation of problems with arbitrary combinatorial sets X ⊆ {0, 1}n under uncorrelated ellipsoidal uncertainty. For this case we embed the O(n log n)-algorithm for the unconstrained binary problems into a Lagrangean decomposition approach. The approach separates the objective function from the combinatorial structure applying Lagrangean relaxation to some artificial connecting constraints. This creates two subproblems, one of which is the linear version of the combinatorial problem and the other one is just the unconstrained binary uncorrelated problem, which can be solved using the O(n log n)-algorithm. The solutions of the subproblems are used to obtain primal and dual bounds which are used in a branch and bound-approach. The approach shows an excellent performance in practice. In the correlated case already the unconstrained binary problem turns out to be strongly NP-hard. Here we also define a branch and bound-approach, now with lower bounds determined by underestimation of the given ellipsoid with certainly defined axis-parallel ellipsoids. We use this idea to extend the decomposition approach to general combinatorial problems under correlated uncertainty. In contrast to the uncorrelated case the uncertain subproblem of the decomposition is here strongly NP-hard in itself. We solve it approximately using the developed underestimators which are determined in a preprocessing step. The approach offers room for improvement concerning in the primal extent a faster computation of the underestimators, which is done by solving semidefinite programs

An Approximation Algorithm for the Exact Matching Problem in Bipartite Graphs

In 1982 Papadimitriou and Yannakakis introduced the Exact Matching problem, in which given a red and blue edge-colored graph G and an integer k one has to decide whether there exists a perfect matching in G with exactly k red edges. Even though a randomized polynomial-time algorithm for this problem was quickly found a few years later, it is still unknown today whether a deterministic polynomial-time algorithm exists. This makes the Exact Matching problem an important candidate to test the RP=P hypothesis. In this paper we focus on approximating Exact Matching. While there exists a simple algorithm that computes in deterministic polynomial-time an almost perfect matching with exactly k red edges, not a lot of work focuses on computing perfect matchings with almost k red edges. In fact such an algorithm for bipartite graphs running in deterministic polynomial-time was published only recently (STACS\u2723). It outputs a perfect matching with k\u27 red edges with the guarantee that 0.5k ? k\u27 ? 1.5k. In the present paper we aim at approximating the number of red edges without exceeding the limit of k red edges. We construct a deterministic polynomial-time algorithm, which on bipartite graphs computes a perfect matching with k\u27 red edges such that k/3 ? k\u27 ? k

Heuristic optimization of electrical energy systems: Refined metrics to compare the solutions

Many optimization problems admit a number of local optima, among which there is the global optimum. For these problems, various heuristic optimization methods have been proposed. Comparing the results of these solvers requires the definition of suitable metrics. In the electrical energy systems literature, simple metrics such as best value obtained, the mean value, the median or the standard deviation of the solutions are still used. However, the comparisons carried out with these metrics are rather weak, and on these bases a somehow uncontrolled proliferation of heuristic solvers is taking place. This paper addresses the overall issue of understanding the reasons of this proliferation, showing a conceptual scheme that indicates how the assessment of the best solver may result in the unlimited formulation of new solvers. Moreover, this paper shows how the use of more refined metrics defined to compare the optimization result, associated with the definition of appropriate benchmarks, may make the comparisons among the solvers more robust. The proposed metrics are based on the concept of first-order stochastic dominance and are defined for the cases in which: (i) the globally optimal solution can be found (for testing purposes); and (ii) the number of possible solutions is so large that practically it cannot be guaranteed that the global optimum has been found. Illustrative examples are provided for a typical problem in the electrical energy systems area – distribution network reconfiguration. The conceptual results obtained are generally valid to compare the results of other optimization problem