The positive semidefinite Grothendieck problem with rank constraint

Given a positive integer n and a positive semidefinite matrix A = (A_{ij}) of size m x m, the positive semidefinite Grothendieck problem with rank-n-constraint (SDP_n) is maximize \sum_{i=1}^m \sum_{j=1}^m A_{ij} x_i \cdot x_j, where x_1, ..., x_m \in S^{n-1}. In this paper we design a polynomial time approximation algorithm for SDP_n achieving an approximation ratio of \gamma(n) = \frac{2}{n}(\frac{\Gamma((n+1)/2)}{\Gamma(n/2)})^2 = 1 - \Theta(1/n). We show that under the assumption of the unique games conjecture the achieved approximation ratio is optimal: There is no polynomial time algorithm which approximates SDP_n with a ratio greater than \gamma(n). We improve the approximation ratio of the best known polynomial time algorithm for SDP_1 from 2/\pi to 2/(\pi\gamma(m)) = 2/\pi + \Theta(1/m), and we show a tighter approximation ratio for SDP_n when A is the Laplacian matrix of a graph with nonnegative edge weights.Comment: (v3) to appear in Proceedings of the 37th International Colloquium on Automata, Languages and Programming, 12 page

The 0-1 inverse maximum stable set problem

Given an instance of a weighted combinatorial optimization problem and its feasible solution, the usual inverse problem is to modify as little as possible (with respect to a fixed norm) the given weight system to make the giiven feasible solution optimal. We focus on its 0-1 version, which is to modify as little as possible the structure of the given instance so that the fixed solution becomes optimal in the new instance. In this paper, we consider the 0-1 inverse maximum stable set problem against a specific (optimal or not) algorithm, which is to delete as few vertices as possible so that the fixed stable set S* can be returned as a solution by the given algorithm in the new instance. Firstly, we study the hardness and approximation results of the 0-1 inverse maximum stable set problem against the algorithms. Greedy and 2-opt. Secondly, we identify classes of graphs for which the 0-1 inverse maximum stable set problem can be polynomially solvable. We prove the tractability of the problem for several classes of perfect graphs such as comparability graphs and chordal graphs.Combinatorial inverse optimization, maximum stable set problem, NP-hardness, performance ratio, perfect graphs.

Solving Vertex Cover in Polynomial Time on Hyperbolic Random Graphs

The VertexCover problem is proven to be computationally hard in different ways: It is NP-complete to find an optimal solution and even NP-hard to find an approximation with reasonable factors. In contrast, recent experiments suggest that on many real-world networks the run time to solve VertexCover is way smaller than even the best known FPT-approaches can explain. Similarly, greedy algorithms deliver very good approximations to the optimal solution in practice. We link these observations to two properties that are observed in many real-world networks, namely a heterogeneous degree distribution and high clustering. To formalize these properties and explain the observed behavior, we analyze how a branch-and-reduce algorithm performs on hyperbolic random graphs, which have become increasingly popular for modeling real-world networks. In fact, we are able to show that the VertexCover problem on hyperbolic random graphs can be solved in polynomial time, with high probability. The proof relies on interesting structural properties of hyperbolic random graphs. Since these predictions of the model are interesting in their own right, we conducted experiments on real-world networks showing that these properties are also observed in practice. When utilizing the same structural properties in an adaptive greedy algorithm, further experiments suggest that, on real instances, this leads to better approximations than the standard greedy approach within reasonable time

Towards Distributed Two-Stage Stochastic Optimization

The weighted vertex cover problem is concerned with selecting a subset of the vertices that covers a target set of edges with the objective of minimizing the total cost of the selected vertices. We consider a variant of this classic combinatorial optimization problem where the target edge set is not fully known; rather, it is characterized by a probability distribution. Adhering to the model of two-stage stochastic optimization, the execution is divided into two stages so that in the first stage, the decision maker selects some of the vertices based on the probabilistic forecast of the target edge set. Then, in the second stage, the edges in the target set are revealed and in order to cover them, the decision maker can augment the vertex subset selected in the first stage with additional vertices. However, in the second stage, the vertex cost increases by some inflation factor, so the second stage selection becomes more expensive. The current paper studies the two-stage stochastic vertex cover problem in the realm of distributed graph algorithms, where the decision making process (in both stages) is distributed among the vertices of the graph. By combining the stochastic optimization toolbox with recent advances in distributed algorithms for weighted vertex cover, we develop an algorithm that runs in time O(log (?) / ?), sends O(m) messages in total, and guarantees to approximate the optimal solution within a (3 + ?)-ratio, where m is the number of edges in the graph, ? is its maximum degree, and 0 < ? < 1 is a performance parameter