Variational approximation of functionals defined on 1-dimensional connected sets: the planar case

In this paper we consider variational problems involving 1-dimensional connected sets in the Euclidean plane, such as the classical Steiner tree problem and the irrigation (Gilbert-Steiner) problem. We relate them to optimal partition problems and provide a variational approximation through Modica-Mortola type energies proving a $\Gamma$-convergence result. We also introduce a suitable convex relaxation and develop the corresponding numerical implementations. The proposed methods are quite general and the results we obtain can be extended to $n$-dimensional Euclidean space or to more general manifold ambients, as shown in the companion paper [11].Comment: 30 pages, 5 figure

Colored Non-Crossing Euclidean Steiner Forest

Given a set of $k$-colored points in the plane, we consider the problem of finding $k$ trees such that each tree connects all points of one color class, no two trees cross, and the total edge length of the trees is minimized. For $k=1$, this is the well-known Euclidean Steiner tree problem. For general $k$, a $k\rho$-approximation algorithm is known, where $\rho \le 1.21$ is the Steiner ratio. We present a PTAS for $k=2$, a $(5/3+\varepsilon)$-approximation algorithm for $k=3$, and two approximation algorithms for general~$k$, with ratios $O(\sqrt n \log k)$ and $k+\varepsilon$

Control for Localization and Visibility Maintenance of an Independent Agent using Robotic Teams

Given a non-cooperative agent, we seek to formulate a control strategy to enable a team of robots to localize and track the agent in a complex but known environment while maintaining a continuously optimized line-of-sight communication chain to a fixed base station. We focus on two aspects of the problem. First, we investigate the estimation of the agent\u27s location by using nonlinear sensing modalities, in particular that of range-only sensing, and formulate a control strategy based on improving this estimation using one or more robots working to independently gather information. Second, we develop methods to plan and sequence robot deployments that will establish and maintain line-of-sight chains for communication between the independent agent and the fixed base station using a minimum number of robots. These methods will lead to feedback control laws that can realize this plan and ensure proper navigation and collision avoidance

Combinatorial Optimization

Combinatorial Optimization is a very active field that benefits from bringing together ideas from different areas, e.g., graph theory and combinatorics, matroids and submodularity, connectivity and network flows, approximation algorithms and mathematical programming, discrete and computational geometry, discrete and continuous problems, algebraic and geometric methods, and applications. We continued the long tradition of triannual Oberwolfach workshops, bringing together the best researchers from the above areas, discovering new connections, and establishing new and deepening existing international collaborations

Variational Approximation of Functionals Defined on 1-dimensional Connected Sets: The Planar Case

In this paper we consider variational problems involving 1-dimensional connected sets in the Euclidean plane, such as the classical Steiner tree problem and the irrigation (Gilbert--Steiner) problem. We relate them to optimal partition problems and provide a variational approximation through Modica--Mortola type energies proving a $Gamma$-convergence result. We also introduce a suitable convex relaxation and develop the corresponding numerical implementations. The proposed methods are quite general and the results we obtain can be extended to $n$-dimensional Euclidean space or to more general manifold ambients, as shown in the companion paper [M. Bonafini, G. Orlandi, and E. Oudet, Variational Approximation of Functionals Defined on 1-Dimensional Connected Sets in $mathbb{R}^n$, preprint, 2018]

Matroids and Integrality Gaps for Hypergraphic Steiner Tree Relaxations

Until recently, LP relaxations have played a limited role in the design of approximation algorithms for the Steiner tree problem. In 2010, Byrka et al. presented a ln(4)+epsilon approximation based on a hypergraphic LP relaxation, but surprisingly, their analysis does not provide a matching bound on the integrality gap. We take a fresh look at hypergraphic LP relaxations for the Steiner tree problem - one that heavily exploits methods and results from the theory of matroids and submodular functions - which leads to stronger integrality gaps, faster algorithms, and a variety of structural insights of independent interest. More precisely, we present a deterministic ln(4)+epsilon approximation that compares against the LP value and therefore proves a matching ln(4) upper bound on the integrality gap. Similarly to Byrka et al., we iteratively fix one component and update the LP solution. However, whereas they solve an LP at every iteration after contracting a component, we show how feasibility can be maintained by a greedy procedure on a well-chosen matroid. Apart from avoiding the expensive step of solving a hypergraphic LP at each iteration, our algorithm can be analyzed using a simple potential function. This gives an easy means to determine stronger approximation guarantees and integrality gaps when considering restricted graph topologies. In particular, this readily leads to a 73/60 bound on the integrality gap for quasi-bipartite graphs. For the case of quasi-bipartite graphs, we present a simple algorithm to transform an optimal solution to the bidirected cut relaxation to an optimal solution of the hypergraphic relaxation, leading to a fast 73/60 approximation for quasi-bipartite graphs. Furthermore, we show how the separation problem of the hypergraphic relaxation can be solved by computing maximum flows, providing a fast independence oracle for our matroids.Comment: Corrects an issue at the end of Section 3. Various other minor improvements to the expositio

Computational Methods for Discrete Conic Optimization Problems

This thesis addresses computational aspects of discrete conic optimization. Westudy two well-known classes of optimization problems closely related to mixedinteger linear optimization problems. The case of mixed integer second-ordercone optimization problems (MISOCP) is a generalization in which therequirement that solutions be in the non-negative orthant is replaced by arequirement that they be in a second-order cone. Inverse MILP, on the otherhand, is the problem of determining the objective function that makes a givensolution to a given MILP optimal.Although these classes seem unrelated on the surface, the proposedsolution methodology for both classes involves outer approximation of a conicfeasible region by linear inequalities. In both cases, an iterative algorithmin which a separation problem is solved to generate the approximation isemployed. From a complexity standpoint, both MISOCP and inverse MILP areNP--hard. As in the case of MILPs, the usual decision version ofMISOCP is NP-complete, whereas in contrast to MILP, we provide the firstproof that a certain decision version of inverse MILP is rathercoNP-complete.With respect to MISOCP, we first introduce a basic outer approximationalgorithm to solve SOCPs based on a cutting-plane approach. As expected, theperformance of our implementation of such an algorithm is shown to lag behindthe well-known interior point method. Despite this, such a cutting-planeapproach does have promise as a method of producing bounds when embedded withina state-of-the-art branch-and-cut implementation due to its superior ability towarm-start the bound computation after imposing branching constraints. Ourouter-approximation-based branch-and-cut algorithm relaxes both integrality andconic constraints to obtain a linear relaxation. This linear relaxation isstrengthened by the addition of valid inequalities obtained by separatinginfeasible points. Valid inequalities may be obtained by separation from theconvex hull of integer solution lying within the relaxed feasible region or byseparation from the feasible region described by the (relaxed) conicconstraints. Solutions are stored when both integer and conic feasibility isachieved. We review the literature on cutting-plane procedures for MISOCP andmixed integer convex optimization problems.With respect to inverse MILP, we formulate this problem as a conicproblem and derive a cutting-plane algorithm for it. The separation problem inthis algorithm is a modified version of the original MILP. We show that thereis a close relationship between this algorithm and a similar iterativealgorithm for separating infeasible points from the convex hull of solutions tothe original MILP that forms part of the basis for the well-known result ofGrotschel-Lovasz-Schrijver that demonstrates the complexity-wiseequivalence of separation and optimization.In order to test our ideas, we implement a number of software librariesthat together constitute DisCO, a full-featured solver for MISOCP. Thefirst of the supporting libraries is OsiConic, an abstract base classin C++ for interfacing to SOCP solvers. We provide interfaces using thislibrary for widely used commercial and open source SOCP/nonlinear problemsolvers. We also introduce CglConic, a library that implements cuttingprocedures for MISOCP feasible set. We perform extensive computationalexperiments with DisCO comparing a wide range of variants of our proposedalgorithm, as well as other approaches. As DisCO is built on top of a libraryfor distributed parallel tree search algorithms, we also perform experimentsshowing that our algorithm is effective and scalable when parallelized