Helly's Theorem: New Variations and Applications

This survey presents recent Helly-type geometric theorems published since the appearance of the last comprehensive survey, more than ten years ago. We discuss how such theorems continue to be influential in computational geometry and in optimization.Comment: 40 pages, 1 figure

Defragmenting the Module Layout of a Partially Reconfigurable Device

Modern generations of field-programmable gate arrays (FPGAs) allow for partial reconfiguration. In an online context, where the sequence of modules to be loaded on the FPGA is unknown beforehand, repeated insertion and deletion of modules leads to progressive fragmentation of the available space, making defragmentation an important issue. We address this problem by propose an online and an offline component for the defragmentation of the available space. We consider defragmenting the module layout on a reconfigurable device. This corresponds to solving a two-dimensional strip packing problem. Problems of this type are NP-hard in the strong sense, and previous algorithmic results are rather limited. Based on a graph-theoretic characterization of feasible packings, we develop a method that can solve two-dimensional defragmentation instances of practical size to optimality. Our approach is validated for a set of benchmark instances.Comment: 10 pages, 11 figures, 1 table, Latex, to appear in "Engineering of Reconfigurable Systems and Algorithms" as a "Distinguished Paper

A counterexample to the Alon-Saks-Seymour conjecture and related problems

Consider a graph obtained by taking edge disjoint union of $k$ complete bipartite graphs. Alon, Saks and Seymour conjectured that such graph has chromatic number at most $k+1$. This well known conjecture remained open for almost twenty years. In this paper, we construct a counterexample to this conjecture and discuss several related problems in combinatorial geometry and communication complexity

Hardness of discrepancy computation and epsilon-net verification in high dimension

Discrepancy measures how uniformly distributed a point set is with respect to a given set of ranges. There are two notions of discrepancy, namely continuous discrepancy and combinatorial discrepancy. Depending on the ranges, several possible variants arise, for example star discrepancy, box discrepancy, and discrepancy of half-spaces. In this paper, we investigate the hardness of these problems with respect to the dimension d of the underlying space. All these problems are solvable in time {n^O(d)}, but such a time dependency quickly becomes intractable for high-dimensional data. Thus it is interesting to ask whether the dependency on d can be moderated. We answer this question negatively by proving that the canonical decision problems are W[1]-hard with respect to the dimension. This is done via a parameterized reduction from the Clique problem. As the parameter stays linear in the input parameter, the results moreover imply that these problems require {n^\Omega(d)} time, unless 3-Sat can be solved in {2^o(n)} time. Further, we derive that testing whether a given set is an {\epsilon}-net with respect to half-spaces takes {n^\Omega(d)} time under the same assumption. As intermediate results, we discover the W[1]-hardness of other well known problems, such as determining the largest empty star inside the unit cube. For this, we show that it is even hard to approximate within a factor of {2^n}.Comment: 20 pages, 5 figure

A combinatorial characterization of higher-dimensional orthogonal packing

Higher-dimensional orthogonal packing problems have a wide range of practical applications, including packing, cutting, and scheduling. Previous efforts for exact algorithms have been unable to avoid structural problems that appear for instances in two- or higher-dimensional space. We present a new approach for modeling packings, using a graph-theoretical characterization of feasible packings. Our characterization allows it to deal with classes of packings that share a certain combinatorial structure, instead of having to consider one packing at a time. In addition, we can make use of elegant algorithmic properties of certain classes of graphs. This allows our characterization to be the basis for a successful branch-and-bound framework. This is the first in a series of papers describing new approaches to higher-dimensional packing.Comment: 21 pages, 8 figures, Latex, to appear in Mathematics of Operations Researc

Evo* 2020 -- Late-Breaking Abstracts Volume

This volume contains the Late-Breaking Abstracts submitted to the Evo* 2020 Conference, that took place online, from 15 to 17 of April 2020. These papers where presented as short talks and also at the poster session of the conference together with other regular submissions. All of them present ongoing research and preliminary results investigating on the application of different approaches of Bioinspired Methods (mainly Evolutionary Computation) to different problems, most of them real world ones.Comment: LBAs accepted in Evo* 2020. Part of the Conference Proceeding

Routing for analog chip designs at NXP Semiconductors

During the study week 2011 we worked on the question of how to automate certain aspects of the design of analog chips. Here we focused on the task of connecting different blocks with electrical wiring, which is particularly tedious to do by hand. For digital chips there is a wealth of research available for this, as in this situation the amount of blocks makes it hopeless to do the design by hand. Hence, we set our task to finding solutions that are based on the previous research, as well as being tailored to the specific setting given by NXP. This resulted in an heuristic approach, which we presented at the end of the week in the form of a protoype tool. In this report we give a detailed account of the ideas we used, and describe possibilities to extend the approach

Higher-Dimensional Packing with Order Constraints

We present a first exact study on higher-dimensional packing problems with order constraints. Problems of this type occur naturally in applications such as logistics or computer architecture and can be interpreted as higher-dimensional generalizations of scheduling problems. Using graph-theoretic structures to describe feasible solutions, we develop a novel exact branch-and-bound algorithm. This extends previous work by Fekete and Schepers; a key tool is a new order-theoretic characterization of feasible extensions of a partial order to a given complementarity graph that is tailor-made for use in a branch-and-bound environment. The usefulness of our approach is validated by computational results.Comment: 23 pages, 14 figures, 5 tables, Latex; revision clarifies various minor points, fixes typos, etc. To appear in SIAM Journal on Discrete Mathematic

A Polynomial Time Algorithm for The Traveling Salesman Problem

The ATSP polytope can be expressed by asymmetric polynomial size linear program.Comment: 8 pages. Simplifie

Scalable Parallel Numerical Constraint Solver Using Global Load Balancing

We present a scalable parallel solver for numerical constraint satisfaction problems (NCSPs). Our parallelization scheme consists of homogeneous worker solvers, each of which runs on an available core and communicates with others via the global load balancing (GLB) method. The parallel solver is implemented with X10 that provides an implementation of GLB as a library. In experiments, several NCSPs from the literature were solved and attained up to 516-fold speedup using 600 cores of the TSUBAME2.5 supercomputer.Comment: To be presented at X10'15 Worksho