Fast movement strategies for a step-and-scan wafer stepper

We describe algorithms for the determination of fast movement strategies for a step-and-scan wafer stepper, a device that is used for the photolithographic processing of integrated circuits. The proposed solution strategy consists of two parts. First, we determine the maximum number of congruent rectangular chips that can be packed on a wafer, subject to the restriction that the chips are placed in a rectangular grid. Second, we find fast movement strategies for scanning all chips of a given packing, given the mechanical restrictions of the wafer stepper. The corresponding combinatorial optimization problem is formulated as a generalized asymmetric traveling salesman problem. We show how feasible scan strategies are determined, and how these strategies are improved by local search techniques, such as iterative improvement based on 2- and 3-exchanges, and simulated annealing based on 2-exchanges.\ud \u

Distributed Connectivity Decomposition

We present time-efficient distributed algorithms for decomposing graphs with large edge or vertex connectivity into multiple spanning or dominating trees, respectively. As their primary applications, these decompositions allow us to achieve information flow with size close to the connectivity by parallelizing it along the trees. More specifically, our distributed decomposition algorithms are as follows: (I) A decomposition of each undirected graph with vertex-connectivity $k$ into (fractionally) vertex-disjoint weighted dominating trees with total weight $\Omega(\frac{k}{\log n})$, in $\widetilde{O}(D+\sqrt{n})$ rounds. (II) A decomposition of each undirected graph with edge-connectivity $\lambda$ into (fractionally) edge-disjoint weighted spanning trees with total weight $\lceil\frac{\lambda-1}{2}\rceil(1-\varepsilon)$, in $\widetilde{O}(D+\sqrt{n\lambda})$ rounds. We also show round complexity lower bounds of $\tilde{\Omega}(D+\sqrt{\frac{n}{k}})$ and $\tilde{\Omega}(D+\sqrt{\frac{n}{\lambda}})$ for the above two decompositions, using techniques of [Das Sarma et al., STOC'11]. Moreover, our vertex-connectivity decomposition extends to centralized algorithms and improves the time complexity of [Censor-Hillel et al., SODA'14] from $O(n^3)$ to near-optimal $\tilde{O}(m)$. As corollaries, we also get distributed oblivious routing broadcast with $O(1)$-competitive edge-congestion and $O(\log n)$-competitive vertex-congestion. Furthermore, the vertex connectivity decomposition leads to near-time-optimal $O(\log n)$-approximation of vertex connectivity: centralized $\widetilde{O}(m)$ and distributed $\tilde{O}(D+\sqrt{n})$. The former moves toward the 1974 conjecture of Aho, Hopcroft, and Ullman postulating an $O(m)$ centralized exact algorithm while the latter is the first distributed vertex connectivity approximation

Approximating the Held-Karp Bound for Metric TSP in Nearly Linear Time

We give a nearly linear time randomized approximation scheme for the Held-Karp bound [Held and Karp, 1970] for metric TSP. Formally, given an undirected edge-weighted graph $G$ on $m$ edges and $\epsilon > 0$, the algorithm outputs in $O(m \log^4n /\epsilon^2)$ time, with high probability, a $(1+\epsilon)$-approximation to the Held-Karp bound on the metric TSP instance induced by the shortest path metric on $G$. The algorithm can also be used to output a corresponding solution to the Subtour Elimination LP. We substantially improve upon the $O(m^2 \log^2(m)/\epsilon^2)$ running time achieved previously by Garg and Khandekar. The LP solution can be used to obtain a fast randomized $\big(\frac{3}{2} + \epsilon\big)$-approximation for metric TSP which improves upon the running time of previous implementations of Christofides' algorithm

Ant colony optimisation and local search for bin-packing and cutting stock problems

The Bin Packing Problem and the Cutting Stock Problem are two related classes of NP-hard combinatorial optimization problems. Exact solution methods can only be used for very small instances, so for real-world problems, we have to rely on heuristic methods. In recent years, researchers have started to apply evolutionary approaches to these problems, including Genetic Algorithms and Evolutionary Programming. In the work presented here, we used an ant colony optimization (ACO) approach to solve both Bin Packing and Cutting Stock Problems. We present a pure ACO approach, as well as an ACO approach augmented with a simple but very effective local search algorithm. It is shown that the pure ACO approach can compete with existing evolutionary methods, whereas the hybrid approach can outperform the best-known hybrid evolutionary solution methods for certain problem classes. The hybrid ACO approach is also shown to require different parameter values from the pure ACO approach and to give a more robust performance across different problems with a single set of parameter values. The local search algorithm is also run with random restarts and shown to perform significantly worse than when combined with ACO

Sequential and Parallel Algorithms for Mixed Packing and Covering

Mixed packing and covering problems are problems that can be formulated as linear programs using only non-negative coefficients. Examples include multicommodity network flow, the Held-Karp lower bound on TSP, fractional relaxations of set cover, bin-packing, knapsack, scheduling problems, minimum-weight triangulation, etc. This paper gives approximation algorithms for the general class of problems. The sequential algorithm is a simple greedy algorithm that can be implemented to find an epsilon-approximate solution in O(epsilon^-2 log m) linear-time iterations. The parallel algorithm does comparable work but finishes in polylogarithmic time. The results generalize previous work on pure packing and covering (the special case when the constraints are all "less-than" or all "greater-than") by Michael Luby and Noam Nisan (1993) and Naveen Garg and Jochen Konemann (1998)

Fast Parallel Fixed-Parameter Algorithms via Color Coding

Fixed-parameter algorithms have been successfully applied to solve numerous difficult problems within acceptable time bounds on large inputs. However, most fixed-parameter algorithms are inherently \emph{sequential} and, thus, make no use of the parallel hardware present in modern computers. We show that parallel fixed-parameter algorithms do not only exist for numerous parameterized problems from the literature -- including vertex cover, packing problems, cluster editing, cutting vertices, finding embeddings, or finding matchings -- but that there are parallel algorithms working in \emph{constant} time or at least in time \emph{depending only on the parameter} (and not on the size of the input) for these problems. Phrased in terms of complexity classes, we place numerous natural parameterized problems in parameterized versions of AC$^0$. On a more technical level, we show how the \emph{color coding} method can be implemented in constant time and apply it to embedding problems for graphs of bounded tree-width or tree-depth and to model checking first-order formulas in graphs of bounded degree

Cut Tree Construction from Massive Graphs

The construction of cut trees (also known as Gomory-Hu trees) for a given graph enables the minimum-cut size of the original graph to be obtained for any pair of vertices. Cut trees are a powerful back-end for graph management and mining, as they support various procedures related to the minimum cut, maximum flow, and connectivity. However, the crucial drawback with cut trees is the computational cost of their construction. In theory, a cut tree is built by applying a maximum flow algorithm for $n$ times, where $n$ is the number of vertices. Therefore, naive implementations of this approach result in cubic time complexity, which is obviously too slow for today's large-scale graphs. To address this issue, in the present study, we propose a new cut-tree construction algorithm tailored to real-world networks. Using a series of experiments, we demonstrate that the proposed algorithm is several orders of magnitude faster than previous algorithms and it can construct cut trees for billion-scale graphs.Comment: Short version will appear at ICDM'1

Parameterized Complexity of Equitable Coloring

A graph on $n$ vertices is equitably $k$-colorable if it is $k$-colorable and every color is used either $\left\lfloor n/k \right\rfloor$ or $\left\lceil n/k \right\rceil$ times. Such a problem appears to be considerably harder than vertex coloring, being $\mathsf{NP\text{-}Complete}$ even for cographs and interval graphs. In this work, we prove that it is $\mathsf{W[1]\text{-}Hard}$ for block graphs and for disjoint union of split graphs when parameterized by the number of colors; and $\mathsf{W[1]\text{-}Hard}$ for $K_{1,4}$-free interval graphs when parameterized by treewidth, number of colors and maximum degree, generalizing a result by Fellows et al. (2014) through a much simpler reduction. Using a previous result due to Dominique de Werra (1985), we establish a dichotomy for the complexity of equitable coloring of chordal graphs based on the size of the largest induced star. Finally, we show that \textsc{equitable coloring} is $\mathsf{FPT}$ when parameterized by the treewidth of the complement graph