Smoothed Analysis of the Successive Shortest Path Algorithm

The minimum-cost flow problem is a classic problem in combinatorial optimization with various applications. Several pseudo-polynomial, polynomial, and strongly polynomial algorithms have been developed in the past decades, and it seems that both the problem and the algorithms are well understood. However, some of the algorithms' running times observed in empirical studies contrast the running times obtained by worst-case analysis not only in the order of magnitude but also in the ranking when compared to each other. For example, the Successive Shortest Path (SSP) algorithm, which has an exponential worst-case running time, seems to outperform the strongly polynomial Minimum-Mean Cycle Canceling algorithm. To explain this discrepancy, we study the SSP algorithm in the framework of smoothed analysis and establish a bound of $O(mn\phi)$ for the number of iterations, which implies a smoothed running time of $O(mn\phi (m + n\log n))$, where $n$ and $m$ denote the number of nodes and edges, respectively, and $\phi$ is a measure for the amount of random noise. This shows that worst-case instances for the SSP algorithm are not robust and unlikely to be encountered in practice. Furthermore, we prove a smoothed lower bound of $\Omega(m \phi \min\{n, \phi\})$ for the number of iterations of the SSP algorithm, showing that the upper bound cannot be improved for $\phi = \Omega(n)$.Comment: A preliminary version has been presented at SODA 201

Smoothed analysis of belief propagation and minimum-cost flow algorithms

Algorithms that have good worst-case performance are not always the ones that perform best in practice. The smoothed analysis framework is a way of analyzing algorithms that usually matches practical performance of these algorithms much better than worst-case analysis. In this thesis we apply smoothed analysis to two classes of algorithms: minimum-cost flow algorithms and belief propagation algorithms. The minimum-cost flow problem is the problem of sending a prescribed amount of flow through a network in the cheapest possible way. It is very well known, and over the last half a century many algorithms have been developed to solve it. We analyze three of these algorithms (the successive shortest path algorithm, the minimum-mean cycle canceling algorithm, and the network simplex algorithm) in the framework of smoothed analysis and show lower and upper bounds on their smoothed running-times. The belief propagation algorithm is a message-passing algorithm for solving probabilistic inference problems. Because of its simplicity, it is very popular in practice. However, its theoretical behavior is not well understood. To obtain a better theoretical understanding of the belief propagation algorithm, we apply it to several well-studied optimization problems. We analyze under which conditions the belief propagation algorithm converges to the correct solution and we analyze its smoothed running-time

Belief propagation for the maximum-weight independent set and minimum spanning tree problems

The belief propagation (BP) algorithm is a message-passing algorithm that is used for solving probabilistic inference problems. In practice, the BP algorithm performs well as a heuristic in many application fields. However, the theoretical understanding of BP is limited. To improve the theoretical understanding of BP, the BP algorithm has been applied to many well-understood combinatorial optimization problems. In this paper, we consider BP applied to the maximum-weight independent set (MWIS) and minimum spanning tree (MST) problems. Sanghavi et al. (2009) [12] applied the BP algorithm to the MWIS problem. We denote their algorithm by BP-MWIS. They showed that if the LP relaxation of the MWIS problem has a unique integral optimal solution and BP-MWIS converges, then BP-MWIS finds the optimal solution. Also, they showed that if the LP relaxation has a non-integral optimal solution, then BP-MWIS does not converge. In this paper, we precisely characterize the graphs for which BP-MWIS is guaranteed to find the optimal solution, regardless of the node weights. Bayati et al. (2008) [2] applied the BP algorithm to the MST problem. We denote their algorithm by BP-MST. They showed that if BP-MST converges, then it finds the optimal solution. In this paper, however, we provide an instance for which BP-MST does not converge. Also, since this instance is small and simple, we believe that BP-MST does not converge for most instances encountered in practice

Comparing and taming the reactivity of HWE and Wittig reagents with cyclic hemiacetals

A practical solution to the formation of mixtures of E/Z and open/cyclic isomers in the reaction of (2R,4S)-4-hydroxy-2-methylpentanal (as its hemiacetal, a lactol) with conjugated phosphoranes (stabilised Wittig reagents) and Horner-Wadsworth-Emmons reagents is disclosed. The HWE reaction has a strong bias to give oxolanes. On the other hand, stabilised Wittig reagents give unsaturated carboxyl derivatives of configuration E (major) and oxolanes (minor); the latter can be avoided by addition of CF3CH2OH or using morpholine amide phosphorane

Approximation algorithms for connected graph factors of minimum weight

Finding low-cost spanning subgraphs with given degree and connectivity requirements is a fundamental problem in the area of network design. We consider the problem of finding d-regular spanning subgraphs (or d-factors) of minimum weight with connectivity requirements. For the case of k-edge-connectedness, we present approximation algorithms that achieve constant approximation ratios for all . For the case of k-vertex-connectedness, we achieve constant approximation ratios for dae -1. Our algorithms also work for arbitrary degree sequences if the minimum degree is at least (for k-edge-connectivity) or 2k-1 (for k-vertex-connectivity). To complement our approximation algorithms, we prove that the problem with simple connectivity cannot be approximated better than the traveling salesman problem. In particular, the problem is APX-hard

Data mining the TopoChip

Pattern Recognition and Bioinformatic