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**1 - 7**of**7**### Bachelor Project Connectivity Algorithms for Undirected Graphs in Sage Author:

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### Strongly polynomial algorithm for generalized flow maximization (Extended Abstract)

, 2013

"... A strongly polynomial algorithm is given for the generalized flow maximization problem. It uses a new variant of the scaling technique, called continuous scaling. The main measure of progress is that within a strongly polynomial number of steps, an arc can be identified that must be tight in every d ..."

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A strongly polynomial algorithm is given for the generalized flow maximization problem. It uses a new variant of the scaling technique, called continuous scaling. The main measure of progress is that within a strongly polynomial number of steps, an arc can be identified that must be tight in every dual optimal solution, and thus can be contracted.

### Counting and Sampling Minimum Cuts in Genus g Graphs

, 2012

"... Let G be a directed graph with n vertices embedded on an orientable surface of genus g with two designated vertices s and t. We show that counting the number of minimum (s, t)-cuts in G is fixed parameter tractable in g. Specially, we give a 2 O(g) n 2 + min { n 2 log n, g O(g) n 3/2} time algorithm ..."

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Let G be a directed graph with n vertices embedded on an orientable surface of genus g with two designated vertices s and t. We show that counting the number of minimum (s, t)-cuts in G is fixed parameter tractable in g. Specially, we give a 2 O(g) n 2 + min { n 2 log n, g O(g) n 3/2} time algorithm for this problem. Our algorithm requires counting sets of cycles in a particular integer homology class. That we can count these cycles is an interesting result in itself as there are few prior results that are fixed parameter tractable and deal directly with integer homology. We also describe an algorithm which, after running our algorithm to count the number of cuts once, can sample repeatedly for a minimum cut in O(g 2 n) time per sample.

### Analyzing Graphs with Node Differential Privacy

, 2013

"... We develop algorithms for the private analysis of network data that provide accurate analysis of realistic networks while satisfying stronger privacy guarantees than those of previous work. We present several techniques for designing node differentially private algorithms, that is, algorithms whose ..."

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We develop algorithms for the private analysis of network data that provide accurate analysis of realistic networks while satisfying stronger privacy guarantees than those of previous work. We present several techniques for designing node differentially private algorithms, that is, algorithms whose output distribution does not change significantly when a node and all its adjacent edges are added to a graph. We also develop methodology for analyzing the accuracy of such algorithms on realistic networks. The main idea behind our techniques is to “project ” (in one of several senses) the input graph onto the set of graphs with maximum degree below a certain threshold. We design projection operators, tailored to specific statistics that have low sensitivity and preserve information about the original statistic. These operators can be viewed as giving a fractional (low-degree) graph that is a solution to an optimization problem described as a maximum flow instance, linear program, or convex program. In addition, we derive a generic, efficient reduction that allows us to apply any differentially private algorithm for bounded-degree graphs to an arbitrary graph. This reduction is based on analyzing the smooth sensitivity of the “naive ” truncation that simply discards nodes of high degree. 1

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"... An efficient technique for solving the scheduling of appliances in smart-homes ..."

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An efficient technique for solving the scheduling of appliances in smart-homes