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Algorithms, Graph Theory, and Linear Equations in Laplacian Matrices
"... Abstract. The Laplacian matrices of graphs are fundamental. In addition to facilitating the application of linear algebra to graph theory, they arise in many practical problems. In this talk we survey recent progress on the design of provably fast algorithms for solving linear equations in the Lapla ..."
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Abstract. The Laplacian matrices of graphs are fundamental. In addition to facilitating the application of linear algebra to graph theory, they arise in many practical problems. In this talk we survey recent progress on the design of provably fast algorithms for solving linear equations in the Laplacian matrices of graphs. These algorithms motivate and rely upon fascinating primitives in graph theory, including lowstretch spanning trees, graph sparsifiers, ultrasparsifiers, and local graph clustering. These are all connected by a definition of what it means for one graph to approximate another. While this definition is dictated by Numerical Linear Algebra, it proves useful and natural from a graph theoretic perspective.
SPECTRAL SPARSIFICATION IN THE SEMISTREAMING SETTING
"... Abstract. Let G be a graph with n vertices and m edges. A sparsifier of G is a sparse graph on the same vertex set approximating G in some natural way. It allows us to say useful things about G while considering much fewer than m edges. The strongest commonlyused notion of sparsification is spectra ..."
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Abstract. Let G be a graph with n vertices and m edges. A sparsifier of G is a sparse graph on the same vertex set approximating G in some natural way. It allows us to say useful things about G while considering much fewer than m edges. The strongest commonlyused notion of sparsification is spectral sparsification; H is a spectral sparsifier of G if the quadratic forms induced by the Laplacians of G and H approximate one another well. This notion is strictly stronger than the earlier concept of combinatorial sparsification. In this paper, we consider a semistreaming setting, where we have only Õ(n) storage space, and we thus cannot keep all of G. In this case, maintaining a sparsifier instead gives us a useful approximation to G, allowing us to answer certain questions about the original graph without storing all of it. In this paper, we introduce an algorithm for constructing a spectral sparsifier of G with O(n log n/ɛ 2) edges (where ɛ is a parameter measuring the quality of the sparsifier), taking Õ(m) time and requiring only one pass over G. In addition, our algorithm has the property that it maintains at all times a valid sparsifier for the subgraph of G that we have received. Our algorithm is natural and conceptually simple. As we read edges of G, we add them to the sparsifier H. Whenever H gets too big, we resparsify it in Õ(n) time. Adding edges to a graph changes the structure of its sparsifier’s restriction to the already existing edges. It would thus seem that the above procedure would cause errors to compound each time that we resparsify, and that we should need to either retain significantly more information or reexamine previously discarded edges in order to construct the new sparsifier. However, we show how to use the information contained in H to perform this resparsification using only the edges retained by earlier steps in nearly linear time. 1.
Improved spectral sparsification and numerical algorithms for sdd matrices
 STACS
, 2012
"... We present three spectral sparsification algorithms that, on input a graph G with n vertices and m edges, return a graph H with n vertices and O(n log n/ɛ 2) edges that provides a strong approximation of G. Namely, for all vectors x and any ɛ> 0, we have (1 − ɛ)x T LGx ≤ x T LHx ≤ (1 + ɛ)x T LGx, ..."
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We present three spectral sparsification algorithms that, on input a graph G with n vertices and m edges, return a graph H with n vertices and O(n log n/ɛ 2) edges that provides a strong approximation of G. Namely, for all vectors x and any ɛ> 0, we have (1 − ɛ)x T LGx ≤ x T LHx ≤ (1 + ɛ)x T LGx, where LG and LH are the Laplacians of the two graphs. The first algorithm is a simple modification of the fastest known algorithm and runs in Õ(m log2 n) time, an O(log n) factor faster than before. The second algorithm runs in Õ(m log n) time and generates a sparsifier with Õ(n log3 n) edges. The third algorithm applies to graphs where m> n log 5 n and runs in Õ(m logm/n log5 n n) time. In the range where m> n1+r for some constant r this becomes Õ(m). The improved sparsification algorithms are employed to accelerate linear system solvers and algorithms for computing fundamental eigenvectors of dense SDD matrices.
An almostlineartime algorithm for approximate max flow in undirected graphs, and its multicommodity generalizations
"... In this paper we present an almost linear time algorithm for solving approximate maximum flow in undirected graphs. In particular, given a graph with m edges we show how to produce a 1−ε approximate maximum flow in time O(m 1+o(1) · ε −2). Furthermore, we present this algorithm as part of a general ..."
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In this paper we present an almost linear time algorithm for solving approximate maximum flow in undirected graphs. In particular, given a graph with m edges we show how to produce a 1−ε approximate maximum flow in time O(m 1+o(1) · ε −2). Furthermore, we present this algorithm as part of a general framework that also allows us to achieve a running time of O(m 1+o(1) ε −2 k 2) for the maximum concurrent kcommodity flow problem, the first such algorithm with an almost linear dependence on m. We also note that independently Jonah Sherman has produced an almost linear time algorithm for maximum flow and we thank him for coordinating submissions.
Sketching as a tool for numerical linear algebra
 Foundations and Trends in Theoretical Computer Science
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Fast Approximation Algorithms for Graph Partitioning Using Spectral and SemidefiniteProgramming Techniques
, 2011
"... Graphpartitioning problems are a central topic of research in the study of approximation algorithms. They are of interest to both theoreticians, for their farreaching connections to different areas of mathematics, and to practitioners, as algorithms for graph partitioning can be used as fundamenta ..."
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Graphpartitioning problems are a central topic of research in the study of approximation algorithms. They are of interest to both theoreticians, for their farreaching connections to different areas of mathematics, and to practitioners, as algorithms for graph partitioning can be used as fundamental building blocks in many applications, such as image segmentation and clustering. While many theoretical approximation algorithms exist for graph partitioning, they often rely on multicommodityflow computations that run in quadratic time in the worst case and are too timeconsuming for the massive graphs that are prevalent in today’s applications. In this dissertation, we study the design of approximation algorithms that yield strong approximation guarantees, while running in subquadratic time and relying on computational procedures that are often fast in practice. The results that we describe encompass two different approaches to the construction of such fast algorithms. Our first result exploits the CutMatching game of Khandekar, Rao and Vazirani [41], an elegant framework for designing graphpartitioning algorithms that rely on singlecommodity, rather than multicommodity, maximum flow. Within this framework, we give two novel algorithms that achieve an O(log n)approximation for the problem of finding the cut of minimum
Hallucination Helps: Energy Efficient Virtual Circuit Routing
, 2013
"... We consider virtual circuit routing protocols, with an objective of minimizing energy, in a network of components that are speed scalable, and that may be shutdown when idle. We assume that the speed s of the router is proportional to its load, and assume the standard model for component power, name ..."
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We consider virtual circuit routing protocols, with an objective of minimizing energy, in a network of components that are speed scalable, and that may be shutdown when idle. We assume that the speed s of the router is proportional to its load, and assume the standard model for component power, namely that the power is some constant static power plus sα, where typically α ∈ [1.1, 3]. We give a polynomialtime offline algorithm that is the combination of three natural combinatorial algorithms, and show that for any fixed α the algorithm has approximation ratio O(logα k), where k is the number of demand pairs. The algorithm extends rather naturally to a randomized online algorithm, which we show has competitive ratio Õ(log3α+1 k). This is the first online result for the problem. We also show that this online algorithm has competitive ratio Õ(logα+1 k) for the case that all connections have a common source. 1