We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

This book gives an elementary treatment of the basic material about graph spectra, both for ordinary, and Laplace and Seidel spectra. The text progresses systematically, by covering standard topics before presenting some new material on trees, strongly regular graphs, two-graphs, association schemes, p-ranks of configurations and similar topics. Exercises at the end of each chapter provide practice and vary from easy yet interesting applications of the treated theory, to little excursions into related topics. Tables, references at the end of the book, an author and subject index enrich the text. Spectra of Graphs is written for researchers, teachers and graduate students interested in graph spectra. The reader is assumed to be familiar with basic linear algebra and eigenvalues, although some more advanced topics in linear algebra, like the Perron-Frobenius theorem and eigenvalue interlacing are included.

Spectra of graphs

https://users.monash.edu.au/~accmcc/talks/TalkMcKay.pdf

Practical graph isomorphism, II

This comprehensive textbook presents a clean and coherent account of most fundamental tools and techniques in Parameterized Algorithms and is a self-contained guide to the area. The book covers many of the recent developments of the field, including application of important separators, branching based on linear programming, Cut & Count to obtain faster algorithms on tree decompositions, algorithms based on representative families of matroids, and use of the Strong Exponential Time Hypothesis. A number of older results are revisited and explained in a modern and didactic way. The book provides a toolbox of algorithmic techniques. Part I is an overview of basic techniques, each chapter discussing a certain algorithmic paradigm. The material covered in this part can be used for an introductory course on fixed-parameter tractability. Part II discusses more advanced and specialized algorithmic ideas, bringing the reader to the cutting edge of current research. Part III presents complexity results and lower bounds, giving negative evidence by way of W[1]-hardness, the Exponential Time Hypothesis, and kernelization lower bounds. All the results and concepts are introduced at a level accessible to graduate students and advanced undergraduate students. Every chapter is accompanied by exercises, many with hints, while the bibliographic notes point to original publications and related work.

Parameterized Algorithms

Power Aware Routing in Mobile Ad Hoc Networks

Today most computer scientists believe that NP-hard problems cannot be solved by polynomial-time algorithms. From the polynomial-time perspective, all NP-complete problems are equivalent but their exponential-time properties vary widely. Why do some NP-hard problems appear to be easier than others? Are there algorithmic techniques for solving hard problems that are significantly faster than the exhaustive, brute-force methods? The algorithms that address these questions are known as exact exponential algorithms.The history of exact exponential algorithms for NP-hard problems dates back to the 1960s. The two classical examples are Bellman, Held and Karps dynamic programming algorithm for the traveling salesman problem and Rysers inclusionexclusion formula for the permanent of a matrix. The design and analysis of exact algorithms leads to a better understanding of hard problems and initiates interesting new combinatorial and algorithmic challenges. The last decade has witnessed a rapid development of the area, with many new algorithmic techniques discovered. This has transformed exact algorithms into a very active research field. This book provides an introduction to the area and explains the most common algorithmic techniques, and the text is supported throughout with exercises and detailed notes for further reading.The book is intended for advanced students and researchers in computer science, operations research, optimization and combinatorics.

Exact Exponential Algorithms

We present a fast algorithm for the subset convolution problem:given functions f and g defined on the lattice of subsets of ann-element set n, compute their subset convolution f*g, defined for S⊆ N by [ (f * g)(S) = [T ⊆ S] f(T) g(S/T),,]where addition and multiplication is carried out in an arbitrary ring. Via Mobius transform and inversion, our algorithm evaluates the subset convolution in O(n2 2n) additions and multiplications, substanti y improving upon the straightforward O(3n) algorithm. Specifically, if the input functions have aninteger range [-M,-M+1,...,M], their subset convolution over the ordinary sum--product ring can be computed in O(2n log M) time; the notation O suppresses polylogarithmic factors.Furthermore, using a standard embedding technique we can compute the subset convolution over the max--sum or min--sum semiring in O(2n M) time.To demonstrate the applicability of fast subset convolution, wepresent the first O(2k n2 + n m) algorithm for the Steiner tree problem in graphs with n vertices, k terminals, and m edges with bounded integer weights, improving upon the O(3kn + 2k n2 + n m) time bound of the classical Dreyfus-Wagner algorithm. We also discuss extensions to recent O(2n)-time algorithms for covering and partitioning problems (Bjorklund and Husfeldt, FOCS 2006; Koivisto, FOCS 2006).

Fourier meets Möbius: fast subset convolution

The problem of canonically labeling a graph is studied. Within the general framework of backtracking algorithms based on individualization and refinement, data structures, subroutines, and pruning heuristics especially for fast handling of large and sparse graphs are developed. Experiments indicate that the algorithm implementation in most cases clearly outperforms existing state-of-the-art tools.

Engineering an efficient canonical labeling tool for large and sparse graphs

/pdf/fourier-meets-mobius-fast-subset-convolution-1zpdn8cpeu.pdf

Fourier meets möbius: fast subset convolution

A new starting-point and a new method are requisite, to insure a complete [classi?cation of the Steiner triple systems of order 15]. This method was furnished, and its tedious and di?cult execution und- taken, by Mr. Cole. F. N. Cole, L. D. Cummings, and H. S. White (1917) [129] The history of classifying combinatorial objects is as old as the history of the objects themselves. In the mid-19th century, Kirkman, Steiner, and others became the fathers of modern combinatorics, and their work on various objects, including (what became later known as) Steiner triple systems led to several classi?cation results. Almost a century earlier, in 1782, Euler [180] published some results on classifying small Latin squares, but for the ?rst few steps in this direction one should actually go at least as far back as ancient Greece and the proof that there are exactly ?ve Platonic solids. One of the most remarkable achievements in the early, pre-computer era is the classi?cation of the Steiner triple systems of order 15, quoted above. An onerous task that, today, no sensible person would attempt by hand calcu- tion. Because, with the exception of occasional parameters for which com- natorial arguments are e?ective (often to prove nonexistence or uniqueness), classi?cation in general is about algorithms and computation.

Petteri Kaski

Papers

Exact Exponential Algorithms

Fourier meets Möbius: fast subset convolution

Engineering an efficient canonical labeling tool for large and sparse graphs

Fourier meets möbius: fast subset convolution

Classification Algorithms for Codes and Designs