Classical spectral graph theory characterizes graphs with logarithmic mixing time. In this work, we present a combinatorial characterization of graphs with constant mixing time. The combinatorial characterization is based on the small-set bipartite density condition, which is weaker than having near-optimal spectral radius and is stronger than having near-optimal small-set vertex expansion.

In the load-balancing problem, we have an n-vertex bipartite graph G = (L, R, E ) between a set of clients and servers. The goal is to find an assignment of all clients to the servers, while minimizing the maximum load on each server, where load of a server is the number of clients assigned to it. Motivated by understanding the streaming complexity of this problem, we study load-balancing in the one-way (two-party) communication model: the edges of the input graph are partitioned between Alice and Bob, and Alice needs to send a short message to Bob for him to output a solution of the entire graph.

Abstract. The classic Cheeger’s inequality relates the edge conductance [Formula: see text] of a graph and the second smallest eigenvalue [Formula: see text] of the Laplacian matrix. Recently, Olesker-Taylor and Zanetti discovered a Cheeger-type inequality [Formula: see text] connecting the vertex expansion [Formula: see text] of a graph [Formula: see text] and the maximum reweighted second smallest eigenvalue [Formula: see text] of the Laplacian matrix. In this work, we first improve their result to [Formula: see text], where [Formula: see text] is the maximum degree in [Formula: see text], which is optimal up to a constant factor. Also, the improved result holds for weighted vertex expansion, answering an open question by Olesker-Taylor and Zanetti. Building on this connection, we then develop a new spectral theory for vertex expansion. We discover that several interesting generalizations of Cheeger inequalities relating edge conductances and eigenvalues have a close analogue in relating vertex expansions and reweighted eigenvalues. These include the following: (1) An analogue of Trevisan’s result that relates the bipartite vertex expansion [Formula: see text] of a graph and the maximum reweighted lower spectral gap [Formula: see text] of the adjacency matrix. This implies the first approximation algorithm for bipartite vertex expansion. (2) An analogue of higher-order Cheeger’s inequalities that relates the [Formula: see text]-way vertex expansion [Formula: see text] of a graph and the maximum reweighted [Formula: see text]th smallest eigenvalue [Formula: see text] of the Laplacian matrix. This implies the first approximation algorithm for [Formula: see text]-way vertex expansion. (3) An analogue of improved Cheeger’s inequality that relates the vertex expansion [Formula: see text] and the reweighted eigenvalues [Formula: see text] and [Formula: see text]. This provides an improved bound for [Formula: see text] using [Formula: see text], when the [Formula: see text]-way vertex expansion [Formula: see text] is large for a small [Formula: see text]. Finally, inspired by this connection, we present negative evidence to the [Formula: see text]-polytope edge expansion conjecture by Mihail and Vazirani. We construct [Formula: see text]-polytopes whose graphs have very poor vertex expansion. This implies that the fastest mixing time to the uniform distribution on the vertices of these [Formula: see text]-polytopes is almost linear in the graph size. This does not provide a counterexample to the conjecture, but this is in contrast with known positive results which proved poly-logarithmic mixing time to the uniform distribution on the vertices of subclasses of [Formula: see text]-polytopes.

We consider a new semidefinite programming relaxation for directed edge expansion, which is obtained by adding triangle inequalities to the reweighted eigenvalue formulation. Applying the matrix multiplicative weight update method on this relaxation, we derive almost linear-time algorithms to achieve O (√log n)- approximation and Cheeger-type guarantee for directed edge expansion, as well as an improved cut-matching game for directed graphs. This provides a primal-dual flow-based framework to obtain the best known algorithms for directed graph partitioning. The same approach also works for vertex expansion and for hypergraphs, providing a simple and unified approach to achieve the best known results for different expansion problems and different algorithmic techniques.

We consider a general $p$-norm objective for experimental design problems that captures some well-studied objectives (D/A/E-design) as special cases. We prove that a randomized local search approach provides a unified algorithm to solve this problem for all $p$. This provides the first approximation algorithm for the general $p$-norm objective, and a nice interpolation of the best known bounds of the special cases.

We derive Cheeger inequalities for directed graphs and hypergraphs using the reweighted eigenvalue approach that was recently developed for vertex expansion in undirected graphs. The goal is to develop a new spectral theory for directed graphs and an alternative spectral theory for hypergraphs.

The classical Cheeger’s inequality relates the edge conductance of a graph and the second smallest eigenvalue of the Laplacian matrix. Recently, Olesker-Taylor and Zanetti discovered a Cheeger-type inequality connecting the vertex expansion of a graph and the maximum reweighted second smallest eigenvalue of the Laplacian matrix.In this work, we first improve their result to a logarithmic dependence on the maximum degree in the graph, which is optimal up to a constant factor. Also, the improved result holds for weighted vertex expansion, answering an open question by Olesker-Taylor and Zanetti. Building on this connection, we then develop a new spectral theory for vertex expansion. We discover that several interesting generalizations of Cheeger inequalities relating edge conductances and eigenvalues have a close analog in relating vertex expansions and reweighted eigenvalues. These include an analog of Trevisan’s result on bipartiteness, an analog of higher order Cheeger’s inequality, and an analog of improved Cheeger’s inequality. Finally, inspired by this connection, we present negative evidence to the 0/1-polytope edge expansion conjecture by Mihail and Vazirani. We construct 0/1-polytopes whose graphs have very poor vertex expansion. This implies that the fastest mixing time to the uniform distribution on the vertices of these 0/1-polytopes is almost linear in the graph size.

We present a local search framework to design and analyze both combinatorial algorithms and rounding algorithms for experimental design problems. This framework provides a unifying approach to match and improve all known results in D/A/E-design and to obtain new results in previously unknown settings. For combinatorial algorithms, we provide a new analysis of the classical Fedorov's exchange method. We prove that this simple local search algorithm works well as long as there exists an almost optimal solution with good condition number. Moreover, we design a new combinatorial local search algorithm for E-design using the regret minimization framework. For rounding algorithms, we provide a unified randomized exchange algorithm to match and improve previous results for D/A/E-design. Furthermore, the algorithm works in the more general setting to approximately satisfy multiple knapsack constraints, which can be used for weighted experimental design and for incorporating fairness constraints into experimental design.

We present a spectral approach to design approximation algorithms for network design problems. We observe that the underlying mathematical questions are the spectral rounding problems, which were studied in spectral sparsification and in discrepancy theory. We extend these results to incorporate additional nonnegative linear constraints, and show that they can be used to significantly extend the scope of network design problems that can be solved. Our algorithm for spectral rounding is an iterative randomized rounding algorithm based on the regret minimization framework. In some settings, this provides an alternative spectral algorithm to achieve constant factor approximation for the classical survivable network design problem, and partially answers a question of Bansal about survivable network design with concentration property. We also show many other applications of the spectral rounding results, including weighted experimental design and spectral network design.

We consider a new problem of designing a network with small s - t effective resistance. In this problem, we are given an undirected graph G = (V,E) , two designated vertices s,t ∈ V , and a budget k . The goal is to choose a subgraph of G with at most k edges to minimize the s - t effective resistance. This problem is an interpolation between the shortest path problem and the minimum cost flow problem and has applications in electrical network design. We present several algorithmic and hardness results for this problem and its variants. On the hardness side, we show that the problem is NP-hard, and the weighted version is hard to approximate within a factor smaller than two assuming the small-set expansion conjecture. On the algorithmic side, we analyze a convex programming relaxation of the problem and design a constant factor approximation algorithm. The key of the rounding algorithm is a randomized path-rounding procedure based on the optimality conditions and a flow decomposition of the fractional solution. We also use dynamic programming to obtain a fully polynomial time approximation scheme when the input graph is a series-parallel graph, with better approximation ratio than the integrality gap of the convex program for these graphs.

We present a local search framework to design and analyze both combinatorial algorithms and rounding algorithms for experimental design problems. This framework provides a unifying approach to match and improve all known results in D/A/E-design and to obtain new results in previously unknown settings. For combinatorial algorithms, we provide a new analysis of the classical Fedorov's exchange method. We prove that this simple local search algorithm works well as long as there exists an almost optimal solution with good condition number. Moreover, we design a new combinatorial local search algorithm for E-design using the regret minimization framework. For rounding algorithms, we provide a unified randomized exchange algorithm to match and improve previous results for D/A/E-design. Furthermore, the algorithm works in the more general setting to approximately satisfy multiple knapsack constraints, which can be used for weighted experimental design and for incorporating fairness constraints into experimental design.

The motivation of this work is to extend the techniques of higher order random walks on simplicial complexes to analyze mixing times of Markov chains for combinatorial problems. Our main result is a sharp upper bound on the second eigenvalue of the down-up walk on a pure simplicial complex, in terms of the second eigenvalues of its links. We show some applications of this result in analyzing mixing times of Markov chains, including sampling independent sets of a graph and sampling common independent sets of two partition matroids.

We present a spectral approach to design approximation algorithms for network design problems. We observe that the underlying mathematical questions are the spectral rounding problems, which were studied in spectral sparsification and in discrepancy theory. We extend these results to incorporate additional linear constraints, and show that they can be used to significantly extend the scope of network design problems that can be solved. Our algorithm for spectral rounding is an iterative randomized rounding algorithm based on the regret minimization framework. In some settings, this provides an alternative spectral algorithm to achieve constant factor approximation for survivable network design, and partially answers a question of Bansal about survivable network design with concentration property. We also show that the spectral rounding results have many other applications, including weighted experimental design and additive spectral sparsification.

We present a spectral analysis of a continuous scaling algorithm for matrix scaling and operator scaling. The main result is that if the input matrix or operator has a spectral gap, then a natural gradient flow has linear convergence. This implies that a simple gradient descent algorithm also has linear convergence under the same assumption. The spectral gap condition for operator scaling is closely related to the notion of quantum expander studied in quantum information theory. The spectral analysis also provides bounds on some important quantities of the scaling problems, such as the condition number of the scaling solution and the capacity of the matrix and operator. These results can be used in various applications of scaling problems, including matrix scaling on expander graphs, permanent lower bounds on random matrices, the Paulsen problem on random frames, and Brascamp--Lieb constants on random operators. In some applications, the inputs of interest satisfy the spectral condition and we prove significantly stronger bounds than the worst case bounds.

$ \def\vecc#1{\boldsymbol{#1}} $We design a polynomial time algorithm that for any weighted undirected graph $G = (V, E,\vecc w)$ and sufficiently large $δ> 1$, partitions $V$ into subsets $V_1, \ldots, V_h$ for some $h\geq 1$, such that $\bullet$ at most $δ^{-1}$ fraction of the weights are between clusters, i.e. \[ w(E - \cup_{i = 1}^h E(V_i)) \lesssim \frac{w(E)}δ;\] $\bullet$ the effective resistance diameter of each of the induced subgraphs $G[V_i]$ is at most $δ^3$ times the average weighted degree, i.e. \[ \max_{u, v \in V_i} \mathsf{Reff}_{G[V_i]}(u, v) \lesssim δ^3 \cdot \frac{|V|}{w(E)} \quad \text{ for all } i=1, \ldots, h.\] In particular, it is possible to remove one percent of weight of edges of any given graph such that each of the resulting connected components has effective resistance diameter at most the inverse of the average weighted degree. Our proof is based on a new connection between effective resistance and low conductance sets. We show that if the effective resistance between two vertices $u$ and $v$ is large, then there must be a low conductance cut separating $u$ from $v$. This implies that very mildly expanding graphs have constant effective resistance diameter. We believe that this connection could be of independent interest in algorithm design.

The Paulsen problem is a basic open problem in operator theory: Given vectors u1, …, un ∈ ℝd that are є-nearly satisfying the Parseval’s condition and the equal norm condition, is it close to a set of vectors v1, …, vn ∈ ℝd that exactly satisfy the Parseval’s condition and the equal norm condition? Given u1, …, un, the squared distance (to the set of exact solutions) is defined as infv ∑i=1n || ui − vi ||22 where the infimum is over the set of exact solutions. Previous results show that the squared distance of any є-nearly solution is at most O(poly(d,n,є)) and there are є-nearly solutions with squared distance at least Ω(d є). The fundamental open question is whether the squared distance can be independent of the number of vectors n.

We design approximation algorithms for Unique Gmeas when the constraint graph admits good low diameter graph decomposition. For the M2Lin(k) problem in K(r)-minor free graphs, when there is an assignment satisfying 1-eps fraction of constraints, we present an algorithm that produces an assignment satisfying 1-O(r*eps) fraction of constraints, with the approximation ratio independent of the alphabet size. A corollary is an improved approximation algorithm for the Min-UnCut problem for K(r)-minor free graphs. For general Unique Games in K(r)-minor free graphs, we provide another algorithm that produces an assignment satisfying 1-O(r *sqrt(eps)) fraction of constraints. Our approach is to round a linear programming relaxation to find a minimum subset of edges that intersects all the inconsistent cycles. We show that it is possible to apply the low diameter graph decomposition technique on the constraint graph directly, rather than to work on the label extended graph as in previous algorithms for Unique Games. The same approach applies when the constraint graph is of genus g, and we get similar results with r replaced by log g in the M2Lin(k) problem and by sqrt(log g) in the general problem. The former result generalizes the result of Gupta-Talwar for Unique Games in the M2Lin(k) case, and the latter result generalizes the result of Trevisan for general Unique Games.

Analyzing the mixing time of random walks is a well- studied problem with applications in random sampling and more recently in graph partitioning. In this work, we present new analysis of random walks and evolving sets using more combinatorial graph structures, and show some implications in approximating small-set expansion. On the other hand, we provide examples showing the limitations of using random walks and evolving sets in disproving the small-set expansion hypothesis. 1. We define a combinatorial analog of the spectral gap, and use it to prove the convergence of non- lazy random walks. A corollary is a tight lower bound on the small-set expansion of graph powers for any graph. 2. We prove that random walks converge faster when the robust vertex expansion of the graph is larger. This provides an improved analysis of the local graph partitioning algorithm using the evolving set process, and also derives an alternative proof of an improved Cheeger's inequality. 3. We give an example showing that the evolving set process fails to disprove the small-set expansion hypothesis. This refutes a conjecture of Oveis Gharan and shows the limitations of all existing local graph partitioning algorithms in approximating small-set expansion.

We prove two generalizations of the Cheeger's inequality. The first generalization relates the second eigenvalue to the edge expansion and the vertex expansion of the graph $G$, $\lambda_2 = \Omega( \phi^V(G) \phi(G) )$, where $\phi^V(G)$ denotes the robust vertex expansion of $G$ and $\phi(G)$ denotes the edge expansion of $G$. The second generalization relates the second eigenvalue to the edge expansion and the expansion profile of $G$, for all $k \geq 2$, $ \lambda_2 = \Omega( \phi_k(G) \phi(G) / k )$, where $\phi_k(G)$ denotes the $k$-way expansion of $G$. These show that the spectral partitioning algorithm has better performance guarantees when $\phi^V(G)$ is large (e.g., planted random instances) or $\phi_k(G)$ is large (instances with few disjoint nonexpanding sets). Both bounds are tight up to a constant factor. Our approach is based on a method to analyze solutions of Laplacian systems, and this allows us to extend the results to local graph partitioning algorithms. In particular, we show that our approach can be used to analyze personal pagerank vectors and to give a local graph partitioning algorithm for the small-set expansion problem with performance guarantees similar to the generalizations of Cheeger's inequality. We also present a spectral approach to prove similar results for the truncated random walk algorithm. These show that local graph partitioning algorithms almost match the performance of the spectral partitioning algorithm, with the additional advantages that they apply to the small-set expansion problem and their running time could be sublinear. Our techniques provide common approaches to analyze the spectral partitioning algorithm and local graph partitioning algorithms.

We prove two generalizations of the Cheeger's inequality. The first generalization relates the second eigenvalue to the edge expansion and the vertex expansion of the graph G, where φV(G) denotes the robust vertex expansion of G and φ(G) denotes the edge expansion of G. The second generalization relates the second eigenvalue to the edge expansion and the expansion profile of G, for all k ≥ 2, where φk(G) denotes the k-way expansion of G. These show that the spectral partitioning algorithm has better performance guarantees when φV(G) is large (e.g. planted random instances) or φk(G) is large (instances with few disjoint non-expanding sets). Both bounds are tight up to a constant factor. Our approach is based on a method to analyze solutions of Laplacian systems, and this allows us to extend the results to local graph partitioning algorithms. In particular, we show that our approach can be used to analyze personal pagerank vectors, and to give a local graph partitioning algorithm for the small-set expansion problem with performance guarantees similar to the generalizations of Cheeger's inequality. We also present a spectral approach to prove similar results for the truncated random walk algorithm. These show that local graph partitioning algorithms almost match the performance of the spectral partitioning algorithm, with the additional advantages that they apply to the small-set expansion problem and their running time could be sublinear. Our techniques provide common approaches to analyze the spectral partitioning algorithm and local graph partitioning algorithms.

In the Minimum Bounded Degree Spanning Tree problem, we are given an undirected graph G = ( V, E ) with a degree upper bound B v on each vertex v ∈ V , and the task is to find a spanning tree of minimum cost that satisfies all the degree bounds. Let OPT be the cost of an optimal solution to this problem. In this article we present a polynomial-time algorithm which returns a spanning tree T of cost at most OPT and d T ( v ) ≤ B v + 1 for all v , where d T ( v ) denotes the degree of v in T . This generalizes a result of Fürer and Raghavachari [1994] to weighted graphs, and settles a conjecture of Goemans [2006] affirmatively. The algorithm generalizes when each vertex v has a degree lower bound A v and a degree upper bound B v , and returns a spanning tree with cost at most OPT and A v - 1 ≤ d T ( v ) ≤ B v + 1 for all v ∈ V . This is essentially the best possible. The main technique used is an extension of the iterative rounding method introduced by Jain [2001] for the design of approximation algorithms.

Given a lazy regular graph G, we prove that the expansion of G^t is at least sqrt(t) times the expansion of G. This bound is tight and can be generalized to small set expansion. We show some applications of this result.

This special section contains five selected papers from the 50th Annual Symposium on Foundations of Computer Science (FOCS 2010) sponsored by the IEEE Technical Committee on Mathematical Foundations of Computing. The conference was held in Las Vegas, Nevada, October 23-26, 2010. The conference program consisted of 81 papers, which the program committee selected from 270 submissions. The program committee was composed of Scott Aaronson, Dorit Aharonov, Eli Ben-Sasson, Julia Chuzhoy, Ryan O'Donnell, Roberto Grossi, Nick Harvey, Adam Kalai, Nicole Immorlica, Yuval Ishai, Lap Chi Lau, James Lee, Tal Malkin, Joe Mitchell, Dana Moshkovitz, S. Muthukrishnan, Christos Papadimitriou, Sofya Raskhodnikova, Steve Skiena, Mikkel Thorup, Luca Trevisan, and Eric Vigoda. Each of the five papers appearing in this issue was subject to the standard refereeing process of the SIAM Journal on Computing. In “The Monotone Complexity of $k$-Clique on Random Graphs," Ben Rossman proves that monotone circuits that solve the $k$-clique problem in random graphs must have size $\omega(n^{k/4})$, establishing the first average-case monotone lower bound. Andreas Björklund, in the paper “Determinant Sums for Undirected Hamiltonicity," develops the first improvement to the $\tilde O(2^n)$ dynamic programming algorithm for Hamiltonian circuit: Björklund's algorithm runs in time $\tilde O(1.657^n)$. The paper “Distance Oracles Beyond the Thorup--Zwick Bound" by Mihai Pătraşcu and Liam Roditty presents the first improvement in ten years for the problem of compactly representing an approximation to all-pairs shortest path distances in a graph. Shaddin Dughmi and Tim Roughgarden show how to convert any approximation algorithm for a certain class of problems into a truthful mechanism in the paper “Black-Box Randomized Reductions in Algorithmic Mechanism Design." Ioannis Koutis, Gary Miller, and Richard Peng, in the paper “Approaching Optimality for Solving SDD Linear Systems," present a new nearly linear time algorithm for the problem of solving systems of linear equations that are symmetric and diagonally dominant. We wish to thank Madhu Sudan and Leonard Schulman, the former and current Editors-in-Chief of SICOMP, for being very generous with their time as they helped us in this project. We also wish to thank Heather Blythe of SIAM and the anonymous referees.

We present fast and simple algebraic algorithms for the linear matroid parity problem and its applications. For the linear matroid parity problem, we obtain a simple randomized algorithm with running time O ( mr ω-1 ), where m and r are the number of columns and the number of rows, respectively, and ω ≈ 2.3727 is the matrix multiplication exponent. This improves the O ( mr ω )-time algorithm by Gabow and Stallmann and matches the running time of the algebraic algorithm for linear matroid intersection, answering a question of Harvey. We also present a very simple alternative algorithm with running time O ( mr 2 ), which does not need fast matrix multiplication. We further improve the algebraic algorithms for some specific graph problems of interest. For the Mader’s disjoint S -path problem, we present an O ( n ω )-time randomized algorithm where n is the number of vertices. This improves the running time of the existing results considerably and matches the running time of the algebraic algorithms for graph matching. For the graphic matroid parity problem, we give an O ( n 4 )-time randomized algorithm where n is the number of vertices, and an O ( n 3 )-time randomized algorithm for a special case useful in designing approximation algorithms. These algorithms are optimal in terms of n as the input size could be Ω ( n 4 ) and Ω ( n 3 ), respectively. The techniques are based on the algebraic algorithmic framework developed by Mucha and Sankowski, Harvey, and Sankowski. While linear matroid parity and Mader’s disjoint S -path are challenging generalizations for the design of combinatorial algorithms, our results show that both the algebraic algorithms for linear matroid intersection and graph matching can be extended nicely to more general settings. All algorithms are still faster than the existing algorithms even if fast matrix multiplication is not used. These provide simple algorithms that can be easily implemented in practice.

Let φ(G) be the minimum conductance of an undirected graph G, and let 0=λ1 ≤ λ2 ≤ ... ≤ λn ≤ 2 be the eigenvalues of the normalized Laplacian matrix of G. We prove that for any graph G and any k ≥ 2, [φ(G) = O(k) l2/√lk,] and this performance guarantee is achieved by the spectral partitioning algorithm. This improves Cheeger's inequality, and the bound is optimal up to a constant factor for any $k$. Our result shows that the spectral partitioning algorithm is a constant factor approximation algorithm for finding a sparse cut if lk is a constant for some constant k. This provides some theoretical justification to its empirical performance in image segmentation and clustering problems. We extend the analysis to spectral algorithms for other graph partitioning problems, including multi-way partition, balanced separator, and maximum cut.

We consider the problem of computing the rank of an m × n matrix A over a field. We present a randomized algorithm to find a set of r = rank( A ) linearly independent columns in Õ (| A | + r ω ) field operations, where | A | denotes the number of nonzero entries in A and ω < 2.38 is the matrix multiplication exponent. Previously the best known algorithm to find a set of r linearly independent columns is by Gaussian elimination, with deterministic running time O ( mnr ω-2 ). Our algorithm is faster when r < max{ m , n }, for instance when the matrix is rectangular. We also consider the problem of computing the rank of a matrix dynamically, supporting the operations of rank one updates and additions and deletions of rows and columns. We present an algorithm that updates the rank in Õ ( mn ) field operations. We show that these algorithms can be used to obtain faster algorithms for various problems in exact linear algebra, combinatorial optimization and dynamic data structure.

We present a new algebraic formulation for computing edge connectivities in a directed graph, using ideas developed in network coding. This reduces the problem of computing edge connectivities to solving systems of linear equations, thus allowing us to use tools in linear algebra to design new algorithms. Using the algebraic formulation, we obtain faster algorithms for computing single source edge connectivities and all pairs edge connectivities. In some settings, the amortized time to compute the edge connectivity for one pair is sublinear. Through this connection, we have also found an interesting use of expanders and superconcentrators to design fast algorithms for some graph connectivity problems.

In the minimum bounded degree Steiner network problem, we are given an undirected graph with an edge cost for each edge, a connectivity requirement $r_{uv}$ for each pair of vertices $u$ and $v$, and a degree upper bound $b_v$ for each vertex $v$. The task is to find a minimum cost subgraph that satisfies all the connectivity requirements and degree upper bounds. Let $r_{\max}:=\max_{u,v} \{r_{uv}\}$ and ${\sc opt}$ be the cost of an optimal solution that satisfies all the degree bounds. We present approximation algorithms that minimize the total cost and the degree violation simultaneously. In the special case when $r_{\max}=1$, there is a polynomial time algorithm that returns a Steiner forest of cost at most $2{\sc opt}$ and the degree of each vertex $v$ is at most $b_v+3$. In the general case, there is a polynomial time algorithm that returns a Steiner network of cost at most $2{\sc opt}$ and the degree of each vertex $v$ is at most $b_v+6r_{\max}+3$. The algorithms are based on the iterative relaxation method, and the analysis of the algorithms is nearly tight.

We present new edge splitting-off results maintaining all-pairs edge-connectivities of an undirected graph. We first give an alternate proof of Mader's theorem, and use it to obtain a deterministic $\tilde{O}(m + {r_{\max}}^2 \cdot n^2)$-time complete edge splitting-off algorithm for unweighted graphs, where $r_{\max}$ denotes the maximum edge-connectivity requirement. This improves upon the best known algorithm by Gabow by a factor of $\tilde{\Omega}(n)$. We then prove a new structural property, and use it to further speed up the algorithm to obtain a randomized $\tilde{O}(m + {r_{\max}}^3 \cdot n)$-time algorithm. These edge splitting-off algorithms can be used directly to speed up various graph algorithms.

We consider the problem of computing the rank of an mxn matrix A over a field. We present a randomized algorithm to find a set of r = rank(A) linearly independent columns in O(|A| + rw) field operations, where |A| denotes the number of nonzero entries in A and w < 2.38 is the matrix multiplication exponent. Previously the best known algorithm to find a set of r linearly independent columns is by Gaussian elimination, with running time O(mnrw). Our algorithm is faster when r < max{m,n}, for instance when the matrix is rectangular. We also consider the problem of computing the rank of a matrix dynamically, supporting the operations of rank one updates and additions and deletions of rows and columns. We present an algorithm that updates the rank in O(mn) field operations. We show that these algorithms can be used to obtain faster algorithms for various problems in numerical linear algebra, combinatorial optimization and dynamic data structure.

We present a new algebraic formulation to compute edge connectivities in a directed graph, using the ideas developed in network coding. This reduces the problem of computing edge connectivities to solving systems of linear equations, thus allowing us to use tools in linear algebra to design new algorithms. Using the algebraic formulation we obtain faster algorithms for computing single source edge connectivities and all pairs edge connectivities, in some settings the amortized time to compute the edge connectivity for one pair is sub linear. Through this connection, we have also found an interesting use of expanders and super concentrators to design fast algorithms for some graph connectivity problems.

We present fast and simple algebraic algorithms for the linear matroid parity problem and its applications. For the linear matroid parity problem, we obtain a simple randomized algorithm with running time O(mrω–1) where m and r are the number of columns and the number of rows and ω ≈ 2.376 is the matrix multiplication exponent. This improves the O(mrω)-time algorithm by Gabow and Stallmann, and matches the running time of the algebraic algorithm for linear matroid intersection, answering a question of Harvey. We also present a very simple alternative algorithm with running time O(mr2) which does not need fast matrix multiplication. We further improve the algebraic algorithms for some specific graph problems of interest. For the Mader's disjoint S-path problem, we present an O(nω)-time randomized algorithm where n is the number of vertices. This improves the running time of the existing results considerably, and matches the running time of the algebraic algorithms for graph matching. For the graphic matroid parity problem, we give an O(n4)-time randomized algorithm where n is the number of vertices, and an O(n3)-time randomized algorithm for a special case useful in designing approximation algorithms. These algorithms are optimal in terms of n as the input size could be Ω(n4) and Ω(n3) respectively. The techniques are based on the algebraic algorithmic framework developed by Mucha and Sankowski, Harvey, and Sankowski. While linear matroid parity and Mader's disjoint S-path are challenging generalizations for the design of combinatorial algorithms, our results show that both the algebraic algorithms for linear matroid intersection and graph matching can be extended nicely to more general settings. All algorithms are still faster than the existing algorithms even if fast matrix multiplication is not used. These provide simple algorithms that can be easily implemented in practice.

Given a complete undirected graph, a cost function on edges, and a degree bound B, the degree bounded network design problem is to find a minimum cost simple subgraph with maximum degree B satisfying given connectivity requirements. Even for a simple connectivity requirement such as finding a spanning tree, computing a feasible solution for the degree bounded network design problem is already NP-hard, and thus there is no polynomial factor approximation algorithm for this problem. In this paper, we show that when the cost function satisfies the triangle inequality, there are constant factor approximation algorithms for various degree bounded network design problems. In global edge-connectivity, there is a $(2+\frac{1}{k})$-approximation algorithm for the minimum bounded degree k-edge-connected subgraph problem. In local edge-connectivity, there is a 4-approximation algorithm for the minimum bounded degree Steiner network problem when $r_{\max}$ is even, and a 5.5-approximation algorithm when $r_{\max}$ is odd, where $r_{\max}$ is the maximum connectivity requirement. In global vertex-connectivity, there is a $(2+\frac{k-1}{n}+\frac{1}{k})$-approximation algorithm for the minimum bounded degree k-vertex-connected subgraph problem when $n\geq2k$, where n is the number of vertices. For spanning tree, there is a $(1+\frac{1}{B-1})$-approximation algorithm for the minimum bounded degree spanning tree problem. These approximation algorithms return solutions with the smallest possible maximum degree, and in most cases the cost guarantee is obtained by comparing to the optimal cost when there are no degree constraints. This demonstrates that degree constraints can be incorporated into network design problems with metric costs. Our algorithms can be seen as a generalization of Christofides' algorithm for the metric traveling salesman problem. The main technical tool is a simplicity-preserving edge splitting-off operation, which is used to “short-cut” vertices with high degree while maintaining connectivity requirements and preserving simplicity of the solutions.

We prove two results on packing common bases of two matroids. First, we show that the computational problem of common base packing reduces to the special case where one of the matroids is a direct sum of uniform matroids. Second, we give a counterexample to a conjecture of Chow, which proposed a sufficient condition for the existence of a common base packing. Chow's conjecture is a generalization of Rota's basis conjecture.

The hypergraph matching problem is to find a largest collection of disjoint hyperedges in a hypergraph. This is a well-studied problem in combinatorial optimization and graph theory with various applications. The best known approximation algorithms for this problem are all local search algorithms. In this paper we analyze different linear and semidefinite programming relaxations for the hypergraph matching problem, and study their connections to the local search method. Our main results are the following: We consider the standard linear programming relaxation of the problem. We provide an algorithmic proof of a result of Füredi, Kahn and Seymour, showing that the integrality gap is exactly k – 1 + 1/k for k-uniform hypergraphs, and is exactly k – 1 for k-partite hypergraphs. This yields an improved approximation algorithm for the weighted 3-dimensional matching problem. Our algorithm combines the use of the iterative rounding method and the fractional local ratio method, showing a new way to round linear programming solutions for packing problems. We study the strengthening of the standard LP relaxation by local constraints. We show that, even after linear number of rounds of the Sherali-Adams lift-and-project procedure on the standard LP relaxation, there are k-uniform hypergraphs with integrality gap at least k – 2. On the other hand, we prove that for every constant k, there is a strengthening of the standard LP relaxation by only a polynomial number of constraints, with integrality gap at most (k + 1)/2 for k-uniform hypergraphs. The construction uses a result in extremal combinatorics. We consider the standard semidefinite programming relaxation of the problem. We prove that the Lovász ϑ-function provides an SDP relaxation with integrality gap at most (k + 1)/2. The proof gives an indirect way (not by a rounding algorithm) to bound the ratio between any local optimal solution and any optimal SDP solution. This shows a new connection between local search and linear and semidefinite programming relaxations.

Graph $G$ is the square of graph $H$ if two vertices $x,y$ have an edge in $G$ if and only if $x,y$ are of distance at most two in $H$. Given $H$ it is easy to compute its square $H^2$, however Motwani and Sudan proved that it is NP-complete to determine if a given graph $G$ is the square of some graph $H$ (of girth $3$). In this paper we consider the characterization and recognition problems of graphs that are squares of graphs of small girth, i.e. to determine if $G=H^2$ for some graph $H$ of small girth. The main results are the following. \begin{itemize} \item There is a graph theoretical characterization for graphs that are squares of some graph of girth at least $7$. A corollary is that if a graph $G$ has a square root $H$ of girth at least $7$ then $H$ is unique up to isomorphism. \item There is a polynomial time algorithm to recognize if $G=H^2$ for some graph $H$ of girth at least $6$. \item It is NP-complete to recognize if $G=H^2$ for some graph $H$ of girth $4$. \end{itemize} These results almost provide a dichotomy theorem for the complexity of the recognition problem in terms of girth of the square roots. The algorithmic and graph theoretical results generalize previous results on tree square roots, and provide polynomial time algorithms to compute a graph square root of small girth if it exists. Some open questions and conjectures will also be discussed.

We present algorithmic and hardness results for network design problems with degree or order constraints. The first problem we consider is the Survivable Network Design problem with degree constraints on vertices. The objective is to find a minimum cost subgraph which satisfies connectivity requirements between vertices and also degree upper bounds $B_v$ on the vertices. This includes the well-studied Minimum Bounded Degree Spanning Tree problem as a special case. Our main result is a $(2,2B_v+3)$-approximation algorithm for the edge-connectivity Survivable Network Design problem with degree constraints, where the cost of the returned solution is at most twice the cost of an optimum solution (satisfying the degree bounds) and the degree of each vertex v is at most $2B_v+3$. This implies the first constant factor (bicriteria) approximation algorithms for many degree constrained network design problems, including the Minimum Bounded Degree Steiner Forest problem. Our results also extend to directed graphs and provide the first constant factor (bicriteria) approximation algorithms for the Minimum Bounded Degree Arborescence problem and the Minimum Bounded Degree Strongly k-Edge-Connected Subgraph problem. In contrast, we show that the vertex-connectivity Survivable Network Design problem with degree constraints is hard to approximate, even when the cost of every edge is zero. A striking aspect of our algorithmic result is its simplicity. It is based on the iterative relaxation method, which is an extension of Jain's iterative rounding method. This provides an elegant and unifying algorithmic framework for a broad range of network design problems. We also study the problem of finding a minimum cost $\lambda$-edge-connected subgraph with at least k vertices, which we call the $(k,\lambda)$-subgraph problem. This generalizes some well-studied classical problems such as the k-MST and the minimum cost $\lambda$-edge-connected subgraph problems. We give a polylogarithmic approximation for the $(k,2)$-subgraph problem. However, by relating it to the Densest k-Subgraph problem, we provide evidence that the $(k,\lambda)$-subgraph problem might be hard to approximate for arbitrary $\lambda$.

Recent research in network coding shows that, joint consideration of both coding and routing strategies may lead to higher information transmission rates than routing only. A fundamental question in the field of network coding is: how large can the throughput improvement due to network coding be? In this paper, we prove that in undirected networks, the ratio of achievable multicast throughput with network coding to that without network coding is bounded by a constant ratio of2, i.e., network coding can at most double the throughput. This result holds for any undirected network topology, any link capacity configuration, any multicast group size, and any source information rate. This constant bound2represents the tightest bound that has been proved so far in general undirected settings, and is to be contrasted with the unbounded potential of network coding in improving multicast throughput in directed networks.

Given a complete undirected graph, a cost function on edges and a degree bound B, the degree bounded network design problem is to find a minimum cost simple subgraph with maximum degree B satisfying given connectivity requirements. Even for simple connectivity requirement such as finding a spanning tree, computing a feasible solution for the degree bounded network design problem is already NP-hard, and thus there is no polynomial factor approximation algorithm for this problem. In this paper, we show that when the cost function satisfies triangle inequalities, there are constant factor approximation algorithms for various degree bounded network design problems.Global edge-connectivity: There is a (2+1/k)-approximation algorithm for the minimum bounded degree k-edge-connected subgraph problem. Local edge-connectivity: There is a 6-approximation algorithm for the minimum bounded degree Steiner network problem. Global vertex-connectivity: there is a (2+(k-1)/n+1/k)-approximation algorithm for the minimum bounded degree k-vertex-connected subgraph problem. Spanning tree: there is an (1+1/(d-1))-approximation algorithm for the minimum bounded degree spanning tree problem. These approximation algorithms return solutions with smallest possible maximum degree, and the cost guarantee is obtained by comparing to the optimal cost when there are no degree constraints. This demonstrates that degree constraints can be incorporated into network design problems with metric costs.Our algorithms can be seen as a generalization of Christofides' algorithm for metric TSP. The main technical tool is a simplicity-preserving edge splitting-off operation, which is used to "short-cut" vertices with high degree while maintaining connectivity requirements and preserving simplicity of the solutions.

We study a general network design problem with additional degree constraints. Given connectivity requirements ruv for all pairs of vertices, a Steiner network is a graph in which there are at least ruv edge-disjoint paths between u and v for all pairs of vertices u,v. In the MINIMUM BOUNDED-DEGREE STEINER NETWORK problem, we are given an undirected graph G with an edge cost for each edge, a connectivity requirement ruv for each pair of vertices u and v, and a degree upper bound for each vertex v. The task is to find a minimum cost Steiner network which satisfies all the degree upper bounds.

We present algorithmic and hardness results for network design problems with degree or order constraints. The first problem we consider is the Survivable Network Design problem with degree constraints on vertices. The objective is to find a minimum cost subgraph which satisfies connectivity requirements between vertices and also degree upper bounds Bv on the vertices. This includes the well-studied Minimum Bounded Degree Spanning Tree problem as a special case. Our main result is a (2, 2Bv +3)-approximation algorithm for the edge-connectivity Survivable Network Design problem with degree constraints, where the cost of the returned solution is at most twice the cost of an optimum solution (satisfying the degree bounds) and the degree of each vertex v is at most 2Bv + 3. This implies the first constant factor (bicriteria) approximation algorithms for many degree constrained network design problems, including the Minimum Bounded Degree Steiner Forest problem. Our results also extend to directed graphs and provide the first constant factor (bicriteria) approximation algorithms for the Minimum Bounded Degree Arborescence problem and the Minimum Bounded Degree Strongly k-Edge-Connected Subgraph problem. In contrast, we show that the vertex-connectivity Survivable Network Design problem with degree constraints is hard to approximate, even when the cost of every edge is zero. A striking aspect of our algorithmic

In the Minimum Bounded Degree Spanning Tree problem, we aregiven an undirected graph with a degree upper bound Bv on eachvertex v, and the task is to find a spanning tree of minimumcost which satisfies all the degree bounds. Let OPT be the costof an optimal solution to this problem. In this paper, we presenta polynomial time algorithm which returns a spanning tree T ofcost at most OPT and dT(v) ≤ Bv+1 for all v, where dT(v) denotes the degree of v in T. This generalizes aresult of Furer and Raghavachari [8] to weighted graphs, andsettles a 15-year-old conjecture of Goemans [10] affirmatively. The algorithm generalizes when each vertex v hasa degree lower bound Av and a degree upper bound Bv, andreturns a spanning tree with cost at most OPT and Av - 1 ≤dT(v) ≤ Bv + 1 for all v. This is essentially the bestpossible. The main technique used is an extension of the iterativerounding method introduced by Jain [12] for the design ofapproximation algorithms.

Given an undirected hypergraph and a subset of vertices S sube V with a specified root vertex r isin S, the Steiner rooted-orientation problem is to find an orientation of all the hyperedges so that in the resulting directed hypergraph the "connectivity" from the root r to the vertices in S is maximized. This is motivated by a multicasting problem in undirected networks as well as a generalization of some classical problems in graph theory. The main results of this paper are the following approximate min-max relations: middot Given an undirected hypergraph H, if S is 2k-hyperedge-connected in H, then H has a Steiner rooted k-hyperarc-connected orientation. middot Given an undirected graph G, if S is 2k-element-connected in G, then G has a Steiner rooted k-element-connected orientation. Both results are tight in terms of the connectivity bounds. These also give polynomial time constant factor approximation algorithms for both problems. The proofs are based on submodular techniques, and a graph decomposition technique used in the Steiner tree packing problem. Some complementary hardness results are presented at the end

Graph H is a root of graph G if there exists a positive integer k such that x and y are adjacent in G if and only if their distance in H is at most k . Motwani and Sudan [1994] proved the NP-completeness of graph square recognition and conjectured that it is also NP-complete to recognize squares of bipartite graphs. The main result of this article is to show that squares of bipartite graphs can be recognized in polynomial time. In fact, we give a polynomial-time algorithm to count the number of different bipartite square roots of a graph, although this number could be exponential in the size of the input graph. By using the ideas developed, we are able to give a new and simpler linear-time algorithm to recognize squares of trees and a new algorithmic proof that tree square roots are unique up to isomorphism. Finally, we prove the NP-completeness of recognizing cubes of bipartite graphs.

The transmission of information within a data network is constrained by the network topology and link capacities. In this paper, we study the fundamental upper bound of information dissemination rates with these constraints in undirected networks, given the unique replicable and encodable properties of information flows. Based on recent advances in network coding and classical modeling techniques in flow networks, we provide a natural linear programming formulation of the maximum multicast rate problem. By applying Lagrangian relaxation on the primal and the dual linear programs (LPs), respectively, we derive a) a necessary and sufficient condition characterizing multicast rate feasibility, and b) an efficient and distributed subgradient algorithm for computing the maximum multicast rate. We also extend our discussions to multiple communication sessions, as well as to overlay and ad hoc network models. Both our theoretical and simulation results conclude that, network coding may not be instrumental to achieve better maximum multicast rates in most cases; rather, it facilitates the design of significantly more efficient algorithms to achieve such optimality.

With the constraints of network topologies and link capacities, achieving the optimal end-to-end throughput in data networks has been known as a fundamental but computationally hard problem. In this paper, we seek efficient solutions to the problem of achieving optimal throughput in data networks, with single or multiple unicast, multicast and broadcast sessions. Although previous approaches lead to solving NP-complete problems, we show the surprising result that, facilitated by the recent advances of network coding, computing the strategies to achieve the optimal end-to-end throughput can be performed in polynomial time. This result holds for one or more communication sessions, as well as in the overlay network model. Supported by empirical studies, we present the surprising observation that in most topologies, applying network coding may not improve the achievable optimal throughput; rather, it facilitates the design of significantly more efficient algorithms to achieve such optimality.

Given an undirected multigraph G and a subset of vertices S /spl sube/ V(G), the Steiner tree packing problem is to find a largest collection of edge-disjoint trees that each connects S. This problem and its generalizations have attracted considerable attention from researchers in different areas because of their wide applicability. This problem was shown to be APX-hard (no polynomial time approximation scheme unless P=NP). In fact, prior to this paper, not even an approximation algorithm with asymptotic ratio o(n) was known despite several attempts. In this work, we close this huge gap by presenting the first polynomial time constant factor approximation algorithm for the Steiner tree packing problem. The main theorem is an approximate min-max relation between the maximum number of edge-disjoint trees that each connects S (i.e. S-trees) and the minimum size of an edge-cut that disconnects some pair of vertices in S (i.e. S-cut). Specifically, we prove that if the minimum S-cut in G has 26k edges, then G has at least k edge-disjoint S-trees; this answers Kriesell's conjecture affirmatively up to a constant multiple. The techniques that we use are purely combinatorial, where matroid theory is the underlying ground work.

In this paper, we study the complexity of recognizing powers of chordal graphs and its subclasses. We present the first polynomial time algorithm to recognize squares of proper interval graphs and give an outline of an algorithm to recognize kth powers of proper interval graphs for every natural number k. These are the first results of this type for a family of graphs that contains arbitrarily large cliques. On the other hand, we show the NP-completeness of recognizing squares of chordal graphs, recognizing squares of split graphs, and recognizing chordal graphs that are squares of some graph.