In this paper, we introduce a new, spectral notion of approximation between directed graphs, which we call singular value (SV) approximation. SV-approximation is stronger than previous notions of spectral approximation considered in the literature, including spectral approximation of Laplacians for undirected graphs [ST04], standard approximation for directed graphs [CKP+17], and unit-circle (UC) approximation for directed graphs [AKM+20]. Further, SV approximation enjoys several useful properties not possessed by previous notions of approximation, e.g., it is preserved under products of randomwalk matrices and bounded matrices. We provide a nearly linear-time algorithm for SV-sparsifying (and hence UC-sparsifying) Eulerian directed graphs, as well as $\ell$-step random walks on such graphs, for any $\ell \leq \operatorname{poly}(n)$. Combined with the Eulerian scaling algorithms of [CKK+18], given an arbitrary (not necessarily Eulerian) directed graph and a set S of vertices, we can approximate the stationary probability mass of the $\left(S, S^{c}\right)$ cut in an $\ell$-step random walk to within a multiplicative error of $1 / \operatorname{polylog}(n)$ and an additive error of $1 / \operatorname{poly}(n)$ in nearly linear time. As a starting point for these results, we provide a simple black-box reduction from SV-sparsifying Eulerian directed graphs to SV-sparsifying undirected graphs; such a directed-to-undirected reduction was not known for previous notions of spectral approximation.

In this paper, we provide a deterministic ~O(log N)-space algorithm for estimating random walk probabilities on undirected graphs, and more generally Eulerian directed graphs, to within inverse polynomial additive error (ε = 1/poly(N)) where N is the length of the input. Previously, this problem was known to be solvable by a randomized algorithm using space O(log N) (following Aleliunas et al., FOCS '79) and by a deterministic algorithm using space O(log3/2N) (Saks and Zhou, FOCS '95 and JCSS '99), both of which held for arbitrary directed graphs but had not been improved even for undirected graphs. We also give improvements on the space complexity of both of these previous algorithms for non-Eulerian directed graphs when the error is negligible (ε = 1/Nω(1)), generalizing what Hoza and Zuckerman (FOCS '18) recently showed for the special case of distinguishing whether a random walk probability is 0 or greater than ε. We achieve these results by giving new reductions between powering Eulerian random-walk matrices and inverting Eulerian Laplacian matrices, providing a new notion of spectral approximation for Eulerian graphs that is preserved under powering, and giving the first deterministic ~O(log N)-space algorithm for inverting Eulerian Laplacian matrices. The latter algorithm builds on the work of Murtagh et al. (FOCS '17) that gave a deterministic ~O(log N)-space algorithm for inverting undirected Laplacian matrices, and the work of Cohen et al. (FOCS '19) that gave a randomized ~O(N)-time algorithm for inverting Eulerian Laplacian matrices. A running theme throughout these contributions is an analysis of “cycle-lifted graphs,” where we take a graph and “lift” it to a new graph whose adjacency matrix is the tensor product of the original adjacency matrix and a directed cycle (or variants of one).

In this paper we provide nearly linear time algorithms for several problems closely associated with the classic Perron-Frobenius theorem, including computing Perron vectors, i.e. entrywise non-negative eigenvectors of non-negative matrices, and solving linear systems in asymmetric M-matrices, a generalization of Laplacian systems. The running times of our algorithms depend nearly linearly on the input size and polylogarithmically on the desired accuracy and problem condition number. Leveraging these results we also provide improved running times for a broader range of problems including computing random walk-based graph kernels, computing Katz centrality, and more. The running times of our algorithms improve upon previously known results which either depended polynomially on the condition number of the problem, required quadratic time, or only applied to special cases. We obtain these results by providing new iterative methods for reducing these problems to solving linear systems in Row-Column Diagonally Dominant (RCDD) matrices. Our methods are related to the classic shift-and-invert preconditioning technique for eigenvector computation and constitute the first alternative to the result in Cohen et al. (2016) for reducing stationary distribution computation and solving directed Laplacian systems to solving RCDD systems.

Motivated by the bus escape routing problem in printed circuit boards (PCBs), we study the following optimization problem: given a set of rectangles attached to the boundary of a rectangular region, find a subset of nonoverlapping rectangles with maximum total weight. We present an efficient algorithm that solves this problem optimally in O(n4) time, where n is the number of rectangles in the input instance. This improves over the best previous O(n6)-time algorithm available for the problem. We also present two efficient approximation algorithms for the problem that find near-optimal solutions with guaranteed approximation factors. The first algorithm finds a 2-approximate solution in O(n2) time, and the second one computes a 4/3-approximation in O(n3) time. The experimental results demonstrate the efficiency of both our exact and approximation algorithms.

We study the problem of computing Nash equilibria of zero-sum games.Many natural zero-sum games have exponentially many strategies, but highly structured payoffs. For example, in the well-studied Colonel Blotto game (introduced by Borel in 1921), players must divide a pool of troops among a set of battlefields with the goal of winning (i.e., having more troops in) a majority. The Colonel Blotto game is commonly used for analyzing a wide range of applications from the U.S presidential election, to innovative technology competitions, toadvertisement, to sports.However, because of the size of the strategy space, standard methods for computing equilibria of zero-sum games fail to be computationally feasible.Indeed, despite its importance, only few solutions for special variants of the problem are known. In this paper we show how to compute equilibria of Colonel Blotto games. Moreover, our approach takes the form of a general reduction: to find a Nash equilibrium of a zero-sum game, it suffices to design a separation oracle for the strategy polytope of any bilinear game that is payoff-equivalent. We then apply this technique to obtain the first polytime algorithms for a variety of games. In addition to Colonel Blotto, we also show how to compute equilibria in an infinite-strategy variant called the General Lotto game; this involves showing how to prune the strategy space to a finite subset before applying our reduction. We also consider the class of dueling games, first introduced by Immorlica et al. (2011). We show that our approach provably extends the class of dueling games for which equilibria can be computed: we introduce a new dueling game, the matching duel, on which prior methods fail to be computationally feasible but upon which our reduction can be applied.

People make decisions and express their opinions according to their communities. A natural idea for controlling the diffusion of a behavior is to find influential people, and employ them to spread a desired behavior. We investigate an influencing problem when individuals' behaviors are affected by their friends in an opinion formation process. Our goal is to design efficient algorithms for finding opinion leaders such that changing their opinions has a great impact on the overall external behaviors in the society. We study directed social networks and define a set of problems like maximizing the sum of individuals' behaviors or maximizing the number of individuals whose external behaviors are above a threshold. We discuss the complexity of the defined problems and design polynomial-time optimum algorithms for the non NP-hard variants of them. We also propose polynomial-time approximation algorithms with guaranteed performances and prove inapproximability results for the NP-hard variants of these problems. Furthermore, we run simulations on real-world social networks and show our proposed algorithm outperforms the classical algorithms such as degree-based algorithm, closeness-based algorithm, and pagerank-based algorithm.