Reconfiguration problems arise when we wish to find a stepby-step transformation between two feasible solutions of a problem such that all intermediate results are also feasible. We demonstrate that a host of reconfiguration problems derived from NP-complete problems are PSPACE-complete, while some are also NP-hard to approximate. In contrast, several reconfiguration versions of problems in P are solvable in polynomial time.

Abstract. Submodular functions are discrete functions that model laws of diminishing returns and enjoy numerous algorithmic applications that have been used in many areas, including combinatorial optimization, machine learning, and economics. In this work we use a learning theoretic angle for studying submodular functions. We provide algorithms for learning submodular functions, as well as lower bounds on their learnability. In doing so, we uncover several novel structural results revealing both extremal properties as well as regularities of submodular functions, of interest to many areas.Submodular functions are a discrete analog of convex functions that enjoy numerous applications and have structural properties that can be exploited algorithmically. They arise naturally in the study of graphs, matroids, covering problems, facility location problems, etc., and they have been extensively studied in operations research and combinatorial optimization for many years [8]. More recently submodular functions have become key concepts both in the machine learning and algorithmic game theory communities. For example, submodular functions have been used to model bidders' valuation functions in combinatorial auctions [12,6,3,14], and for solving feature selection problems in graphical models [11] or for solving various clustering problems [13]. In fact, submodularity has been the topic of several tutorials and workshops at recent major conferences in machine learning [1,9,10,2].Despite the increased interest on submodularity in machine learning, little is known about the topic from a learning theory perspective. In this work, we provide a statistical and computational theory of learning submodular functions in a distributional learning setting.Our study has multiple motivations. From a foundational perspective, submodular functions are a powerful, broad class of important functions, so studying their learnability allows us to understand their structure in a new way. To draw a parallel to the Boolean-valued case, a class of comparable breadth would be the class of monotone Boolean functions; the learnability of such functions has been intensively studied [4,5]. From an applications perspective, algorithms for learning submodular functions may be useful in some of the applications where these functions arise. For example, in the context of algorithmic game theory This note summarizes several results in the paper "Learning Submodular Functions", by Maria Florina Balcan and Nicholas Harvey, which appeared The 43rd ACM Symposium on Theory of Computing (STOC) 2011.

Submodular functions are a key concept in combinatorial optimization. Algorithms that involve submodular functions usually assume that they are given by a (value) oracle. Many interesting problems involving submodular functions can be solved using only polynomially many queries to the oracle, e.g., exact minimization or approximate maximization.In this paper, we consider the problem of approximating a non-negative, monotone, submodular function f on a ground set of size n everywhere, after only poly(n) oracle queries. Our main result is a deterministic algorithm that makes poly(n) oracle queries and derives a functionf such that, for every set S,f (S) approximates f (S) within a factor α(n), where α(n) = √ n + 1 for rank functions of matroids and α(n) = O( √ n log n) for general monotone submodular functions. Our result is based on approximately finding a maximum volume inscribed ellipsoid in a symmetrized polymatroid, and the analysis involves various properties of submodular functions and polymatroids.Our algorithm is tight up to logarithmic factors. Indeed, we show that no algorithm can achieve a factor better than Ω( √ n/ log n), even for rank functions of a matroid.

Abstract-An outer bound on the rate region of noise-free information networks is given. This outer bound combines properties of entropy with a strong information inequality derived from the structure of the network. This blend of information theoretic and graph theoretic arguments generates many interesting results. For example, the capacity of directed cycles is characterized. Also, a gap between the sparsity of an undirected graph and its capacity is shown. Extending this result, it is shown that multicommodity flow solutions achieve the capacity in an infinite class of undirected graphs, thereby making progress on a conjecture of Li and Li. This result is in sharp contrast to the situation with directed graphs, where a family of graphs are presented in which the gap between the capacity and the rate achievable using multicommodity flows is linear in the size of the graph.

No abstract

No abstract

No abstract

We present new algebraic approaches for two well-known combinatorial problems: non-bipartite matching and matroid intersection. Our work yields new randomized algorithms that exceed or match the efficiency of existing algorithms. For nonbipartite matching, we obtain a simple, purely algebraic algorithm with running time O(n ω ) where n is the number of vertices and ω is the matrix multiplication exponent. This resolves the central open problem of Mucha and Sankowski (2004). For matroid intersection, our algorithm has running time O(nr ω−1 ) for matroids with n elements and rank r that satisfy some natural conditions.