We present a new algorithm for generating a uniformly random spanning tree in an undirected graph. Our algorithm samples such a tree in expected O(m At a high level, our result stems from carefully exploiting the interplay of random spanning trees, random walks, and the notion of effective resistance, as well as from devising a way to algorithmically relate these concepts to the combinatorial structure of the graph. This involves, in particular, establishing a new connection between the effective resistance metric and the cut structure of the underlying graph.

In this paper we study dynamics inspired by Physarum polycephalum (a slime mold) for solving linear programs ([NYT00, IJNT11, JZ12]). These dynamics are arrived at by a local and mechanistic interpretation of the inner workings of the slime mold and a global optimization perspective has been lacking even in the simplest of instances. Our first result is an interpretation of the dynamics as an optimization process. We show that Physarum dynamics can be seen as a steepest-descent type algorithm on a certain Riemannian manifold. Moreover, we prove that the trajectories of Physarum are in fact paths of optimizers to a parametrized family of convex programs, in which the objective is a linear cost function regularized by an entropy barrier. Subsequently, we rigorously establish several important properties of solution curves of Physarum. We prove global existence of such solutions and show that they have limits, being optimal solutions of the underlying LP. Finally, we show that the discretization of the Physarum dynamics is efficient for a class of linear programs, which include unimodular constraint matrices. Thus, together, our results shed some light on how nature might be solving instances of perhaps the most complex problem in P: linear programming. ContentsDamian Straszak,École Polytechnique Fédérale de Lausanne (EPFL). Nisheeth K. Vishnoi,École Polytechnique Fédérale de Lausanne (EPFL). N p = O(ε). Moreover we can write x J = y J + τ J , where τ J < ε. This gives a decomposition of W J into W Pickp: any optimal solution to the dual of the linear program (1). From complementary slackness we have Y (A ⊤p − c) = 0. In particular A ⊤ Jp = c J . We obtain: A We want to show thatit is a symmetric PSD matrix. Moreover the kernels of A ⊤ J and K coincide, since y J > 0. Pick a vector v ∈ R m so that A ⊤ J (p −p) = A ⊤ J v and v is orthogonal to the kernel of A ⊤ J . (In other words v is the orthogonal projection of p −p onto the orthogonal complement of the kernel of A ⊤ J .) 23Using the fact that y i c i > d c i ≥ d ′ for i ∈ J and some absolute positive constant d ′ , we get:where denotes the PSD ordering. This implies in particular that d ′ λ + ≤ λ + K , where λ + , λ + K are the smallest positive eigenvalues of A J A ⊤ J and K respectively. Since v is orthogonal to the kernel of K and K is PSD we have:In consequence A ⊤ J (p −p) = A ⊤ J v = O(ε) and hence W J A ⊤ J (p −p) = O(ε), because y = O(1) (the set of optimal solution is bounded). Finally:

A great variety of fundamental optimization and counting problems arising in computer science, mathematics and physics can be reduced to one of the following computational tasks involving polynomials and set systems: given an oracle access to an m-variate real polynomial g and to a family of (multi-)subsets B of [m], (1) find S ∈ B such that the monomial in g corresponding to S has the largest coefficient in g, or (2) compute the sum of coefficients of monomials in g corresponding to all the sets that appear in B. Special cases of these problems, such as computing permanents and mixed discriminants, sampling from determinantal point processes, and maximizing subdeterminants with combinatorial constraints have been topics of much recent interest in theoretical computer science.In this paper we present a very general convex programming framework geared to solve both of these problems. Subsequently, we show that roughly, when g is a real stable polynomial with non-negative coefficients and B is a matroid, the integrality gap of our convex relaxation is finite and depends only on m (and not on the coefficients of g) -in fact, in most interesting cases it is never worse than e m . Prior to our work, such results were known only in important but sporadic cases that relied heavily on the structure of either g or B; it was not even a priori clear if one could formulate a convex relaxation that has a finite integrality gap beyond these special cases. Two notable examples are a result by Gurvits [Gur06] on the van der Waerden conjecture for all real stable g when B contains one element, and a result by Nikolov and Singh [NS16] for a family of multilinear real stable polynomials when B is the partition matroid. Our work, which encapsulates almost all interesting cases of g and B, benefits from both -we were inspired by the latter in coming up with the right convex programming relaxation and the former in deriving the integrality gap. However, proving our results requires significant extensions of both; in that process we come up with new notions and connections between real stable polynomials and matroids which should be of independent and wide interest.

Ranking algorithms are deployed widely to order a set of items in applications such as search engines, news feeds, and recommendation systems. Recent studies, however, have shown that, left unchecked, the output of ranking algorithms can result in decreased diversity in the type of content presented, promote stereotypes, and polarize opinions. In order to address such issues, we study the following variant of the traditional ranking problem when, in addition, there are fairness or diversity constraints. Given a collection of items along with 1) the value of placing an item in a particular position in the ranking, 2) the collection of sensitive attributes (such as gender, race, political opinion) of each item and 3) a collection of fairness constraints that, for each k, bound the number of items with each attribute that are allowed to appear in the top k positions of the ranking, the goal is to output a ranking that maximizes the value with respect to the original rank quality metric while respecting the constraints. This problem encapsulates various well-studied problems related to bipartite and hypergraph matching as special cases and turns out to be hard to approximate even with simple constraints. Our main technical contributions are fast exact and approximation algorithms along with complementary hardness results that, together, come close to settling the approximability of this constrained ranking maximization problem. Unlike prior work on the approximability of constrained matching problems, our algorithm runs in linear time, even when the number of constraints is (polynomially) large, its approximation ratio does not depend on the number of constraints, and it produces solutions with small constraint violations. Our results rely on insights about the constrained matching problem when the objective function satisfies certain properties that appear in common ranking metrics such as discounted cumulative gain (DCG), Spearman's rho or Bradley-Terry, along with the nested structure of fairness constraints.

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