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We prove a strengthening of the trickle down theorem for partite complexes. Given a (d + 1)-partite d-dimensional simplicial complex, we show that if "on average" the links of faces of co-dimension 2 are [EQUATION]-(one-sided) spectral expanders, then ...

We give a randomized 1+5.06/√k-approximation algorithm for the minimum k-edge connected spanning multi-subgraph problem, k-ECSM.

We prove tight mixing time bounds for natural random walks on bases of matroids, determinantal distributions, and more generally distributions associated with log-concave polynomials. For a matroid of rank k on a ground set of n elements, or more ...

For some > 10⁻³⁶ we give a randomized 3/2− approximation algorithm for metric TSP.

We say a probability distribution $\mu$ is spectrally independent if an associated pairwise influence matrix has a bounded largest eigenvalue for the distribution and all of its conditional distributions. We prove that if $\mu$ is spectrally independent, ...

We design a 1.49993-approximation algorithm for the metric traveling salesperson problem (TSP) for instances in which an optimal solution to the subtour linear programming relaxation is half-integral. These instances received significant attention over ...

We study a generalization of classical combinatorial graph spanners to the spectral setting. Given a set of vectors V ⊆ R^d, we say a set U ⊆ V is an α-spectral k-spanner, for k ≤ d, if for all v ∈ V there is a probability distribution μ_v supported on U ...

We study the bias of random bounded-degree polynomials over odd prime fields and show that, with probability exponentially close to 1, $n$-variate polynomials of degree $d$ over $\mathbb{F}_p$ have bias at most $p^{-\Omega(n/d)}$. This also yields an ...

We design an FPRAS to count the number of bases of any matroid given by an independent set oracle, and to estimate the partition function of the random cluster model of any matroid in the regime where 0<q<1. Consequently, we can sample random spanning ...

Stable matching is a classical combinatorial problem that has been the subject of intense theoretical and empirical study since its introduction in 1962 in a seminal paper by Gale and Shapley. In this paper, we provide a new upper bound on f(n), the ...

Recently Cole and Gkatzelis [10] gave the first constant factor approximation algorithm for the problem of allocating indivisible items to agents, under additive valuations, so as to maximize the Nash social welfare (NSW). We give constant factor ...

We study the problem of approximating the largest root of a real-rooted polynomial of degree n using its top k coefficients and give nearly matching upper and lower bounds. We present algorithms with running time polynomial in k that use the top k ...

We present a randomized O log n /log log n -approximation algorithm for the asymmetric traveling salesman problem ATSP. This provides the first asymptotic improvement over the long-standing Îï log n -approximation bound stemming from the work of ...

A polynomial pΕℝ[z₁,…,z_n] is real stable if it has no roots in the upper-half complex plane. Gurvits's permanent inequality gives a lower bound on the coefficient of the z₁z₂…z_n monomial of a real stable polynomial p with nonnegative coefficients. This ...

We initiate the study of approximating the largest induced expander in a given graph G. Given a Δ-regular graph G with n vertices, the goal is to find the set with the largest induced expansion of size at least δ · n. We design a bi-criteria ...

Spectral partitioning is a simple, nearly linear time algorithm to find sparse cuts, and the Cheeger inequalities provide a worst-case guarantee for the quality of the approximation found by the algorithm. A local graph partitioning algorithm finds a ...

We show that the integrality gap of the natural LP relaxation of the Asymmetric Traveling Salesman Problem is polyloglog(n). In other words, there is a polynomial time algorithm that approximates the value of the optimum tour within a factor of ...

A basic fact in spectral graph theory is that the number of connected components in an undirected graph is equal to the multiplicity of the eigenvalue zero in the Laplacian matrix of the graph. In particular, the graph is disconnected if and only if ...

Let G = (V, E) be an undirected graph, λ_k be the k^th smallest eigenvalue of the normalized laplacian matrix of G. There is a basic fact in algebraic graph theory that λ_k > 0 if and only if G has at most k -- 1 connected components. We prove a robust ...

We consider the online stochastic matching problem proposed by Feldman et al. [Feldman J, Mehta A, Mirrokni VS, Muthukrishnan S (2009) Online stochastic matching: Beating 1-1/e. Annual IEEE Sympos. Foundations Comput. Sci. 117--126] as a model of ...