In this paper, we disprove a conjecture of Goemans [23] and Linial [36] (also see [6,38]); namely, that every negative type metric embeds into ℓ 1 with constant distortion. We show that for an arbitrarily small constant δ > 0, for all large enough n, there is an n-point negative type metric which requires distortion at least (log log n) 1/6−δ to embed into ℓ 1 . Surprisingly, our construction is inspired by the Unique Games Conjecture (UGC) of Khot [28], establishing a previously unsuspected connection between probabilistically checkable proof systems (PCPs) and the theory of metric embeddings. We first prove that the UGC implies a super-constant hardness result for the (non-uniform) SPARSESTCUT problem. Though this hardness result relies on the UGC, we demonstrate, nevertheless, that the corresponding PCP reduction can be used to construct an "integrality gap instance" for SPARSESTCUT. Towards this, we first construct an integrality gap instance for a natural SDP relaxation of UNIQUEGAMES. Then we "simulate" the PCP reduction and "translate" the integrality gap instance of UNIQUEGAMES to an integrality gap instance of SPARSESTCUT. This enables us to prove a (log log n) 1/6−δ integrality gap for SPARSESTCUT, which is known to be equivalent to the metric embedding lower bound.

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We give a novel spectral approximation algorithm for the balanced separator problem that, given a graph G, a constant balance b ∈ (0, 1/2], and a parameter γ, either finds an Ω(b)balanced cut of conductance O( √ γ) in G, or outputs a certificate that all b-balanced cuts in G have conductance at least γ, and runs in timeÕ(m). This settles the question of designing asymptotically optimal spectral algorithms for balanced separator. Our algorithm relies on a variant of the heat kernel random walk and requires, as a subroutine, an algorithm to compute exp(−L)v where L is the Laplacian of a graph related to G and v is a vector. Algorithms for computing the matrix-exponential-vector product efficiently comprise our next set of results. Our main result here is a new algorithm which computes a good approximation to exp(−A)v for a class of symmetric positive semidefinite (PSD) matrices A and a given vector u, in time roughlyÕ(m A ), where m A is the number of non-zero entries of A. This uses, in a non-trivial way, the breakthrough result of Spielman and Teng on inverting symmetric and diagonally-dominant matrices inÕ(m A ) time. Finally, we prove that e −x can be uniformly approximated up to a small additive error, in a non-negative interval [a, b] with a polynomial of degree roughly √ b − a. While this result is of independent interest in approximation theory, we show that, via the Lanczos method from numerical analysis, it yields a simple algorithm to compute exp(−A)v for symmetric PSD matrices that runs in time roughly O(t A · A ), where t A is time required for the computation of the vector Aw for given vector w. As an application, we obtain a simple and practical algorithm, with output conductance O( √ γ), for balanced separator that runs in timeÕ( m / √ γ). This latter algorithm matches the running time, but improves on the approximation guarantee of the Evolving-Sets-based algorithm by Andersen and Peres for balanced separator.

In this paper, we disprove a conjecture of Goemans [23] and Linial [36] (also see [6,38]); namely, that every negative type metric embeds into ℓ 1 with constant distortion. We show that for an arbitrarily small constant δ > 0, for all large enough n, there is an n-point negative type metric which requires distortion at least (log log n) 1/6−δ to embed into ℓ 1 .Surprisingly, our construction is inspired by the Unique Games Conjecture (UGC) of Khot [28], establishing a previously unsuspected connection between probabilistically checkable proof systems (PCPs) and the theory of metric embeddings. We first prove that the UGC implies a super-constant hardness result for the (non-uniform) SPARSESTCUT problem. Though this hardness result relies on the UGC, we demonstrate, nevertheless, that the corresponding PCP reduction can be used to construct an "integrality gap instance" for SPARSESTCUT. Towards this, we first construct an integrality gap instance for a natural SDP relaxation of UNIQUEGAMES. Then we "simulate" the PCP reduction and "translate" the integrality gap instance of UNIQUEGAMES to an integrality gap instance of SPARSESTCUT. This enables us to prove a (log log n) 1/6−δ integrality gap for SPARSESTCUT, which is known to be equivalent to the metric embedding lower bound.

Multiwinner voting rules are used to select a small representative subset of candidates or items from a larger set given the preferences of voters. However, if candidates have sensitive attributes such as gender or ethnicity (when selecting a committee), or specified types such as political leaning (when selecting a subset of news items), an algorithm that chooses a subset by optimizing a multiwinner voting rule may be unbalanced in its selection -it may under or over represent a particular gender or political orientation in the examples above. We introduce an algorithmic framework for multiwinner voting problems when there is an additional requirement that the selected subset should be "fair" with respect to a given set of attributes. Our framework provides the flexibility to (1) specify fairness with respect to multiple, non-disjoint attributes (e.g., ethnicity and gender) and (2) specify a score function. We study the computational complexity of this constrained multiwinner voting problem for monotone and submodular score functions and present several approximation algorithms and matching hardness of approximation results for various attribute group structure and types of score functions. We also present simulations that suggest that adding fairness constraints may not affect the scores significantly when compared to the unconstrained case.