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Series: ACO Student Seminar

In this talk, we study solvers for geometrically embedded graph structured block linear systems. The general form of such systems, PSD-Graph-Structured Block Matrices (PGSBM), arise in scientific computing, linear elasticity, the inner loop of interior point algorithms for linear programming, and can be viewed as extensions of graph Laplacians into multiple labels at each graph vertex. Linear elasticity problems, more commonly referred to as trusses, describe forces on a geometrically embedded object.We present an asymptotically faster algorithm for solving linear systems in well-shaped 3-D trusses. Our algorithm utilizes the geometric structures to combine nested dissection and support theory, which are both well studied techniques for solving linear systems. We decompose a well-shaped 3-D truss into balanced regions with small boundaries, run Gaussian elimination to eliminate the interior vertices, and then solve the remaining linear system by preconditioning with the boundaries.On the other hand, we prove that the geometric structures are ``necessary`` for designing fast solvers. Specifically, solving linear systems in general trusses is as hard as solving general linear systems over the real. Furthermore, we give some other PGSBM linear systems for which fast solvers imply fast solvers for general linear systems.Based on the joint works with Robert Schwieterman and Rasmus Kyng.

Series: ACO Student Seminar

The random to random shuffle on a deck of cards is given by at each
step choosing a random card from the deck, removing it, and replacing it
in a random location. We show an upper bound for the total variation
mixing time of the walk of 3/4n log(n) +cn steps. Together with matching
lower bound of Subag (2013), this shows the walk mixes with cutoff at
3/4n log(n) steps, answering a conjecture of Diaconis. We use the
diagonalization of the walk by Dieker and Saliola (2015), which relates
the eigenvalues to Young tableaux.
Joint work with Evita Nestorid.

Series: ACO Student Seminar

Beginning with Szemerédi’s regularity lemma, the theory of graph
decomposition and graph limits has greatly increased our understanding
of large dense graphs and provided a framework for graph approximation.
Unfortunately, much of this work does not meaningfully extend to
non-dense graphs. We present preliminary work towards our goal of
creating tools for approximating graphs of intermediate degree (average
degree o(n) and not bounded). We give a new random graph model that
produces a graph of desired size and density that approximates the
number of small closed walks of a given sparse graph (i.e., small
moments of its eigenspectrum). We show how our model can be applied to
approximate the hypercube graph. This is joint work with Santosh
Vempala.

Series: ACO Student Seminar

The concentration of measure phenomenon is of great importance in probabilistic combinatorics and theoretical computer science. For example, in the theory of random graphs, we are often interested in showing that certain random variables are concentrated around their expected values. In this talk we consider a variation of this theme, where we are interested in proving that certain random variables remain concentrated around their expected trajectories as an underlying random process (or random algorithm) evolves. In particular, we shall give a gentle introduction to the differential equation method popularized by Wormald, which allows for proving such dynamic concentration results. This method systematically relates the evolution of a given random process with an associated system of differential equations, and the basic idea is that the solution of the differential equations can be used to approximate the dynamics of the random process. If time permits, we shall also sketch a new simple proof of Wormalds method.

Series: ACO Student Seminar

We study stable marriage where individuals strategically submit private preference information to a publicly known stable marriage algorithm. We prove that no stable marriage algorithm ensures actual stability at every Nash equilibrium when individuals are strategic. More specifically, we show that any rational marriage, stable or otherwise, can be obtained at a Nash equilibrium. Thus the set of Nash equilibria provides no predictive value nor guidance for mechanism design. We propose the following new minimal dishonesty equilibrium refinement, supported by experimental economics results: an individual will not strategically submit preference list L if there exists a more honest L' that yields as preferred an outcome. Then for all marriage algorithms satisfying monotonicity and IIA, every minimally dishonest equilibrium yields a sincerely stable marriage. This result supports the use of algorithms less biased than the (Gale-Shapley) man-optimal, which we prove yields the woman-optimal marriage in every minimally dishonest equilibrium. However, bias cannot be totally eliminated, in the sense that no monotonic IIA stable marriage algorithm is certain to yield the egalitarian-optimal marriage in a minimally dishonest equilibrium – thus answering a 28-year old open question of Gusfield and Irving's in the negative. Finally, we show that these results extend to student placement problems, where women are polygamous and honest, but not to admissions problems, where women are both polygamous and strategic.
Based on joint work with Craig Tovey at Georgia Tech.

Series: ACO Student Seminar

Using some classical results of invariant theory of finite reflection groups, and Lagrange multipliers, we prove that low degree or sparse real homogeneous polynomials
which are invariant under the action of a finite reflection group
$G$ are nonnegative if they are nonnegative on the hyperplane arrangement
$H$ associated to $G$. That makes $H$ a test set for the above kind of
polynomials. We also prove that under stronger sparsity conditions, for
the symmetric group and other reflection groups, the test set can
be much smaller. One of the main questions is deciding if certain
intersections of some simply constructed real $G$-invariant varieties are
empty or not.

Series: ACO Student Seminar

Optimization problems arising in decentralized multi-agent systems have gained significant attention in the context of cyber-physical, communication, power, and robotic networks combined with privacy preservation, distributed data mining and processing issues. The distributed nature of the problems is inherent due to partial knowledge of the problem data (i.e., a portion of the cost function or a subset of the constraints is known to different entities in the system), which necessitates costly communications among neighboring agents. In this talk, we present a new class of decentralized first-order methods for nonsmooth and stochastic optimization problems which can significantly reduce the number of inter-node communications. Our major contribution is the development of decentralized communication sliding methods, which can skip inter-node communications while agents solve the primal subproblems iteratively through linearizations of their local objective functions.This is a joint work with Guanghui (George) Lan and Yi Zhou.

Series: ACO Student Seminar

Spielman and Teng (2004) showed that linear systems in Laplacian matrices can be solved in nearly linear time. Since
then, a major research goal has been to develop fast solvers for linear
systems in other classes of matrices. Recently, this has led to fast
solvers for directed Laplacians (CKPPRSV'17) and connection Laplacians
(KLPSS'16).Connection
Laplacians are a special case of PSD-Graph-Structured Block Matrices
(PGSBMs), block matrices whose non-zero structure correspond to a graph,
and which additionally can be expressed as a sum of positive
semi-definite matrices each corresponding to an edge in the graph. A
major open question is whether fast solvers can be obtained for all
PGSBMs (Spielman, 2016). Fast solvers for Connection Laplacians provided
some hope for this. Other important
families of matrices in the PGSBM class include truss stiffness
matrices, which have many applications in engineering, and
multi-commodity Laplacians, which arise when solving multi-commodity
flow problems. In
this talk, we show that multi-commodity and truss linear systems are
unlikely to be solvable in nearly linear time, unless general linear
systems (with integral coefficients) can be solved in nearly linear
time. Joint work with Rasmus Kyng.

Series: ACO Student Seminar

We present an algorithm that, with high probability, generates a random spanning treefrom an edge-weighted undirected graph in O (n^{5/3}m^{1/3}) time. The tree is sampled from adistribution where the probability of each tree is proportional to the product of its edge weights.This improves upon the previous best algorithm due to Colbourn et al. that runs in matrix multiplication time, O(n^\omega). For the special case of unweighted graphs, this improves upon thebest previously known running time of ˜O(min{n^\omega, mn^{1/2}, m^{4/3}}) for m >> n^{7/4} (Colbourn et al. ’96, Kelner-Madry ’09, Madry et al. ’15).The effective resistance metric is essential to our algorithm, as in the work of Madry et al., butwe eschew determinant-based and random walk-based techniques used by previous algorithms.Instead, our algorithm is based on Gaussian elimination, and the fact that effective resistance ispreserved in the graph resulting from eliminating a subset of vertices (called a Schur complement).As part of our algorithm, we show how to compute -approximate effective resistances for a set Sof vertex pairs via approximate Schur complements in O (m + (n + |S|)/\eps^{ −2}) time, without usingthe Johnson-Lindenstrauss lemma which requires eO(min{(m+|S|) \eps{−2},m+n /eps^{−4} +|S|/eps^{ −2}}) time. We combine this approximation procedure with an error correction procedure for handing edges where our estimate isn’t sufficiently accurate.Joint work with Rasmus Kyng, John Peebles, Anup Rao, and Sushant Sachdeva

Series: ACO Student Seminar

We give the first polynomial upper bound on the mixing time of the edge-flip Markov chain for unbiased dyadic tilings, resolving an open problem originally posed by Janson, Randall, and Spencer in 2002. The technique used, adapted from spin system analysis in statistical physics and not widely used in computer science literature, involves a multilevel decomposition of the state space and is of independent interest. A dyadic tiling of size n is a tiling of the unit square by n non-overlapping dyadic rectangles, each of area 1/n, where a dyadic rectangle is any rectangle that can be written in the form [a2^{-s}, (a+1)2^{-s}] x [b2^{-t}, (b+1)2^{-t}] for non-negative integers a,b,s,t. The edge-flip Markov chain selects a random edge of the tiling and replaces it with its perpendicular bisector if doing so yields a valid dyadic tiling. Specifically, we show that the relaxation time of the edge-flip Markov chain for dyadic tilings is at most O(n^{4.09}), which implies that the mixing time is at most O(n^{5.09}). We complement this by showing that the relaxation time is at least \Omega(n^{1.38}), improving upon the previously best lower bound of \Omega(n log n) coming from the diameter of the chain. This is joint work with David Levin and Alexandre Stauffer.