James R. Lee

James R. Lee

Professor
Computer Science
University of Washington

Paul G. Allen Center, Room 640
jrl [at] cs [dot] washington [dot] edu

UW THEORY GROUP

Spring 2016: Computability and complexity theory
Winter 2016: Entropy optimality
Winter 2015: Randomized algorithms
Foundations of Computing I [AU'15, SP'15]

Associate Editor, SIDMA.
Associate Editor, SICOMP.
I will be on the PC for STOC 2017.
Recent PCs: SODA 2014, ICALP 2014, FOCS 2014.

Research interests:
Algorithms, complexity, the theory of computation. Geometry and analysis at the interface between the continuous and discrete. Probability and stochastic processes.

Students: Jeffrey Hon
Ben Eggers, Yuegi Sheng, Austrin Stromme

Talks and events:   [earlier | later]

Selected recent works:   [ click on authors for abstract; expand / collapse all ]

  • A region intersection graph over a base graph G0 is a graph G whose vertices correspond to connected subsets of G0 with an edge between two vertices if the corresponding regions intersect. We show that if G0 excludes the complete graph Kh as a minor, then every region intersection graph over G0 with m edges has a balanced separator with O(sqrt{m}) nodes. If G has uniformly bounded vertex degrees, we show the separator is found by spectral partitioning.

    A string graph is the intersection graph of continuous arcs in the plane. The preceding result implies that every string graph with m edges has a balanced separator of size O(sqrt{m}). This bound is optimal, as it generalizes the planar separator theorem. It confirms a conjecture of Fox and Pach (2010) and improves over the O(sqrt{m} log m) bound of Matousek (2013).

  • We show that if the random walk on a graph has positive coarse Ricci curvature in the sense of Ollivier, then the stationary measure satisfies a W1 transport-entropy inequality. Peres and Tetali have conjectured a stronger consequence, that a modified log-Sobolev inequality (MLSI) should hold, in analogy with the setting of Markov diffusions. We discuss how the entropic interpolation approach suggests a natural attack on the MLSI conjecture.

  • Chang's Lemma is a widely employed result in additive combinatorics. It gives optimal bounds on the dimension of the large spectrum of probability distributions on finite abelian groups. In this note, we show how Chang's Lemma and a powerful variant due to Bloom both follow easily from an approximation theorem for probability measures in terms of generalized Riesz products. The latter result involves no algebraic structure. The proofs are correspondingly elementary.

  • [credit: Bernd Sturmfels]

    We introduce a method for proving lower bounds on the efficacy of semidefinite programming (SDP) relaxations for combinatorial problems. In particular, we show that the cut, TSP, and stable set polytopes on n-vertex graphs are not the linear image of the feasible region of any SDP (i.e., any spectrahedron) of dimension less than 2n^c, for some constant c>0. This result yields the first super-polynomial lower bounds on the semidefinite extension complexity of any explicit family of polytopes.

    Our results follow from a general technique for proving lower bounds on the positive semidefinite rank of a matrix. To this end, we establish a close connection between arbitrary SDPs and those arising from the sum-of-squares SDP hierarchy. For approximating maximum constraint satisfaction problems, we prove that SDPs of polynomial-size are equivalent in power to those arising from degree-O(1) sum-of-squares relaxations. This result implies, for instance, that no family of polynomial-size SDP relaxations can achieve better than a 7/8-approximation for MAX-3-SAT.

    [ PPT slides ]
    Best paper award, STOC 2015.

  • Consider a discrete-time martingale {Xt} taking values in a Hilbert space. We show if E[|Xt+1-Xt|2] = 1 and |Xt+1-Xt| ≤ L are satisfied for times t ≥ 0, then {Xt} satisfies a small-ball estimate: P[|Xt| < R] ≤ O(R/t1/2). Following [Lee-Peres 2013], this has applications to diffusive estimates for random walks on vertex-transitive graphs.

  • We show that under the Ornstein-Uhlenbeck semigroup (i.e., the natural diffusion process) on n-dimensional Gaussian space, any nonnegative, measurable function exhibits a uniform tail bound better than that implied by Markov's inequality and conservation of mass. This confirms positively the Gaussian limiting case of Talagrand's convolution conjecture (1989).

    Video: Talagrand's convolution conjecture and geometry via coupling (IAS)
    [ PPT slides ]

  • We prove super-polynomial lower bounds on the size of linear programming relaxations for approximation versions of constraint satisfaction problems. We show that for these problems, polynomial-sized linear programs are exactly as powerful as programs arising from a constant number of rounds of the Sherali-Adams hierarchy. In particular, any polynomial-sized linear program for Max Cut has an integrality gap of 1/2 and any such linear program for Max 3-Sat has an integrality gap of 7/8.

    Video: Linear programming and constraint satisfaction (Simons Institute)

  • We show that if a metric space X threshold-embeds into a Hilbert space, then X has Markov type 2. As a consequence, planar graph metrics and doubling metrics have Markov type 2, answering questions of Naor, Peres, Schramm, and Sheffield. More generally, if a metric space X threshold-embeds into a p-uniformly smooth Banach space, then X has Markov type p. This suggests some non-linear analogs of Kwapien's theorem. For instance, a subset of L^1 threshold-embeds into Hilbert space if and only if it has Markov type 2.

Some older selected papers:   [ click on authors for abstract; expand / collapse all ]

My research has been generously supported by the National Science Foundation, the Sloan Foundation, Microsoft, and the Simons Institute.