Solved problems, open problems, and re-solved problems

Two open problems from this blog were solved recently. First, there is a solution by Bartal, Gottlieb, and Krauthgamer of the Doubling TSP problem. Next, Raghu Meka gave a deterministic procedure to compute the expected supremum of a Gaussian process.

If you’re thirsting for more open problems, I suggest this list on analysis of Boolean functions, compiled by Ryan O’Donnell in conjunction with the 2012 Simons Symposium.

Finally, in the realm of re-proving theorems, let me mention this absolutely gorgeous proof by Lovett and Meka of Spencer’s six standard deviations suffice theorem in discrepancy theory. The proof is constructive (i.e., there exists a polynomial-time algorithm to construct the 2-coloring) as in recent groundbreaking work of Bansal, but unlike Bansal’s proof is independent of Spencer’s bound. It also happens to be elementary and beautiful; my thanks to Aravind Srinivasan for pointing it out.

Reading for a rainy day

Today it’s raining in both Seattle and Paris. Here are a few things I think are worth reading while the weather clears up.

  • Stop collecting coupons. Recently, Batson, Spielman, and Srivastava gave a beautiful sparsification result for graphs: Every graph has a linear-sized spectral sparsifier. Here is one statement of their result (taken from Srivastava’s thesis):

    Theorem [BSS]: For every \varepsilon > 0 and m,n \in \mathbb N, the following holds. For every x_1, x_2, \ldots, x_m \in \mathbb R^n, there are non-negative numbers s_1, s_2, \ldots, s_m of which at most O(n/\varepsilon^2) are non-zero, and such that for all y \in \mathbb R^n,

    \displaystyle (1-\varepsilon)^2 \sum_{i=1}^m \langle x_i, y \rangle^2 \leq \sum_{i=1}^m s_i \langle x_i,y\rangle^2 \leq (1+\varepsilon)^2 \sum_{i=1}^m \langle x_i, y\rangle^2.

    Assaf Naor has given a very nice account of some recent geometric applications of this idea. These involves breaking the n \log n “coupon collecting” barrier from a number of results which were previously based on random sampling. There is still an outstanding open problem left open here: Embedding n-dimensional subspaces of L_1 into \ell_1^{O(n)}.

  • Lecture notes that are lecture notes. Some people write lecture notes like they are preliminary book drafts. These lecture notes by Ryan O’Donnell for an undergraduate course on “Probability and Computing” are like transcribed lectures. They’re conversational, with philosophical asides–a great example of the style.

  • Metaphors in systolic geometry. Larry Guth and Nets Katz recently made awesome progress on the Erdos distance problem in the plane. On a different note, here is a great informal survey of Guth on Gromov’s systolic inequality (which itself answers the question—when would an isometric embedding into L_{\infty} ever be employed for the usefulness of the ambient space?!)

  • An important epsilon. Finally, there is the recent result of Gharan, Saberi, and Singh giving a 3/2 - \varepsilon approximation for the Traveling Salesman Problem in unweighted graphs. I haven’t gotten a chance to digest the paper yet. Here’s hoping someone else will write an overview of the key ideas.