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The Batson-Spielman-Srivastava barrier function method has had a number of applications to geometry and functional analysis, arose in computer science, and is a very original form of potential functio …

Bollobas showed that for any $d$ and any $g$, there exists a $d$-regular graph $G$ of girth at least $g$ such that $$ \alpha(G) < \frac{2n\log d}{d}. $$ So you cannot hope for more than a factor 16 …

This is a very popular heuristic method in machine learning, known as alternating minimization. You can easily find tons of papers using it. Often the setting in which it is used is when $f(x,y)$ is c …

This is a special case of non-monotone submodular maximization with a cardinality constraint, and constant factor approximation algorithms are known. For example, Feldman, Naor, and Schwartz get a fac …

In the indexing problem Alice has a vector $x \in \{0,1\}^d$ and Bob has a number $i$, and Bob wants to learn $x_i$. The randomized one-way communication complexity of this problem is $\Omega(d)$ (see …

The optimization problems seems to be equivalent to shortest common supersequence as well. The two results I found related to approximating this problem (it is NP-hard in general) are this and this. T …

Notice that approximating sparsest cut to within $\alpha$ gives a $2\alpha$ approximation for the Cheeger constant as defined. Here are some papers that give constant approximation algorithms for spar …

I didn't mean this to be an answer but it would require too many comments. Hope it's useful. As Tsuyoshi points out, it's tempting to conjecture that all "non-trivial" properties are hard (NP-hard fo …

NP is equal to coNP if and only if there are efficiently verifiable proofs of unsatisfiability. I.e., if and only if there exists a polynomial time turing machine $M$, which given any SAT formula $\ph …

Yes, this has been studied. It was called the multiple intents re-ranking problem by Azar, Gamzu, and Yin who gave a $\log n$ approximation using a cleverly modified greedy algorithm (the point is bei …

A relatively simple setting to illustrate the method of random restrictions is Subbotovskaya's original application of the method to prove an $\Omega(n^{1.5})$ lower bound on the formula size of the p …

This is the geometric median problem. There is a nearly linear time algorithm based on interior point methods due to Cohen et al.: to find a $(1+\varepsilon)$-approximation their algorithm runs in tim …

I think the proper reference is: @article{Dudley1967290, Author = {R.M Dudley}, Doi = {http://dx.doi.org/10.1016/0022-1236(67)90017-1}, Issn = {0022-1236}, Journal = {Journal of Funct …

Applications of Ramsey theory to CS, Gasarch

You can look at the survey paper by Bogdanov and Trevisan, and this survey talk by Luca. The main open question is whether $\mathsf{P} \neq \mathsf{NP}$ implies that there exist hard on average proble …