In this paper, we develop new tools and connections for . In this setting, we are given a problem instance and an integer , and the goal is to design an approximation algorithm with the fastest possible running time. We give randomized algorithms that establish an approximation ratio of for maximum independent set in time, for chromatic number in time, for minimum vertex cover in time, and for minimum -hypergraph vertex cover in time. (Throughout, and omit and factors polynomial in the input size, respectively.) The best known time bounds for all problems were (Bourgeois et al. in Discret Appl Math 159(17):1954-1970, 2011; Cygan et al. in Exponential-time approximation of hard problems, 2008). For maximum independent set and chromatic number, these bounds were complemented by lower bounds (under the Exponential Time Hypothesis (ETH)) (Chalermsook et al. in Foundations of computer science, FOCS, pp. 370-379, 2013; Laekhanukit in Inapproximability of combinatorial problems in subexponential-time. Ph.D. thesis, 2014). Our results show that the naturally-looking bounds are not tight for all these problems. The key to these results is a procedure that reduces a problem to a bounded-degree variant, allowing the use of approximation algorithms for bounded-degree graphs. To obtain the first two results, we introduce a new . Finally, we show a connection between PCP parameters and exponential-time approximation algorithms. This connection together with our independent set algorithm refute the possibility to overly reduce the size of Chan's PCP (Chan in J. ACM 63(3):27:1-27:32, 2016). It also implies that a (significant) improvement over our result will refute the gap-ETH conjecture (Dinur in Electron Colloq Comput Complex (ECCC) 23:128, 2016; Manurangsi and Raghavendra in A birthday repetition theorem and complexity of approximating dense CSPs, 2016).