  We present a local search framework to design and analyze both combinatorial
algorithms and rounding algorithms for experimental design problems. This
framework provides a unifying approach to match and improve all known results
in D/A/E-design and to obtain new results in previously unknown settings.
  For combinatorial algorithms, we provide a new analysis of the classical
Fedorov's exchange method. We prove that this simple local search algorithm
works well as long as there exists an almost optimal solution with good
condition number. Moreover, we design a new combinatorial local search
algorithm for E-design using the regret minimization framework.
  For rounding algorithms, we provide a unified randomized exchange algorithm
to match and improve previous results for D/A/E-design. Furthermore, the
algorithm works in the more general setting to approximately satisfy multiple
knapsack constraints, which can be used for weighted experimental design and
for incorporating fairness constraints into experimental design.
