CS266: Parameterized Algorithms and Complexity

These problems can be solved trivially by trying all subsets of vertices/integers of size k, and checking whether they satisfy the condition in the problem. For k-SUM we get an O(n ·k) time algorithm and for k-CLIQUE we get O(n ·k) whereas for k-DOMINATING-SET we get O(n · n · k). We will show better algorithms for these problems. For k-SUM we’ll get an O(nd k 2 e) time algorithm. For k-CLIQUE we’ll also get an improvement by a multiplicative factor in the exponent by giving an algorithm that runs in time O(n0.78·k) for k divisible by 3. And for k-DOMINATING-SET we’ll only get O(n) time, for k > 6 which is an improvement by almost 1 in the exponent. There are some hardness results for these problems. It is known that a n time algorithm for all k, for any of these problems, would refute ETH. And for k-DOMINATING-SET, even an O(nk−ε) algorithm, for some ε > 0, would refute Strong ETH. We will see some of these results later on in the course.