picture shows kernelisation for the k-WEIGHTED MARBLES problem, which is presented below

k-WEIGHTED MARBLES Input: A sequence of marbles M = (m,...,mn), along with an integer weight w(m) and a colour c(mi) for every marble mi e M, and an integer k. Parameter: k. Question: Is there a set M' of marbles with w(M') = Imepw(m) 2k +1. [2 points] From the previous two statements we can infer a suitable upper bound on the number of marbles for any reduced instance. However, the size of the instance is also determined by the weights assigned to those marbles, which means that for a proper kernel we need to bound the weights in terms of the parameter value as well. (1) Consider the third reduction rule: if a marble has weight more than k + 1, set its weight to k. Show that this rule is valid as well. [2 points] (k) Argue why exhaustively applying the three reduction rules described above takes time at most polynomial in the original input size. [4 points) (1) Conclude that k-WEIGHTED MARBLEs admits kernels of size (k log k). [2 points] k-WEIGHTED MARBLES Input: A sequence of marbles M = (m,...,mn), along with an integer weight w(m) and a colour c(mi) for every marble mi e M, and an integer k. Parameter: k. Question: Is there a set M' of marbles with w(M') = Imepw(m) 2k +1. [2 points] From the previous two statements we can infer a suitable upper bound on the number of marbles for any reduced instance. However, the size of the instance is also determined by the weights assigned to those marbles, which means that for a proper kernel we need to bound the weights in terms of the parameter value as well. (1) Consider the third reduction rule: if a marble has weight more than k + 1, set its weight to k. Show that this rule is valid as well. [2 points] (k) Argue why exhaustively applying the three reduction rules described above takes time at most polynomial in the original input size. [4 points) (1) Conclude that k-WEIGHTED MARBLEs admits kernels of size (k log k). [2 points]