In the derivation of the spectral graph partitioning algorithm, we relax a combinatorial optimization problem to a continuous optimization problem. This relaxation has the following effects. The combinatorial problem requires an ex- act bisection of the graph, but the continuous al- gorithm can produce (after rounding) partitions that aren't perfectly balanced The combinatorial problem cannot be modi- fied to accommodate vertices that have different masses, whereas the continuous problem can The combinatorial problem requires finding eigenvectors, whereas the continuous problem re- quires only matrix multiplication ■The combinatorial problem is NP-hard, but the continuous problem can be solved in polyno- mial time

Please help with algorithm no coding needed

Design a dynamic programming algorithm for the bigger-is-smarterelephant problem by comparing it, as done previously, with the problem of finding thelongest weighted path within a directed level graph problem.

5. Consider the following modification of the independent set problem. As input, you receive an undirected graph G = (V, E). As output, you must decide if G contains an independent set UCV such that U contains at least 99% of the vertices in V (that is, JU| > 0.99|V|). Prove that this modification of the independent set problem is NP-Complete.

Describe the security implications of jailbreaking or rooting a mobile device.

Could you also do question one please

Q. For a given graph G = (V,E), a subset SCVis called an independent set if (u,v) & E for every pair of vertices u, vES. The three independent set problems ISD, ISO and ISF all have as input a graph G=(V,E). ISD also has as part of the input a natural number k. ISD is a decision problem: the question is whether or not there is a subset S of V with at least k elements such that (u,v)EE for every pair of vertices u, v ES? ISO and ISF do not have yes/no answers. The answer to ISO is an integer n – size of the largest independent set in graph G. The answer to ISF is subset S – largest independent set in graph G. Show that all three of these problems are polynomial-time Turing equivalent. I.e. show the solution of ISO and ISF that uses ISD as a subroutine. Also show a solution to ISD that uses ISO as a subroutine. Provide detailed pseudocode solutions with justification of correctness and running time.

Please show written work with answer please!!   A vertex cover of a graph is a set of vertices that includes at least one endpoint of every edge of the graph. Finding the minimum size vertex cover set is a NP problem, find an greedy approximation algorithm to find a close to optimum solution in polynomial time.