1. Review for E&CE 602

2. 3 3 1 4 4 2 2 2 5 2 Find the minimal cost spanning tree for the graph below (where Values on edges represent the costs). 3 Ans. 18 3

3. 2 2 Find the maximum flow from s to t of the graph below. Each forward (s to t) arc Has capacity of 1 unless labeled otherwise. s t

4. Find maximum flow 5,5 2,4 3,6 1,5 0,1 3,3 6,4,6 3,1,7 4,2,5 ST

5. X1 x2 x3 x4 x5 x6 y1 y2 y3 y4 y5 y6 Given the initial matching on the graph, provide the H tree And the next matching, using the maximum matching algorithm. (use curvy lines to identify the next matching on the graph). x4 y3 x5 x1 y1 x2 y4 x3 y6 y5 x6 y2 Path to flip is x4-y6-x3-y2-x6-y5

6. 2 1 3 3 5 2 2 4 7 1 1 3 3 6 4 2 Using shortest path algorithm (used during lectures), identify all Shortest paths from node u to all other nodes in the graph. Specifically Define S={u}, S={u, }, S={u,, }… through the shortest path algorithm Stages. u f a b g c e d S={u}={u,a}={u,a,e}={u,a,e,f,d,b}={u,a,e,f,d,b,c}={u,a,e,f,d,b,c,g}

7. Formulate the peicewise linear function f(x) shown in graph below. So that it can be minimized in the objective function of the IP given below. a1 a2 a3 a4 x Min f(x) | Ax <= b Slope is m1 f(x)

9. You are given a set of M tasks, t=t1,t2,…,M and N processors, n=1,2,…N. each task must be assigned to one processor and each Processor can be assigned at most two tasks. Assume Represents the proposition that task t is assigned to processor n (use binary variables x t,n ) a)Formulate the following constraint: b)Formulate the following constraint: c) Give a formulation of the objective function which is to minimize The number of processors being used. You may define new constraints Or variables as required

10. (a) (b)

11. (c)

12. Give a good IP formulation for the following problem: The problem is the maximum cardinality node packing problem On the graph below constrained with the following inequality: 7 5 4 6 3 1 2

13. Is a knapsack inequality So we can generate facets or Stronger inequalities to improve The formulation 7 5 4 6 3 1 2 We can use graph to extract node packing Facets/inequalities

14. Describe how you would solve the following problem in general: Find the minimum execution time for 2 processor to execute a set of task (defined using a directed acyclic graph, where arcs represent data transfers between tasks). Assume one task at a time can be executed on each processor, tasks can be executed in parallel one different processors, and there is no communication delay. (all tasks have the same duration)

15. Reformulate the following optimization problem, so that one can implement it with a LP solver which only supports min or max objective functions not the min- max objective below

17. Mid august (9) projects due (noon, 22 august, in DC 3514) final exam (2:00-5:00, 15 august),