Errol Lloyd Design and Analysis of Algorithms Approximation Algorithms for NP-complete Problems Bin Packing Networks

2 What is bin packing? unlimited supply of bins (capacity 1) Items of sizes s 1, s 2, … s n 0 < s i < 1 Given Objective Pack the items into a minimum number of bins Restated Assign each item to a bin such that: Number of nonempty bins is minimized Number of nonempty bins is minimized For each bin B, the sum of the sizes of the items in B does For each bin B, the sum of the sizes of the items in B does not exceed 1 not exceed

3 Approximation Methods One difficulty: Bin packing is NP-complete Cannot guarantee optimal packings in polynomial time (unless P = NP) Settle for: Packings that are close to optimal What is close? No more than a constant factor larger than optimal. Example: Packings using twice as many bins as optimal.

4 Any Fit Packing Open one bin for each item S i if there is an open bin where S i will fit thenplace S i into that bin else open a new bin place S i into that new bin Recall: Optimal used 4 bins

5 How bad can Any Fit be? How many bins might Any Fit use compared with optimal? Items:18 items of size 1/2 18 items of size 1/18 Any Fit packing is 1.8 times optimal Examples exist with Any Fit up to 2 times optimal

6 Can Any Fit do worse than 2*optimal? optimal packing uses at least b/2 bins Any Fit never uses more than twice optimal bins Competitive Ratio of Any Fit: 2 Running time of Any Fit: O(n log n)

7 First Fit Open one bin for each item S i if there is an open bin where S i will fit thenplace S i into the leftmost such bin else open a new bin place S i into that new bin Competitive Ratio of First Fit: 1.7 Running Time: O(n log n)

8 First Fit Decreasing A really good bin packing algorithm Sort the items in decreasing order by size Do a First Fit Packing using that sorted list Competitive Ratio: 11/9 (i.e …) Running Time: O(n log n) [Johnson, Demers, Ullman, Garey, Graham, 1974] Loooooooooong proof (originally 100+ pages).3

9 Changing the Rules – Fully Dynamic Bin Packing NOT given the items all at once Instead: given items one at a time – INSERTs (DELETES) Upon an INSERT/DELETE, update the packing NO apriori limitations on this update – the contents of the bins may be changed at will Maintain a good packing at all times O(log n) time per INSERT/DELETE (to update) So, n INSERTs in O(n log n) time

10 Mostly Myopic Packing (MMP) MMP Competitive Ratio: 5/4 = 1.25 Looong proof MMP running time: O(log n) per INSERT or DELETE O(n log n) for sequence of n INSERT/DELETEs A fully dynamic bin packing algorithm

11 Topology Control for Ad-hoc Networks What is an ad hoc network? A collection of nodes that communicate with one another over a wireless medium

12 Applications Military environments Search and rescue Sensor networks

13 High Transmission Power High channel contention High power consumption Low throughput Low Transmission Power Partition Topology What is the topology of an ad hoc network? IETF MANET Group Definition: The topology of the network is described by a graph G = (V, E), where V is the set of nodes and E is the set of links in the network.

14 Our problem Input: Nodes, power thresholds and a graph property Power threshold for x and y: Min power for a signal from x to reach y Output: A power assignment f(u) for each node u, s.t. the induced graph satisfies the specified graph property, and that minimizes: Max u (f(u)) (Max Power) OR Sum u (f(u)) (Total Power)

15 Minimizing Max Power Gave a general framework to compute an optimal solution in polynomial time for where P is monotone and efficiently testable Monotone property: Property unaffected by edge additions Monotone: 1-connectivity Non-monotone: Tree in O(n 2 logn) in O(n 3 logn)

16 Two related problems What if a property is non-monotone? Property: “Tree” Complexity: NP-complete: even with only three power thresholds even if the minimization objective is dropped! Can number of nodes using max power be minimized? NP-complete even for the “1- Connected ” property Approximation algorithm with: Competitive Ratio: 5/3 Running time: O(n e  (n))

17 Minimizing Total Power Problems are NP-complete, even Developed a general approximation framework for monotone and efficiently testable properties Competitive ratio: 2(2-2/n)(2+1/n)

18 Experimental results Implemented and compared: the approximation algorithm for the exact algorithm for

19 TRANSIMS networks TRANSIMS project of LANL and IBM Traffic in the Portland, OR metropolitan area 1716 nodes in a 3x3 km 2 area Uniform transmission range of 75 meters Area 1: 1 km x 1km, 284 nodes Area 2: 0.6 km x 1.65 km, 271 nodes Area 1 Area 2

20 TRANSIMS networks Area 1 Area 2 Area 1 Max Range Avg. Range Max Degree Avg. Degree MaxP158m67.75m TotalP193m55.07m52.72 Area 2 Max Range Avg. Range Max Degree Avg. Degree MaxP153m73.59m TotalP222m51.95m72.73

21 Additional Information