Approximation Algorithms for NP-complete Problems Errol Lloyd Design and Analysis of Algorithms Approximation Algorithms for NP-complete Problems Bin Packing Computer Networks Part tutorial, part our results

What is bin packing? unlimited supply of bins (capacity 1) Items of sizes s1, s2, … sn 0 < si < 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 For each bin B, the sum of the sizes of the items in B does not exceed 1 The problem we consider is topology control in ad hoc networks. So, what is an ad hoc network? An ad hoc network is a collection of nodes[click] that communicate with one other over [click] wireless medium. When the destination node can not be reached directly by the source, the nodes in the middle can act as routers to forward the transmission. Such networks require no fixed infrastructure, so can be quick deployed. it has application in battle field communication, disaster recovery, sensors networks, etc. .5 .67 .25 .33 .4 .2 .2 .4 .75 .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. The problem we consider is topology control in ad hoc networks. So, what is an ad hoc network? An ad hoc network is a collection of nodes[click] that communicate with one other over [click] wireless medium. When the destination node can not be reached directly by the source, the nodes in the middle can act as routers to forward the transmission. Such networks require no fixed infrastructure, so can be quick deployed. it has application in battle field communication, disaster recovery, sensors networks, etc.

Recall: Optimal used 4 bins Any Fit Packing Open one bin for each item Si if there is an open bin where Si will fit then place Si into that bin else open a new bin place Si into that new bin .5 .67 .25 .33 .4 .2 .2 .4 .75 .3 The problem we consider is topology control in ad hoc networks. So, what is an ad hoc network? An ad hoc network is a collection of nodes[click] that communicate with one other over [click] wireless medium. When the destination node can not be reached directly by the source, the nodes in the middle can act as routers to forward the transmission. Such networks require no fixed infrastructure, so can be quick deployed. it has application in battle field communication, disaster recovery, sensors networks, etc. .2 .33 .25 .2 .3 .5 .67 .4 .4 .75 Recall: Optimal used 4 bins

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 The problem we consider is topology control in ad hoc networks. So, what is an ad hoc network? An ad hoc network is a collection of nodes[click] that communicate with one other over [click] wireless medium. When the destination node can not be reached directly by the source, the nodes in the middle can act as routers to forward the transmission. Such networks require no fixed infrastructure, so can be quick deployed. it has application in battle field communication, disaster recovery, sensors networks, etc. Any Fit packing is 1.8 times optimal Examples exist with Any Fit up to 2 times optimal

Can Any Fit do worse than 2*optimal? optimal packing uses at least b/2 bins The problem we consider is topology control in ad hoc networks. So, what is an ad hoc network? An ad hoc network is a collection of nodes[click] that communicate with one other over [click] wireless medium. When the destination node can not be reached directly by the source, the nodes in the middle can act as routers to forward the transmission. Such networks require no fixed infrastructure, so can be quick deployed. it has application in battle field communication, disaster recovery, sensors networks, etc. Any Fit never uses more than twice optimal bins Competitive Ratio of Any Fit: 2 Running time of Any Fit: O(n log n)

First Fit Open one bin for each item Si if there is an open bin where Si will fit then place Si into the leftmost such bin else open a new bin place Si into that new bin .5 .67 .25 .33 .4 .2 .2 .4 .75 .3 The problem we consider is topology control in ad hoc networks. So, what is an ad hoc network? An ad hoc network is a collection of nodes[click] that communicate with one other over [click] wireless medium. When the destination node can not be reached directly by the source, the nodes in the middle can act as routers to forward the transmission. Such networks require no fixed infrastructure, so can be quick deployed. it has application in battle field communication, disaster recovery, sensors networks, etc. .2 .4 .25 .33 .2 .5 .67 .4 Competitive Ratio of First Fit: 1.7 Running Time: O(n log n)

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 .75 .67 .5 .4 .4 .33 .3 .25 .2 .2 .2 .25 .33 .4 .3 .75 .67 .5 .4 .2 Absolute performance ratio of FFD is 3/2, and asymptotic performance ratio is 11/9 Competitive Ratio: 11/9 (i.e. 1.2222…) Running Time: O(n log n) [Johnson, Demers, Ullman, Garey, Graham, 1974] Loooooooooong proof (originally 100+ pages)

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 .5 .67 .25 .33 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

Mostly Myopic Packing (MMP) A fully dynamic bin packing algorithm 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

Some bin packing papers Z. Ivkovich and E.L. Lloyd, "Fully dynamic algorithms for bin packing: Being mostly myopic helps," SIAM Journal on Computing, 28(1998), 574-611. Z. Ivkovich and E.L. Lloyd, "Partially dynamic bin packing can be solved within 1+e in (amortized) polylogarithmic time", Information Processing Letters 63(1997), 45-50. Z. Ivkovich and E.L. Lloyd, "A fundamental restriction on fully dynamic maintenance of bin packing," Information Processing Letters, 59(1996), 229-232.

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 The problem we consider is topology control in ad hoc networks. So, what is an ad hoc network? An ad hoc network is a collection of nodes[click] that communicate with one other over [click] wireless medium. When the destination node can not be reached directly by the source, the nodes in the middle can act as routers to forward the transmission. Such networks require no fixed infrastructure, so can be quick deployed. it has application in battle field communication, disaster recovery, sensors networks, etc.

Applications Military environments Search and rescue Sensor networks The problem we consider is topology control in ad hoc networks. So, what is an ad hoc network? An ad hoc network is a collection of nodes[click] that communicate with one other over [click] wireless medium. When the destination node can not be reached directly by the source, the nodes in the middle can act as routers to forward the transmission. Such networks require no fixed infrastructure, so can be quick deployed. it has application in battle field communication, disaster recovery, sensors networks, etc.

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. So, what is the topology of an ad hoc network? In our research, we modeled topology in the following way. Given the distribution of nodes in an ad hoc network, its topology is determined by the transmission power at each node. For example, the node at the center of the circle, its transmission can be received by all nodes inside that circle. If all node in this network transmit at the same high power level. The topology of the network will look like this. which may cause ... But from power conservation point of view, the topology is not good. We can change the topology by adjusting transmission power at each node. For saving power, we may want node to transmit at a low power level, but this may cause network partitioning. This brings us the topology control problems. High Transmission Power High channel contention High power consumption Low throughput Low Transmission Power Partition

Why topology control? Desirable topology & minimize energy consumption Topology property guarantee implies: Improving network survivability, performance, etc Lower transmission power implies: Conserving battery power Increasing spatial frequency reuse Topology control is to assign transmission power level to each node, such that the topology of the network satisfies some properties, and the power consumption is minimized. ...

A formal definition 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: Maxu(f(u)) (Max Power) OR Sumu(f(u)) (Total Power)

Minimizing Max Power Gave a general framework to compute an optimal solution in polynomial time for <P, Max power> where P is monotone and efficiently testable Monotone property: Property unaffected by edge additions Monotone: 1-connectivity Non-monotone: Tree <1-connected, Max Power> in O(n2logn) <Diameter K, Max Power> in O(n3logn)

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 a(n))

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

Experimental results Implemented and compared: the approximation algorithm for <2-connected, Total Power> the exact algorithm for <2-connected, Max Power> To gain another perspective of the algorithm, we also conducted experimental studies. We implemented approximation algorithm for ... Since there is not other algorithm for this problem we are aware of, we compared it with R R’s algorithm for ... In the experiment, we used ... as radio propagation model. The experiment scenarios are generated by varying node density. Two types of nodes distributions are simulated, ...

TRANSIMS networks TRANSIMS project of LANL and IBM Traffic in the Portland, OR metropolitan area 1716 nodes in a 3x3 km2 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 In addition to randomly generated networks, we also conducted experiments on this more realistic network. It is obtained from a TRANSIMS data set which simulates the traffic in Portland metropolitan area. There are totally 1716 nodes. When every node has a transmission range of 75 meters. Since the running of our algorithm will be prohibitively high if run on the entire 1716 nodes, we selected to areas, each has around 300 nodes. A1: 55 meters A2: 52 meters Area 2

TRANSIMS networks Max Range Avg. Range Max Degree Avg. Degree MaxP Area 1 Area 1 Max Range Avg. Range Max Degree Avg. Degree MaxP 158m 67.75m 12 4.80 TotalP 193m 55.07m 5 2.72 Area 2 Max Range Avg. Range Max Degree Avg. Degree MaxP 153m 73.59m 28 7.94 TotalP 222m 51.95m 7 2.73 Compare topologies. For tables, compares avg. range, degree. Conclusion, it is consistent with the random graph results. Area 2

Some topology control papers "Topology Control in Constant Rate Mobile Ad Hoc Networks,” Wireless Networks, 16, 2010, 467-480 (with L. Zhao, S.S. Ravi). “Improved topology control algorithms for Simple Mobile Networks," Ad Hoc Nets '09, (with F. Che, L. Zhao). “Topology control problems for wireless ad-hoc networks,” Handbook of Approximation Algorithms and Metaheuristics, 2007, 67-1 to 67-20 (with S.S. Ravi). "Topology Control for Simple Mobile Networks”, IEEE Globecom 2006 (with L. Zhao and S.S. Ravi). "Approximating the minimum number of maximum power users in ad hoc networks", ACM J. on Mobile Networks and Applications (MONET) 11, 2006, 129-142 (with R. Liu and S.S. Ravi). "Algorithmic aspects of topology control problems for ad hoc networks," ACM Journal on Mobile Networks and Applications (MONET), 10(1-2) 2005, 19-34 (with M. Marathe, R. Ramanathan, S.S. Ravi, and R. Liu). "CLTC: A cluster-based topology control framework for ad-hoc networks," IEEE Trans. on Mobile Computing, 3(1) 2004, 18-32 (C.-C. Shen, C. Srisathatpornphat, R. Liu, Z. Huang, C. Jaikaeo).

Additional Information www.cis.udel.edu/~elloyd