Tung-Wei Kuo, Kate Ching-Ju Lin, and Ming-Jer Tsai

Academia Sinica, TaiwanNational Tsing Hua University, Taiwan

Maximizing Submodular Set Function

with Connectivity Constraint: Theory and Application to Networks

• Mesh network deployment

Motivation

• Mesh network deployment

Motivation

How should we deploy the network?

Candidate

location

• Mesh network deployment

Motivation

Candidate

location

The budget is limited!

• Only one router can access the Internet

• Mesh networks exploit multi-hop relays

Connectivity

Candidate

location

• Only one router can access the Internet

• Mesh networks exploit multi-hop relays

Connectivity

Candidate

location

• Only one router can access the Internet

• Mesh networks exploit multi-hop relays

Connectivity

The network must be connected!

Various Performance Metrics • A variety of performance metrics– The number of covered users, total

throughput, the size of the coverage area, …

Given limited resources (routers or budget),

deploy a connected mesh that optimizes the performance metric

𝐺=(𝑉 ,𝐸)

Mesh Deployment Problem• Given:

1. routers, where one of them is a gateway2. The set of candidate locations, 3. The set of connection edges, 4. The optimization goal (e.g., the number of covered users)

𝑘=3This is the

optimal solution

A graph

GOAL: Construct a connected network such that the optimization goal is

achieved

{¿

Design an algorithm for each of the various

optimization goals?Many optimization goals can be modeled as submodular set

functions

Our goal: A universal algorithm for a family of problems whose objective can be modeled

as asubmodular set function

Submodular Set Function

A function is a submodular set function if

Example: Number of covered users

𝑓 (𝑆 )+ 𝑓 (𝑇 )≥ 𝑓 (𝑆∩𝑇 )+ 𝑓 (𝑆∪𝑇 ) ,∀ 𝑆 ,𝑇⊆𝑉𝑎

𝑏𝑐 𝑑

𝑆={𝑎 ,𝑏 ,𝑐 } 𝑇={𝑐 ,𝑑 }

𝑑

𝑎

𝑏𝑐

𝑓 (𝑆 )=6𝑆={𝑎 ,𝑏 ,𝑐 } 𝑇={𝑐 ,𝑑 }

Example: Number of covered users

𝑏

𝑎

𝑐 𝑑

𝑓 (𝑇 )=4𝑓 (𝑆 )=6𝑆={𝑎 ,𝑏 ,𝑐 } 𝑇={𝑐 ,𝑑 }

Example: Number of covered users

𝑏

𝑎

𝑐 𝑑 𝑓 (𝑆∩𝑇 )= 𝑓 ( {𝑐 } )=3

𝑓 (𝑇 )=4𝑓 (𝑆 )=6𝑆={𝑎 ,𝑏 ,𝑐 } 𝑇={𝑐 ,𝑑 }

Example: Number of covered users

𝑏

𝑎

𝑐 𝑑𝑓 (𝑆∪𝑇 )= 𝑓 ( {𝑎 ,𝑏 ,𝑐 ,𝑑} )=6

6+4>3+6𝑓 (𝑇 )=4𝑓 (𝑆 )=6

𝑆={𝑎 ,𝑏 ,𝑐 } 𝑇={𝑐 ,𝑑 }

Example: Number of covered users

𝑓 (𝑆∩𝑇 )= 𝑓 ( {𝑐 } )=3

Example: Total Data Rate

100100

11001

1

𝑏

𝑎

𝑐 𝑑1

100

Example: Total Data Rate

𝑓 ({𝑎 ,𝑏 ,𝑐 })=303

100100

1100

1

𝑏

𝑎

𝑐 𝑑1

100

1

Formal Problem Definition• Given:1. A graph 2. A positive integer 3. A nondecreasing submodular set function

on the set of subsets of with

• Goal: Find a subset such that1. Connectivity: is connected with respect to 2. Limited resources:3. Optimization goal: is maximized

Formal Problem Definition• Given:1. A graph 2. A positive integer 3. A nondecreasing submodular set function

on the set of subsets of with

• Goal: Find a subset such that1. Connectivity: is connected with respect to 2. Limited resources:3. Optimization goal: is maximized

The problem is NP-hard.An approximation algorithm will be

given

Our Algorithm

For every candidate location, , generate a solution in the following

way:Step 1. Find an area, , centered at Step 2. Deploy some routers on Step 3. Use the remaining routers to make the solution connected

The Idea

The best solution is then the final output

The Solution-Step 11. Find an area centered at with a radius of hops

𝑘=16

𝐺=(𝑉 ,𝐸)

𝑟

The Solution-Step 1

𝐺=(𝑉 ,𝐸)

radius : hops 𝑘=16

1. Find an area centered at with a radius of hops

𝑟

The Solution-Step 2

𝐺=(𝑉 ,𝐸)

radius : hops 𝑘=16

2. Deploy routers, where one of them is at the center

𝑟

The Solution-Step 2

𝐺=(𝑉 ,𝐸)

radius : hops 𝑘=16

𝑟

# of covered users2. Deploy routers, where one of them is at the center

The Solution-Step 2

𝐺=(𝑉 ,𝐸)

User

Candidate

location

radius : hops

# of covered users

𝑘=16

𝑟

2. Deploy routers, where one of them is at the center

The Solution-Step 2

𝐺=(𝑉 ,𝐸)

User

Candidate

location

# of covered users

𝑘=16radius : hops

𝑟

2. Deploy routers, where one of them is at the center

The Solution-Step 3

𝐺=(𝑉 ,𝐸)

User

Candidate

location

3. Use shortest paths to connect routers to the center

# of covered users

𝑘=16radius : hops

This is a feasible solution

The Solution-Step 3

𝐺=(𝑉 ,𝐸)

User

Candidate

location

radius : hops

# of covered users

𝑘=16

3. Use shortest paths to connect routers to the center

The AlgorithmFor every candidate location, ,

generate a solution in the following way:Step 1. Find an area, , centered

at , with radius Step 2. Deploy routers on Step 3. Use the remaining routers to make the solution connected

The best solution is then the final output.How, exactly, should we deploy the routers?

How to Deploy the Routers?• Solve a subproblem that is similar to the

main problem, except that① The solution can be disconnected② The center of the given area must be

chosen

• It is still NP-hard–When is dropped, Nemhauser et al.

propose an -approximation algorithm [9]

–We modify Nemhauser’s algorithm to satisfy[9] G. L. Nemhauser, L. A. Wolsey, and M. L. Fisher, “An analysis of approximations for maximizing submodular set functions-I,” Mathematical Programming, vol. 14, pp. 265–294, 1978.

Approximation Ratio

– is the optimal solution when only routers can be used

Our algorithm is an -approximation algorithm

The Problem with Heterogeneous Deployment Costs

Different locations might have different deployment costs

Formal Problem Definition

• Given:1. A vertex-weighted graph 2. A nondecreasing submodular set function on the set of subsets of with 3. A positive integer

• Find a subset such that:1. Connectivity: is connected with respect to 2. Limited budget: The total weight of 3. Optimization goal: is maximized

Approximation Ratio• ,

where is the maximum degree of • A special case: Unit disk graph ⇒

Simulation Results-Use Synthesis Data

Simulation Setting• Field size: 1200 m × 1200 m • User:– # of users: 200– Zipf’s law– 802.11b

• Candidate locations:– Grid network– Grid size: 100 m × 100 m

• Communication range: 150 m• Channel error model: 802.11b PHY Simulink

Model

Another Common Scenario• In some applications, a specific

location may need to be included in the solution

• We modify our algorithm accordingly:How to findthe center?

Our algorithm Try all the possible centers and choose the best one

Our algorithmw/ specific center

Let the specificlocation be the desired center

Comparison Schemes• Two greedy heuristics: – Try all the possible starting locations– Add one neighboring vertex at a time–Minimum deployment cost or

maximum performance gain

• When

[17] F. Vandin, E. Upfal, and B. J. Raphael, “Algorithms for detecting significantly mutated pathways in cancer,” Journal of Computational Biology, vol. 18, pp. 507–522, 2011.

Goal = maximum number of covered usersHomogeneous costs

We compare with Vandin’s algorithm [17]

Simulation Scenarios• Two types of deployment costs:

1. Homogeneous costs2. Heterogeneous costs

• Two performance metrics:1. Total data rate2. The number of covered users

Maximum Total Data Rate• Homogeneous costs

0 10 20 30 40 50 60 70 80 90 1000200400600800

10001200140016001800

Tota

l dat

a ra

te o

f cov

ered

us

ers (

Mb/

sec)

Number of routers, k

Upper boundArbitrary solutionGreedy: max date rateGreedy: max data rate w/ specific centerOur algorithmOur algorithm w/ specific center

• Heterogeneous costs

0 100 200 300 400 500 600 7000200400600800

10001200140016001800

Tota

l dat

a ra

te o

f cov

ered

us

ers (

Mb/

sec)

Total budget for deployment, B

Upper boundArbitrary solutionGreedy: min costGreedy: min cost w/ specific centerGreedy: max data rateGreedy: max data rate w/ specific centerOur algorithmOur algorithm w/ specific center

Maximum Total Data Rate

Maximum Number of Covered Users

0 5 10 15 20 25 300

50

100

150

200

Num

ber o

f cov

ered

use

rs

Upper boundArbitrary solutionVandin’s algorithmVandin’s algorithm w/ specific centerOur algorithmOur algorithm w/ specific center

• Homogeneous costs

Number of routers, k

• Heterogeneous costs

0 100 200 300 400 500 600 7000

50

100

150

200

Num

ber o

f cov

ered

use

rs

Total budget for deployment, B

Upper boundArbitrary solutionGreedy: min costGreedy: min cost w/ specific centerGreedy: max coverageGreedy: max coverage w/ specific center Our algorithmOur algorithm w/ specific center

Maximum Number of Covered Users

Summary of the simulation results

1. Our algorithm can be applied to different optimization goals

2. The ratio between the upper bound and our algorithm matches the approximation ratio

3. Our algorithms perform better than the greedy heuristics

Simulation Results-Use the Census of Taipei

Use the Census of Taipei

• Use the census to locate the users• Heterogeneous deployment costs:– Higher costs are assigned to locations

with higher population density• Goal: Maximize the number of

covered users

Input8 km

12 kmTotal cost of all locations: 60053

Number of users: 7126

Output

The output when the available budget = 15000Number of covered users: 6600 (≈93% of the total users)

8 km

12 km

The Results

0 5000 10000 15000 200000

1000

2000

3000

4000

5000

6000

7000

Num

ber o

f cov

ered

use

rs

Total budget for deployment, B

Upper boundArbitrary solutionGreedy: min costGreedy: min cost w/ specific centerGreedy: max coverageGreedy: max coverage w/ specific centerOur algorithmOur algorithm w/ specific center

Conclusion• We study the problem of finding a

connected set that maximizes some submodular set function under a limited budget

• We propose a universal algorithm for the mesh deployment problem

• We prove that the approximation ratio of the universal algorithm is

Thank you