Transcript Fast Algorithms for Hard Graph Problems: Bidimensionality

Algorithms for
Wireless Network Design
MohammadTaghi HajiAghayi
Labs – Research
Purposes of this Talk
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Real-world applications with deep algorithmic
underpinnings and consequences
Present several problems motivated by (wireless
sensor) networks
Show how we can tie together theory and practice
Demonstrate nice intersections of wireless multi-hop
networks, algorithmic graph theory, probability
theory, computational geometry, computational
economics and finally computational complexity
Outline
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Focus on two real-world applications
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Power optimization in fault-tolerant topology
control and related problems
The low coverage problem and related
problems
Conclusion
Power Optimization in
Fault-Tolerant Topology Control
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Wireless multihop networks
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Power is the main limitation
Power assignment
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Simple low-power devices
A power setting for each device
Defines possible communication links
Power versus distance: It takes power rc
to transmit a message to distance r for
some power attenuation exponent c
between 2 and 4.
Power Optimization in
Fault-Tolerant Topology Control
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Goal: Minimize power usage while maintaining
key network properties
Connectivity: There is a communication path
between any pair of nodes
k-Fault tolerance: Connectivity is maintained in
light of at most k-1 failures
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Device failures (our focus)
Communication link failures
By k-Fault tolerance, we also have k-disjoint
paths and thus higher network capacity
Model
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A wireless network is modeled as a graph
G(V,E) with cost functions d and p on E
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V is the set of mobile devices
E is the set of pairs of devices which can
communicate bi-directionally
duv is the distance between device u and v
puv is the power needed to transmit between
device u and v (usually it is distance to the
power attenuation exponent)
Model
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Conversely, a subgraph H=(V,E’) of the
network graph G defines an assignment of
power settings: device u transmits at
p(u) = max {(u,v) in E’} puv
The power used by a wireless network
with power settings defined by H is
P(H) = Σ u in V p(u)
Problem Formulation
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Given
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A wireless network
Find
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An assignment of power settings that
guarantees k-fault tolerance while minimizing
power usage
Recall k-fault tolerance means the network
remains connected even when up to k-1
devices (or communication links) fail
Related Results for Power Minimization
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Connectivity
 Cone-based local heuristics
[Rodoplu, Meng ’99; Wattenhofer, Li, Bahl, Halpern, Wang ’02]
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A 2-approximation based on minimum weight
spanning tree
[Kerousis, Kranakis, Krizanc, Pelc ’00]
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A 1.69-approximation based on minimum
weight Steiner tree and a more practical 1.875approximation [Calinescu, Mandoiu, Zelikovsky ’02]
Related Results for
Power Minimization
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2-Fault tolerance
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Heuristic to minimize maximum transmit power
[Ramanathan, Rosales-Hain ’00] (the only
previous result)
Fault tolerance for general k
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Pioneered in [Bahramgiri, H., Mirrokni, WINET]
and [H., Immorlica, Mirrokni,IEEE/ACM TON]
More than 100 references
Cone-Based Heuristic
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Algorithm:
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Input: A set of nodes on the plane, with max. power P
Each node increases its power until the angle between any
two consecutive neighbors is less than some threshold or it
reaches its maximum power P.
Output: two nodes are connected if both can hear each
other with the new power assignment
Theorem [BHM’02]: If the network of max. powers is
k-connected and the angle between any pair of
adjacent neighbors is at most 2π/3k, then the new
network is k-connected (2π/3k is almost tight)
Main disadvantage: The algorithm is local and thus
does not give any bound on the global goal of
minimizing sum of the powers (or the average power)
Approximating Connectivity
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Recall the power P(H) of subgraph H is
P(H) = Σ u in V p(u)
where p(u) = max {(u,v) in H(E)} puv
Define the weight W(H) of subgraph H as
W(H) = Σ (u,v) in H(E) puv
Approximating Connectivity
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Theorem [KKKP ’00]: The minimum weight
spanning tree MST of G uses at most twice as
much power as the minimum power connected
subgraph OPT of G.
Lemma 1: For any graph G, P(G) ≤ 2W(G).
Lemma 2: For any tree T, W(T) ≤ P(T).
Lemma 3: OPT is a tree
Proof (of Thm): From the above lemmas,
P(MST) ≤ 2W(MST) ≤ 2W(OPT) ≤ 2P(OPT).
Approximating k-Connectivity
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Theorem: The minimum weight
k-connected subgraph of G is a
2k-approximation to minimum power
k-connected subgraph
Proof sketch: use Mader’s theorem to
decompose a graph into k forests and then use
the previous results for forests (trees)
Minimum weight k-connected subgraph:
an LP-based algorithm gives a solution of weight
at most O(log k) times optimal weight (n is at
least 6k2) [Cheriyan, Vempala, Vetta, STOC’02],
[Kortsarz, Nutov, STOC’04]
Approximating k-Connectivity
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Thus there is an O(k log k) approximation for
minimum power k-connected subgraph
A more complicated combinatorial algorithm
gives an O(k) approximation
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First we find a 2-approximation to the minimum
weight k-outconnected sub-graph using matroid
matching
We can add k-2 disjoint paths to this graph in
order to make it k-connected via a min-cost
k-flow algorithm
We pay a factor O(k) in each of the above two
steps to go from weight to power
Approximating k-Connectivity
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Furthermore, a more subtle approach gives O(log4n)
approximation, an improvement when k is large [H.,
Kortsarz, Mirrokni, Nutov, IPCO’05, also Math. Prog.’07]
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Theorem: If there are
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c-approximation for min. weight k-connected sub.
d-approximation for min. power k-edge cover
Then we have 2c+d-approximation for min power
 k-edge cover: sub-graph with min. degree ≥ k
 c is in O(log n) by previous results
4
 d is in O(log n) by a new involved algorithm
 There are some hardness results for the k-edgeconnectivity case (the best factor is in O(√n))
Distributed (Local) Approximation
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Algorithm: [H., Immorlica, Mirrokni, MOBICOM’03]
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Construct minimum weight spanning tree with O(n log n +
m) messages [Gallager, Humbler, Spira, ’83]
Use local augmentation to create a
k-connected sub-graph with O(n) messages
Theorem: If puv= (duv)c for all pairs of nodes and we
have metricity of distances, then the algorithm is an
O(1)-approximation when k is constant.
Comparing via simulations on random or real inputs
Some other variants [Bredin, Demaine, H., Rus,
MobiHoc’05], [Demaine, H., Mahini, Oveis, Sayedi,
Zadi,SODA’07]
A Related Problem:
Repairing Fault-Tolerance
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Problem Formulation: given an initial placement
of nodes (the unit disk graph model)
Objective:
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Add minimum number of nodes to obtain
connectivity
Or minimize maximum/average movement of the
current nodes without adding any new node
More generally obtaining k-fault tolerance or
k-connectivity of the whole network
Approximation Algorithms for
Repairing the Network
Minimum number of added nodes to obtain connectivity:
5/2-approximation [Du, Wang, Xu, 2001]
 More generally obtaining k-fault tolerance or kconnectivity: O(k4)-approximation (Simple Algorithm,
Complicated Analysis)
[Bredin, Demaine, H., Rus, MobiHoc’05]
 More generally minimizing movement to obtain a new
configuration with a property P (e.g. being connected
being independent, having a perfect matching, etc.)
 More formally the goal of movement problem is to move
the agents into a configuration containing at most h
vertices that contain all k agents and induce a “good”
target patterns, i.e., an induced graph, in the given set
G. Agents can have even different colors.
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Minimizing Movement:
Fixed-Parameter Tractability
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Fixed-parameter algorithms:
Parameterize problem by parameter P
(typically, the cost of the optimal solution)
and aim for f(P) nO(1) (or even f(P) + nO(1))
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There is an FPT algorithm for Vertex Cover
There is no FPT algorithm for Dominating Set
Minimizing Movement: FPT (cont’d)
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CONNECTIVITY. Move the pebbles (agents) so
that they are connected and on distinct vertices
GRID. Move the k2 pebbles so that they form a
k × k grid.
s-t CONNECTIVITY. Move the pebbles to form a
path of pebbled vertices between fixed vertices
s and t.
STEINER CONNECTIVITY. Connect the red
pebbles by moving the blue pebbles to form a
Steiner tree.
Minimizing Movement: FPT (cont’d)
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2-CONNECTIVITY. Move the pebbles so that they induce a 2connected graph and the pebbles are on distinct vertices.
s-t d-CONNECTIVITY. Move the pebbles so that there are d
vertex-disjoint paths using pebbled vertices between two fixed
vertices s and t.
s-t d-EDGE-CONNECTIVITY. Move the pebbles so that there are
d edge-disjoint paths of pebbled vertices between s and t.
FACILITY LOCATION. Move client and facility pebbles so that
each client pebble is within distance at most d from at least one
facility pebble.
MATCHING. Move the pebbles so that the pebbles are on
distinct vertices and there is a perfect matching in the graph
induced by the pebbles.
Minimizing Movement
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Essentially all these problems are polynomialy
hard to approximate or at least there is no
algorithm better than √n-approximatoin
(for max connectivity and max path there are
such algorithnms) [Demaine, H., Mahini, Oveis,
Sayedi, Zadi, SODA’07]
But all of them except GIRD, 2-CONNECTIVITY,
and FACILITY LOCATION with unbounded
number of clients have FPT algorithms which are
quite surprising and interesting for practical
purposes [Demaine, H.,Marx, Submitted’07]
Outline
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Focus on two real-world applications
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Power optimization in fault-tolerant topology
control and related problems
The low coverage problem and related
problems
Conclusion
Interference Problem
The Low Coverage Problem, Motivation
[Demaine, H., Feige, Salavatipur, SODA’06, SICOMP]
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The problem is a bit simplified formulation of a realworld application in Bell-Labs
Input:
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Decomposition of the plane into regions with various
client densities
n base station locations, each with a set of options for
power/directional cones/etc and a cost for each one
Budget limiting the total cost of options
Satisfaction sk for covering a region by k> 0 base
stations where s1≥ s2 ≥ s3 ≥ . . . ≥ 0 (simplified version
s1=1 and s2=s3=. . .=0 is called budgeted unique cover.)
Goal: Find a set of base stations and options within
the total budget which maximizes the total
satisfaction weighted by client densities
The Unique Coverage Problem
[Demaine, H., Feige, Salavatipur, SODA’06]
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The Unique Coverage Problem:
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Given a universe U of n elements, and
Given a collection S of subsets of U.
Find a sub-collection S’ , a subset of S, which maximizes
the number of elements that are uniquely covered, i.e.,
appear in exactly one set of S’
The Budgeted Unique Coverage Problem:
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Given profits for elements and costs for subsets, and
Given also a budget B
Find a sub-collection S’, a subset of S, whose total cost is
at most B and maximizes the total profit of elements that
are uniquely covered
The Unique Coverage Problem
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The budgeted case is almost equivalent to the
wireless network problem
It has the same flavor of the maximum
coverage problem (has e/(e-1)-approximation)
Unique coverage is a generalization of MAX-CUT
It has very close connections to the radio
broadcast problem (considered extensively)
For simplicity, we focus on the (un-budgeted)
unique coverage problem in the rest of the talk
The Unique Coverage Problem
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Simple O(log n) approximation algorithm
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Partition the elements into log n classes according
to their degrees, i.e., the number of sets that cover
an element
Let i be the class of maximum cardinality
Choose a set in S to be in S’ with prob. 1/2 i
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Proof Sketch: we uniquely cover 1/e2 fraction
of elements of class i in expectation
Can be de-randomized by the standard method
of conditional expectation
Can be generalized for budgets, real-world
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O(log B) where B is the max size of a subset (nontrivial)
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Hardness Result
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The algorithm seems naïve
Several other problems have the same solution
Can we do better?
It seems combination sometimes can be hard
Theorem: This problem is hard to approximate
within a factor better than O(logc n), 0<c<1,
unless NP has a sub-exponential algorithm.
O(log1/3 n) hard or even O(log n) hard under
stronger but plausible complexity assumptions
Proof Ideas
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B: Elements
A bad instance that we cannot
uniquely cover > 1/log n fraction
A: Sets
2 i random
subsets
In this graph, at most
O(n) elements of B
can be uniquely
covered by sets of A
Bi : Class i
Proof Ideas
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Bipartite Independent Set (BIS) problem:
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Given a bipartite graph G(A UB,E) where |A|=|B|=n
Find a bipartite sub-graph G’(A’ UB’, E’) where |A’|=a, |B’|=b and E’ is an
empty set.
Theorem: Unless NP has sub-exponential algorithm, it is hard
to decide between (n c,n/logdn)-BIS and (n c’ ,n/logd’ n)-BIS
where 0<c’<c ≤1 and 0 ≤ d<d’ ≤1
Now between A and Bi we put a random matching and an
instance of BIS where the edges remain with probability 1/2i-1
For Yes instance, we have a unique cover of size Ω(nlog1-d n)
For No instance, we have a unique cover of size at most
O(nlog1-d’n)
The inapproximability threshold can be improved under other
stronger but still plausible complexity assumptions
Other Aspects of the Results
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Unique coverage is the first natural maximization problem with
almost logarithmic-hardness
We believe that it can be a central problem for maximization
problems like set cover for minimization problems
Toward this end, we obtain the same almost logarithmic
hardness for envy-free pricing, an important problem in
computational economics considered by [Guruswami, Hartline,
Karlin, Kempe, Kenyon, McSherry, SODA’05]
Other maximization candidates: deadline TSP [BBCM, STOC’04]
and budgeted connected sub-graph [Moss, Rabani, STOC’01]
where both have O(log n) approximation but not better so far
Finally, better models of interference for the real-world
application also has been considered via some primal-dual
schemas and also simulations on real-world inputs [Bahl, H.,
Mirrokni, Qui, Saberi,IEEE TMC]
Application of Market Equilibrium in
Distributed Load Balancing
[Bahl, H., Jain, Mirrokni, Qui, Saberi,IEEE TMC]
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Wireless devices
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Cell-phones, laptops with WiFi cards
Referred as clients or users interchangeably
Demand connections to access points
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Uniform for cell-phones (voice connection)
Non-uniform for laptops (application
dependent)
Application of Market Equilibrium in
Distributed Load Balancing
[Bahl, H., Jain, Mirrokni, Qui, Saberi,IEEE TMC]
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Access points
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Capacities
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Cell-towers, Base stations, Wireless routers
Total traffic they can serve
Integer for cell-towers
Variable transmission power
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Capable of operating at various power levels
Assume levels are continuous real numbers
Clients to APs assignment
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Assign clients to APs in an efficient way
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No over-loading of APs
Assigning the maximum number of clients and
thus satisfying maximum demand
One Heuristic Solution
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A client connects to the AP with reasonable
signal and then the lightest load
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Requires support both from AP and Clients
APs have to communicate their current load
Clients have WiFi cards from various vendors
running legacy software
Overall it has limited benefit in practice
Ideal Case
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We would like a client connects to the AP with
the best received signal strength
If an AP j transmitting at power level Pj then a
client i at distance dij receives signal with
strength
Pij = a.Pj.dij-c
where a and c are constants capturing various
models of power attenuation
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Cell Breathing Heuristic
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An overloaded AP decreases its communication
radius by decreasing power
A lightly loaded AP increases its communication
radius by increasing power
Hopefully an equilibrium would be reached
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Will show that an equilibrium exist
Can be computed in polynomial time
Can be reached by a tatonnement process
Lets start with economics and game theory
Market Equilibrium – A distributed
load balancing mechanism.
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Fisher setting with linear Utilities:
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m buyers (each with budget Bi) and n goods for sale
(each with quantity qj)
Each buyer has linear utility ui, i.e. utility of i is
sumj uij xij where uij>= 0 is the utility of buyer i for good j
and xij is the amount of good j bought by i.
A market equilibrium or market clearance is a price
vector p that
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maximizes utility sumj uij xij of buyer i subject to his
budget sumj pj xij <= Bi
The demand and supply for each good j are equal
sumj xij = qj (and thus the budgets are totally spent).
Fisher Setting with Linear Utilities
Goods
Buyers
M1 , u1   u1 j x1 j
q1
j
xij
M i , ui   uij xij
j
M n , un   unj xnj
j
uij xij
qj
qm
Market Equilibrium – A distributed
load balancing mechanism.
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Static supply
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Prices
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Analogous to received signal strength function
Either all clients are served or all APs are saturated
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corresponding to powers at APs
Utilities
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corresponding to capacities of APs
Analogous to the market clearance(equiblirum) condition
Thus our situation is analogous to Fisher setting
with linear utilities
Clients assignment to APs
APs
Clients
D1 , u1  max P1 j x1 j
C1
j
xij
Pij xij
Di , ui  max Pij xij
j
Dn , un  max Pnj xnj
j
Cj
Cm
Analogousness Is Only Inspirational
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Get inspiration from various algorithms for
the Fisher setting and develop algorithms
for our setting
Though we do not know any reduction –
in fact there are some key differences
Differences from the Market
Equilibrium setting
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Demand
 Price dependent in Market equilibrium setting
 Power independent in our setting
Is demand splittable?
 Yes for the Market equilibrium setting
 No for our setting
Market equilibrium clears both sides but our solution
requires clearance on either client side or AP side
 This also means two separate linear programs for
these two separate cases
Three Approaches for
Market Equilibrium
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Convex Programming Based
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Primal-Dual Based
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Eisenberg, Gale 1957
Devanur, Papadimitriou, Saberi, Vazirani 2004
Auction Based
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Garg, Kapoor 2003
Three Approaches for
Load Balancing
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Linear Programming
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Primal-Dual
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Minimum weight complete matching
Uses properties of bipartite graph matching
Auction
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Useful in dynamically changing situation
Linear Programming Based Solution
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Create a complete bipartite graph
One side is the set of all clients
The other side is the set of all APs,
conceptually each AP is repeated as many
times as its capacity (unit demand)
The weight between client i and AP j is
wij = c.ln(dij) – ln(a)
Find the minimum weight complete
matching
Theorem
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Minimum weight matching is supported by
a power assignment to APs
Power assignment are the dual variables
Two cases for the primal program which is
known at the beginning
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Solution can satisfy all clients
Solution can saturate all APs
Case 1 – Complete matching
covers all clients
minimize

iC , j A
wij xij
subject to
i  C
x
ij
1
x
ij
 Cj
jA
j  A
iC
i  C , j  A
xij  0
Case 1 – Pick Dual Variables
minimize

iC , jA
wij xij
subject to
i  C
x
ij
1
 i
x
ij
 Cj
j
jA
j  A
iC
i  C , j  A
xij  0
Write Dual Program
maximize
  C 
iC
i
jA
j
j
subject to
i  C , j  A
j  A
i   j  wij
j 0
xij
Optimize the dual program
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Choose Pj = e π
Using the complementary slackness
condition we will show that the minimum
weight complete matching is supported by
these power levels
j
Proof
Dual feasibility gives:
-λi ≥ πj – wij= ln(Pj) – c.ln(dij) + ln(a) =
ln(a.Pj.dij-c)
 Complementary slackness gives:
xij=1 implies -λi = ln(a.Pj.dij-c)

(Remember if an AP j transmitting at power level Pj then
a client i at distance dij receives signal with strength Pij =
a.Pj.dij-c)
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Together they imply that i is connected to the
AP with the strongest received signal strength
Case 2 – Complete matching
saturates all APs
minimize

iC , jA
wij xij
subject to
i  C
x
ij
1
x
ij
 Cj
jA
j  A
iC
i  C , j  A
xij  0
Case 2 – The rest of the proof is
similar
Unsplittable Demand
minimize

iC , jA
wij xij
subject to
i  C
x
jA
j  A
1
D x
iC
i  C , j  A
ij
i ij
 Cj
xij  0
Unsplittable Demand
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The integer program is APX-hard in
general (because of knapsack)
Assuming that the number of clients is
much larger than the number of APs, a
realistic assumption, we can obtain a nice
approximation heuristic.
First we compute a basic feasible solution
Analysis of Basic Feasible
Solution
minimize

iC , jA
wij xij
subject to
i  C
x
jA
j  A
1
D x
iC
i  C , j  A
ij
i ij
 Cj
xij  0
Approximate Solution
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All xij’s but a small number of xij’s are integral
Theorem: Number of xij which are integral is at
least the number of clients – the number APs
Most clients are served unsplittably
Clients which are served splittably – do not
serve them
The algorithm is almost optimal
Discrete Power Levels
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Over the shelf APs have only fixed number
of discrete power levels
Equilibrium may not exist
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In fact it is NP-hard to test whether it exists
or not
If every client has a deterministic tie
breaking rule then we can compute the
equilibrium – if exists under the tie
breaking rule
Discrete Power Levels
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Start with the maximum power levels for
each AP
Take any overloaded AP and decrease its
power level by one notch
If an equilibrium exist then it will be
computed in time mk, where m is the
number of APs and k is the number of
power levels
This is a distributed tatonnement process!
Proof.
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Suppose Pj is an equilibrium power level for the
jth AP.
Inductively prove that when j reaches the
power level Pj then it will not be overloaded
again.

Here we use the deterministic tie breaking rule.
Reconstruction Problem

Geometric graph structure
Sensor network

Given: Information about local
geometry


E.g., approximate distances
Goal: Reconstruct global geometry
Motivation: Cricket Location System
[MIT SLAM Project since 2000]
[Badoiu, Demaine, H., Indyk, SOCG’04, DCG]

Cricket devices “chirp” radio & ultrasound





Radio travels at speed of light
Ultrasound travels at speed of sound
?
?
? ?
Measure
distances
Localized
Accuracy ~1cm
Cricket v2
General Embedding Problem
Given graph with edge weights
 Embed into desired metric space
(e.g., 2D or 3D) with edge lengths
matching specified weights
 Weights may be exact or
approximate
 Simple only if we know all
the distances exactly

Embedding with Approx. Distances


Exact embedding likely impossible
Goal: Minimize maximum distortion



Sample results:



Additive:
|embed length − true length|
Multiplicative: |embed length / true length|
[Badoiu, Demaine, H., Indyk, SOCG’04], Embed any
metric on n points into Euclidean 2D plane with
multiplicative/additive distortion (by knowing angles) or
quasi-polynomial time if we do not have the angles
[Alon, Badoiu, Demaine, Farach-Colton, H., Sidiropoulos,
SODA’05], Embed any metric on n points into low
dimensions such that we preserve the order
approximately
Approximating best embedding into plane in
polynomial time is a very important open problem
Conclusions


Introduction and motivations
Focused on two real-world applications



Several other related areas to wireless networks




Power optimization in fault-tolerant topology
control and related problems
The low coverage problem and related problems
Geometric embedding & Reconstruction Problem
Market equiblirum & Load Balancing
Game theory & Low Coverage
Questions ?
‫تشکر‬
Thanks forObrigado
your attention…