Min-Max Coverage in
Multi-Interface Networks
Gianlorenzo D’Angelo, Gabriele Di Stefano
Dept. Electrical and Information Engineering
University of L’Aquila, Italy
{gianlorenzo.dangelo, gabriele.distefano}@ing.univaq.it
Alfredo Navarra
Dept. Mathematics and Computer Science
University of Perugia, Italy
[email protected]
Outline
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Introduction and Motivations
The Model
 Coverage problem
Explanatory example
Obtained results
 Hardness
 Approximation
 Special cases
Conclusion
Alfredo Navarra,University of Perugia,
Italy. [email protected]
Introduction & Motivation
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
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Heterogeneous Networks
Multi-Interface (multi-frequencies) devices
Limited power (both computational and battery)
Required services/connections
Alfredo Navarra,University of Perugia,
Italy. [email protected]
The Multi-Interface Model

Given a graph G = (V,E) with |V | = n and |E| = m, which
models a set of wireless devices (nodes V) connected by
multiple radio interfaces (edges E), the aim is to switch on
a minimum cost set of interfaces at each node in order to
satisfy some required connections
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Every node holds a subset of all the possible k interfaces
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A connection is satisfied when the endpoints of the corresponding
edge share at least one active interface
k might be set a priori (bounded case)
k might depend on the given instance (unbounded case)
The cost of maintaining an active interface is considered
(cost in terms of power percentage required by an interface)
 unit cost (i.e., the same for all the interfaces)
 non-unit cost (i.e., each type of interface has its own cost)
Alfredo Navarra,University of Perugia,
Italy. [email protected]
Min-Max Coverage, MMCC

Definition 1. A function W : V→2{1,…,k} is said to cover
G=(V,E) if for each {u,v} in E, the set W(u) ∩ W(v)≠Ø.
Alfredo Navarra,University of Perugia,
Italy. [email protected]
Example, MMCC
costs
: .6
: .75
: 1.2
: 1.4
: 1.8
:2
: 3.1
+
+
= 3.35
Alfredo Navarra,University of Perugia,
Italy. [email protected]
Cheaper solution
+
= 2.6
Alfredo Navarra,University of Perugia,
Italy. [email protected]
MMCC, complexity
Theorem 1. MMCC is NP-hard even when restricted to the
bounded unit cost case, for any fixed Δ ≥ 5 and k ≥ 16.
Sketch: Polynomial transformation from Satisfiability (with at most 3
literals for each clause and a variable appears, negated or not, in at
most 3 clauses) to the underlying decisional problem of MMCC
(bounding the cost to B=3).
Example:
q = (¬u + v + w), r = (v + ¬z),
s= (v+ ¬w + z),
Correspond to graph with:
W(eq) = {Fu, Tv, Tw},
W(er) = {Tv, Fz},
W(es) = {Tv, Fw, Tz},
W(dq)={Tu, Fu, Tv, Fv, Tw, Fw},
W(au)={Tu, Fu, B, C}
···
Alfredo Navarra,University of Perugia,
Italy. [email protected]
MMCC, complexity
Theorem 2. In the unit cost case with k ≤ 3, MMCC is
optimally solvable in O(m) time.
Sketch: One interface is shared by all the nodes, or each node activates
at most 2 interfaces (it is sufficient to check whether nodes with 3
interfaces can be connected with the nodes holding less interfaces by
means of only 2 interfaces), or at least one node must activates 3
interfaces.
Theorem 3. If the input graph is a tree and k = O(1) or Δ =
O(1), MMCC can be optimally solved in O(n) or O(k2Δn)
time, respectively.
(Dynamic Programming technique)
Theorem 4. If the input graph is a cycle, MMCC is optimally
solved in O(k6n) time.
Alfredo Navarra,University of Perugia,
Italy. [email protected]
MMCC, approximation
Theorem 5. Unless P = NP, MMCC in the unit cost
unbounded case cannot be approximated within an η ln(Δ)
factor for a certain constant η, even when the input graph is
a tree.
Proof (sketch from COCOA’10):
• reduction from Set Cover (SC) to MMCC
• the input graph is a star of n+1 nodes
• each node but the center encodes one element of SC
• each subset from SC is encoded by one interface
• the center holds all the interfaces
(it results that all the nodes reachable from the center by
means of a specific interface represent one subset of SC)
Alfredo Navarra,University of Perugia,
Italy. [email protected]
MMCC, approximation
Theorem 6. In the unit cost case, MMCC is k/2approximable in O(n) time.
Theorem 7. In the unit cost case MMCC is Δ/2approximable in O(n+m) time.
Theorem 8. Let I be an instance of MMCC where the input
graph admits a b-bounded ownership function, then there
exists an algorithm which guarantees a
(ln(Δ)+1+ b · min{ln(Δ)+1, cmax})-approximation factor, with
cmax = maxi∈{1,...k} c(i).
Alfredo Navarra,University of Perugia,
Italy. [email protected]
MMCC, approximation
Given a graph G = (V,E), an ownership function Own : E → V assigns
each edge {u, v} to an owner node between u or v. The set of nodes
connected to node u by the edges owned by u is
Own′(u) = {v | Own({u, v}) = u}.
Function Own is said to be b-bounded if |Own′(u)| ≤ b for each u ∈ V.
Alfredo Navarra,University of Perugia,
Italy. [email protected]
Conclusion
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We have considered the Min-Max Coverage problem in
Multi-Interface Networks studying hardness and
approximation factors in general and more specific settings
Other interesting variations deserve investigation
Further work includes the improvement of the achieved
results and the challenging study of the distributed version
of the problem
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
practical heuristics and experimental studies might be a first step
collaborative or selfish environments
Alfredo Navarra,University of Perugia,
Italy. [email protected]
Thank You!
Alfredo Navarra,University of Perugia,
Italy. [email protected]
Referencies
1. Caporuscio M., Charlet D., Issarny V., Navarra A.: Energetic Performance of Service-oriented Multi-radio
Networks: Issues and Perspectives. In 6th Int. Workshop on Software and Performance (WOSP), ACM Press,
42—45, 2007
2. Klasing R., Kosowski A., Navarra A.: Cost minimisation in multi-interface networks. In 1st EuroFGI Int. Conf.
on Network Control and Optimization (NetCooP). Volume 4465 of LNCS, Springer, 276—285, 2007
3. Kosowski A., Navarra A.: Cost minimisation in unbounded multi-interface networks. In 2nd PPAM Workshop on
Scheduling for Parallel Computing (SPC). Volume 4967 of LNCS, Springer 1039—1047, 2007
4. Kosowski A., Navarra A., Pinotti M. C.: Connectivity in Multi-Interface Networks. In 4th Symp. on Trustworthy
Global Computing (TGC). LNCS 5474, Springer, pp. 157—170, 2008
5. Barsi F., Navarra A., Pinotti M. C.: Cheapest Paths in Multi-Interface Networks. In 10th Int. Conf. on Distributed
Computing and Networking (ICDCN). LNCS 5408, Springer, pp. 37—42, 2009
6. Athanassopoulos S., Caragiannis I., Kaklamanis C., Papaioannou E.: Energy-efficient communication in
multi-interface wireless networks. In 34th Int. Symp. on Mathematical Foundations of Computer Science
(MFCS), LNCS 5743, Springer 102–111, 2009
7. Klasing R., Kosowski A., Navarra A.: Cost minimisation in wireless networks with bounded and unbounded
number of interfaces. In Networks, Vol. 54(1), pp. 12—19, 2009
8. Kosowski A., Navarra A., Pinotti, M.C.: Exploiting Multi-Interface Networks: Connectivity and Cheapest Paths.
In Wireless Networks. Vol. 16(4), pp. 1063—1073, 2010
9. D’Angelo G., Di Stefano G., Navarra A.: Minimizing the Maximum Duty for Connectivity in Multi-Interface
Networks, In 4th Int. Conf. on Combinatorial Optimization and Applications (COCOA). LNCS 6509, Springer 254267, 2010
10.D’Angelo G., Di Stefano G., Navarra A.: Min-Max Coverage in Multi-Interface Networks, In 37th Int. Conf. on
Current Trends in Theory and Practice of Computer Science (SOFSEM). LNCS 6543, Springer 190-201, 2011
11.D’Angelo G., Di Stefano G., Navarra A.: Bandwidth Constrained Multi-Interface Networks, In 37th Int. Conf. on
Current Trends in Theory and Practice of Computer Science (SOFSEM). LNCS 6543, Springer 202-213, 2011
12.D’Angelo G., Di Stefano G., Navarra A.: Maximum Flow and Minimum-Cost Flow in Multi-Interface Networks,
In 5th Int. Conf. on Ubiquitous Information Management and Communication (ICUIMC), 2011
13.Bertossi A., Navarra A., Pinotti M.C.: Maximum Bandwidth Broadcast in Single and Multi-Interface Networks,
In 5th Int. Conf. on Ubiquitous Information Management and Communication (ICUIMC), 2011
MMCC, approximation
Given a graph G = (V,E), an ownership function Own : E → V assigns
each edge {u, v} to an owner node between u or v. The set of nodes
connected to node u by the edges owned by u is
Own′(u) = {v | Own({u, v}) = u}.
Function
Own is said to be b-bounded if |Own′(u)| ≤ b for each u ∈ V.
The genus of a graph is the minimum number of handles that must be
added to the plane to embed the graph without any crossings
Alfredo Navarra,University of Perugia,
Italy. [email protected]
MMCC, approximation
Given a graph G = (V,E), an ownership function Own : E → V assigns
each edge {u, v} to an owner node between u or v. The set of nodes
connected to node u by the edges owned by u is
Own′(u) = {v | Own({u, v}) = u}.
Function Own is said to be b-bounded if |Own′(u)| ≤ b for each u ∈ V.
The arboricity of an undirected graph is the minimum number
of forest into which its edges can be partitioned.
Alfredo Navarra,University of Perugia,
Italy. [email protected]
MMCC, approximation
Given a graph G = (V,E), an ownership function Own : E → V assigns
each edge {u, v} to an owner node between u or v. The set of nodes
connected to node u by the edges owned by u is
Own′(u) = {v | Own({u, v}) = u}.
Function Own is said to be b-bounded if |Own′(u)| ≤ b for each u ∈ V.
The pagenumber of a graph is the minimum number of pages
required to embed the graph in a book, i.e., if the vertices are
rearranged along the spine of a book, the pagenumber is the
number of pages required to draw the edges without crossing.
Alfredo Navarra,University of Perugia,
Italy. [email protected]
MMCC, approximation
Given a graph G = (V,E), an ownership function Own : E → V assigns
each edge {u, v} to an owner node between u or v. The set of nodes
connected to node u by the edges owned by u is
Own′(u) = {v | Own({u, v}) = u}.
Function Own is said to be b-bounded if |Own′(u)| ≤ b for each u ∈ V.
The treewidth measures the number of graph vertices mapped onto
any tree node in an optimal tree decomposition
(i.e., a mapping of a graph into a tree).
Alfredo Navarra,University of Perugia,
Italy. [email protected]