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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 - PowerPoint PPT Presentation
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Min-Max Coverage in Multi-Interface Networks
Gianlorenzo D’Angelo, Gabriele Di StefanoDept. Electrical and Information EngineeringUniversity of L’Aquila, Italy {gianlorenzo.dangelo, gabriele.distefano}@ing.univaq.it
Alfredo NavarraDept. Mathematics and Computer ScienceUniversity of Perugia, Italy [email protected]
Outline
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
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 A connection is satisfied when the endpoints of the corresponding
edge share at least one active interface Every node holds a subset of all the possible k interfaces
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]
+ + = 3.35
Example, MMCC costs
: .6
: .75
: 1.2
: 1.4
: 1.8
: 2
: 3.1
Alfredo Navarra,University of Perugia, Italy. [email protected]
MMCC, complexity
Alfredo Navarra,University of Perugia, Italy. [email protected]
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}
···
MMCC, complexity
Alfredo Navarra,University of Perugia, Italy. [email protected]
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.
MMCC, approximation
Alfredo Navarra,University of Perugia, Italy. [email protected]
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)
MMCC, approximation
Alfredo Navarra,University of Perugia, Italy. [email protected]
Theorem 6. In the unit cost case, MMCC is k/2-approximable in O(n) time.
Theorem 7. In the unit cost case MMCC is Δ/2-approximable 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).
MMCC, approximation
Alfredo Navarra,University of Perugia, Italy. [email protected]
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 isOwn′(u) = {v | Own({u, v}) = u}. Function Own is said to be b-bounded if |Own′(u)| ≤ b for each u V∈ .
Conclusion 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 practical heuristics and experimental studies might be a first step collaborative or selfish environments
Alfredo Navarra,University of Perugia, Italy. [email protected]
Referencies1. 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 254-267, 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
Alfredo Navarra,University of Perugia, Italy. [email protected]
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
MMCC, approximation
Alfredo Navarra,University of Perugia, Italy. [email protected]
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.
MMCC, approximation
Alfredo Navarra,University of Perugia, Italy. [email protected]
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.
MMCC, approximation
Alfredo Navarra,University of Perugia, Italy. [email protected]
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).
