Min-Max Coverage in Multi-Interface Networks

1. Min-Max Coverage in Multi-Interface Networks Gianlorenzo D’Angelo, Gabriele Di Stefano Dept. Electrical and Information Engineering Universityof L’Aquila, Italy {gianlorenzo.dangelo, gabriele.distefano}@ing.univaq.it Alfredo Navarra Dept. Mathematics and Computer Science Universityof Perugia, Italy navarra@dmi.unipg.it

2. Outline • Introduction and Motivations • The Model • Coverage problem • Explanatory example • Obtained results • Hardness • Approximation • Special cases • Conclusion Alfredo Navarra,Universityof Perugia, Italy. navarra@dmi.unipg.it

3. Introduction & Motivation • Heterogeneous Networks • Multi-Interface (multi-frequencies) devices • Limited power (both computational and battery) • Required services/connections Alfredo Navarra,Universityof Perugia, Italy. navarra@dmi.unipg.it

4. 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,Universityof Perugia, Italy. navarra@dmi.unipg.it

5. Min-Max Coverage, MMCC • Definition 1. A function W : V→2{1,…,k} is said to coverG=(V,E) if for each {u,v} in E, the set W(u)∩ W(v)≠Ø. Alfredo Navarra,Universityof Perugia, Italy. navarra@dmi.unipg.it

6. Example, MMCC costs : .6 : .75 : 1.2 : 1.4 : 1.8 : 2 : 3.1 + + = 3.35 Alfredo Navarra,Universityof Perugia, Italy. navarra@dmi.unipg.it

7. Cheaper solution + = 2.6 Alfredo Navarra,Universityof Perugia, Italy. navarra@dmi.unipg.it

8. 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), Correspondtographwith: 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,Universityof Perugia, Italy. navarra@dmi.unipg.it

9. MMCC, complexity • Theorem 2. In the unitcostcase withk ≤ 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. • (DynamicProgrammingtechnique) • Theorem 4. If the input graphis a cycle, MMCC is optimally solved in O(k6n) time. Alfredo Navarra,Universityof Perugia, Italy. navarra@dmi.unipg.it

10. 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,Universityof Perugia, Italy. navarra@dmi.unipg.it

11. MMCC, approximation 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 beaninstanceof MMCC where the input graphadmits a b-boundedownershipfunction, thenthereexistsanalgorithmwhichguaranteesa (ln(Δ)+1+ b · min{ln(Δ)+1, cmax})-approximation factor, with cmax = maxi∈{1,...k} c(i). Alfredo Navarra,Universityof Perugia, Italy. navarra@dmi.unipg.it

12. 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 uis Own′(u) = {v | Own({u, v}) = u}. Function Own is said to be b-bounded if |Own′(u)| ≤ b foreachu ∈ V. Alfredo Navarra,Universityof Perugia, Italy. navarra@dmi.unipg.it

13. 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,Universityof Perugia, Italy. navarra@dmi.unipg.it

14. Thank You! Alfredo Navarra,Universityof Perugia, Italy. navarra@dmi.unipg.it

15. Referencies • 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 • 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 • 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 • 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 • 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 • 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 • 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 • Kosowski A., Navarra A., Pinotti, M.C.: Exploiting Multi-Interface Networks: Connectivity and Cheapest Paths. InWireless Networks. Vol. 16(4), pp. 1063—1073, 2010 • 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 • 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 • 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 • 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 • 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

16. 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 foreachu ∈ 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,Universityof Perugia, Italy. navarra@dmi.unipg.it

17. 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 foreachu ∈ V. The arboricity of an undirected graph is the minimum number of forest into which its edges can be partitioned. Alfredo Navarra,Universityof Perugia, Italy. navarra@dmi.unipg.it

18. 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 foreachu ∈ 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,Universityof Perugia, Italy. navarra@dmi.unipg.it

19. 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 foreachu ∈ 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,Universityof Perugia, Italy. navarra@dmi.unipg.it