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Gianpiero Monaco MASCOTTE INRIA-I3S-CNRS Sophia Antipolis

Gianpiero Monaco MASCOTTE INRIA-I3S-CNRS Sophia Antipolis Optimization and Non-Cooperative Issues in Communication Networks. Research activities. C ombinatorial O ptimization Problems ( from a centralized point of view ): - problem complexity - analysis of algorithms

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Gianpiero Monaco MASCOTTE INRIA-I3S-CNRS Sophia Antipolis

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  1. Gianpiero Monaco MASCOTTE INRIA-I3S-CNRS Sophia Antipolis Optimization and Non-Cooperative Issues in Communication Networks Sophia-Antipolis, 17 thNovember 2009

  2. Researchactivities • CombinatorialOptimizationProblems (from a centralizedpointofview): - problemcomplexity - analysisofalgorithms • Algorithmic Game Theory: - non cooperative game with complete knowledge. - Nash equilibria (convergence, existence and performance). Sophia-Antipolis, 17 thNovember 2009

  3. CombinatorialOptimizationProblems: WDM opticalnetworks • The network is modeled by a graph. • Large bandwidth exploited by wavelength division multiplexing (WDM). • Communication between a pair of nodes is done via a lightpath, which is assigned a certain wavelength (color). • Lightpaths sharing an edge must use different colors (with g=1). Sophia-Antipolis, 17 thNovember 2009

  4. An example b c d e a h g f Sophia-Antipolis, 17 thNovember 2009

  5. ..… • ADMs and OADMs are hardware components. • Each lightpath needs 2 ADMs, one at each endpoint, and an OADM for each intermediate node. • If two adjacent lightpaths are assigned the same color, they can share an ADM. • The Hardware cost is the number of ADMs and/or OADMs. Sophia-Antipolis, 17 thNovember 2009

  6. Grooming • The network operator can put together (groom) low capacity requests into high capacity colors. • More formally, colors can be assigned such that at most g lightpaths with the same color can share an edge. • If at most g lightpaths with the same color enter through the same edge to a node, they can share an ADM, thus saving g-1 ADMs. • If at most g lightpaths with the same color go through the same intermediate node, they can share an OADM, thus saving g-1 OADMs. Sophia-Antipolis, 17 thNovember 2009

  7. ADM grooming g=2 b c d e a h g f Sophia-Antipolis, 17 thNovember 2009

  8. OADM grooming g=2 b c d e a h g f Sophia-Antipolis, 17 thNovember 2009

  9. Previous Works (ADMs) • The problem of minimizing the number of ADMs was introduced by O. Gerstel and others (INFOCOM 1998). • T. Eliam and others (IEEE Journal of Selected Area on Communication, 2002), and A.L. Chiu and E.H. Modiano(Journal of Lightwave Technology, 2000), proved the hardness of the problem for g=1 and general g, respectively, in the case of (only) ADMs. • A approximation algorithm has been proposed by Călinescu and others(IEEE Journal of Selected Area on Communications, 2002) forg=1 and generaltopologies. Sophia-Antipolis, 17 thNovember 2009

  10. ….. • A log g approximationalgorithmhasbeenproposedbyFlammini and others(Journal of discrete algorithm) forevery g and ring topologies. • Amini and others (Theoretical Computer Science) provedthatfor g>=1 (for ring topologies) and g>=2 (forpathtopologies) the problemisAPX-complete. Sophia-Antipolis, 17 thNovember 2009

  11. Mycontribution M. Flammini, G. Monaco, L. Moscardelli, M. Shalom, e S. Zaks: Approximating the TrafficGroomingProblem in Tree and Star Networks. Journal ofParallel and DistributedComputing, 2008 (a preliminaryversionappeared in WG 2006). • Np-Completeness for star networks for any fixed g>2 . • Polynomial time algorithm for star networks for g≤2 . • a 2 ln(δg)+o(2 ln(δg)) approximation algorithms for the minimization of ADMs in bounded tree (for any fixed node degree bound δ). • a 2 lng+o(ln g) for unbounded directed tree. • The main open problem is the determination of an approximation algorithm for general trees. Sophia-Antipolis, 17 thNovember 2009

  12. ..… • M. Flammini, G. Monaco, L. Moscardelli, M. Shalom, e S. Zaks. Approximating the TrafficGroomingProblemwithrespecttoADMs and OADMs. International Conference on Parallel and Distributed Computing (Euro-Par) 2008. • Cost function ƒ(α)= α|OADM|+ (1-α)|ADM| 0≤ α≤1 • NP-completeness chain network for g=2 and for any α. • a approximation algorithms for chain and ring topology. Sophia-Antipolis, 17 thNovember 2009

  13. CombinatorialOptimizationProblems: maximum-coverproblem • Instance: • S: collectionofsets • V: ground set ofelements • w: a positive integer • Solution: • a collection of at most w sets of maximum benefit, i.e. whose union covers the maximum number of elements in V. Sophia-Antipolis, 17 thNovember 2009

  14. Greedy algorithm for Cover problem • Greedyalgorithmhasapproximationratio The resultis tight [Feige 98]) • An interestingspecial case iswhen the sizeof the setsofSissmall (the inapproximabilityresultdoesnothold!) • k-cover problem, kdenotes the maximumsizeofeach set in S Sophia-Antipolis, 17 thNovember 2009

  15. 3-cover problem • This is the simplest variant of maximum cover which is still APX-Hard • maximum 2-cover problem can be solved in polynomial time • Maximum 1-cover problem is trivial Sophia-Antipolis, 17 thNovember 2009

  16. ….. • Application: toWDM opticalnetworksand in generaltoResourceallocation scenario. • Knownapproximationalgorithm: A 9/7approximationalgorithmfor the 3-cover problemhasbeenproposedbyCaragiannis (STACS 2007) . • I. Caragiannis and G. Monaco : A 6/5 approximationalgorithmfor the maximum 3-cover problem. International Symposium on Mathematical Foundations of Computer Science (MFCS) 2008 (an extended version submitted toTheoretical Computer Science). Sophia-Antipolis, 17 thNovember 2009

  17. Job j sj cj CombinatorialOptimizationProblems: Parallel Scheduling with Application to Optical Networks • A job scheduling problem. • Parallel machines. • n jobs, each job given by an interval. • Parallelism parameter g 1 : A machine can process at most g jobs at any given point in time. Sophia-Antipolis, 17 thNovember 2009

  18. ….. • Example: A feasible schedule for g=2 • A machine is busy at time t if it processes at least one job at time t. • The goal: Minimize the total busy time of all the machines. Sophia-Antipolis, 17 thNovember 2009

  19. Solution 1: Cost =busygreen + busyred + busyblue ….. Sophia-Antipolis, 17 thNovember 2009

  20. Previous work • Batch Scheduling: • The number of machines is given. • The cost function is different • Maximum completion time as opposed to • Total busy time • The problem is NP-Hard even for g=2 • Winkler & Zhang (SODA 2003) Sophia-Antipolis, 17 thNovember 2009

  21. My contribution • M.Flammini, G.Monaco, L.Moscardelli, H. Shachnai, M.Shalom, T. Tamir and S.Zaks: Minimizing Total Busy Time in Parallel Scheduling with Application to Optical Networks. IEEE International Parallel and Distributed Processing Symposium (IPDPS) 2009. • An algorithm with approximation ratio between 3 and 4 for the general case. • A 2-approximation algorithm for proper interval graphs (no job is totally contained in another one) • A (2+e)-approximation algorithm for the case in which all the jobs have in common a point of the time. Sophia-Antipolis, 17 thNovember 2009

  22. CombinatorialOptimizationProblems: Regenerator Placement Problem in Optical Networks • Instance: - Optical Network (a GraphG) - set ofrequests, i.e. paths in G - integerd There is a need to put a regenerator every certain distance d because of a decrease in the power of the signal. • Goal:minimizing the number of locations to place the regenerators Sophia-Antipolis, 17 thNovember 2009

  23. example Suppose d=4 b c d e a h g f Sophia-Antipolis, 17 thNovember 2009

  24. example Suppose d=2 b c d e a h g f Sophia-Antipolis, 17 thNovember 2009

  25. Previous works • Works discuss the technological aspects of the problem and include heuristic algorithms for the problem: -S. Chen and S. Raghavan: The Regenerator Location Problem. In Proceedings of the International Network Optimization Conference (INOC 2007). - S. Pachnicke, T. Paschenda and P. M. Krummrich.Physical Impairment Based Regenerator Placement and Routing in Translucent Optical Networks. Proceedings of the Optical Fiber communication/National Fiber Optic Engineers Conference, (OFC/NFOEC 2008). - K. Sriram, D. Griffith, R. Su and N. Golmie. Static vs. dynamic regenerator assignment in optical switches: models and cost trade-offs. Proceedings of the Workshop on High Performance Switching and Routing, (HPSR 2004). - X. Yang and B. Ramamurthyn. Sparse Regeneration in Translucent Wavelength-Routed Optical Networks: Architecture, Network Design and Wavelength Routing. Photonic Network Communications, 10(1), 2005. Sophia-Antipolis, 17 thNovember 2009

  26. ….. • k/rtthereis a bound on the numberofregenerators at eachnode • ∞/rt • k/reqthereis a bound on the numberofregenerators and onlyrequests are given (and part of the solution is also to determine the actual routing) • ∞/req Sophia-Antipolis, 17 thNovember 2009

  27. Mycontribution • M. Flammini, A. Marchetti Spaccamela, G.Monaco, L. Moscardelli, S. Zaks: On the Complexity of the Regenerator Placement Problem in Optical Networks. ACM Symposium on Parallelism in Algorithms and Architectures (SPAA) 2009. Sophia-Antipolis, 17 thNovember 2009

  28. Open problems • The main open problem is solving the RPP/∞/reqproblem. • Considering the objectivefunction of minimizing the total number of regenerators. • Solving the on-line version of any of these problems and dealing with specific network topologies. • Considering the general case where each edge e has a weight w(e) (we assumed w(e) = 1 for every edge e), and the constraint is that the signal never travels a path whose weight (that is the sum of weights of its edges) is greater than d. Sophia-Antipolis, 17 thNovember 2009

  29. Algorithmic Game Theory • A scenario in which users pursue an own selfish strategy and the system evolves as a consequence of the interactions among them. • The interesting arising scenario is thus characterized by the conflicting needs of the users aiming to maximize their personal profit and of the system wishing to compute a socially efficient solution. • Algorithmic Game theory is considered the most powerful tool dealing with such non-cooperative environments in which the lack of coordination yields inefficiencies. • In such a scenario we consider the pure Nash equilibrium as the outcome of the game and in turn as the concept capturing the notion of stable solution of the system. Sophia-Antipolis, 17 thNovember 2009

  30. ….. • A pure Nash equilibrium is a stable outcome of a game in the sense that it is a state in which all players are satisfied with their payoff since none of them can improve it by unilaterally changing his strategy. • The price of anarchy is the ratio between the cost of the worst Nash equilibrium and the one of anoptimalcentralizedsolution (is a classical worst-case analysis and it measures the loss of performance due to the selfish behavior of players). • The price of stability is the ratio between the cost of the best Nash equilibrium and the one of anoptimalcentralizedsolution (itgives us information on the minimum loss of performance a non-cooperative system has to suffer). Sophia-Antipolis, 17 thNovember 2009

  31. Algorithmic Game Theory:ADMsMinimizationProblem • Instance: - G=(V,E); - P={p1,p2,…,pn} n simplepaths • A wavelengthassignmentis a function suchthat w(pi)≠w(pj) foranypairofpathssharinganedge. • Each lightpath needs 2 ADMs, one at each endpoint, If two adjacent lightpaths are assigned the same color, they can share an ADM. Sophia-Antipolis, 17 thNovember 2009

  32. ….. • We assume thateverypath piisisuued and handledby a player. • The strategy set of a player is the collection of all the possible subsets of at most two other adjacent (not overlapping) paths, one per endpoint (player chooses the ADMs). • Shapley (agentsusingan ADM pay for it by equally splitting its cost) and Egalitarian (the whole hardware cost is equally split among all the players) cost sharing . • The social function (the whole system’s objective function) is the whole hardware cost, i.e. the total number of ADMs. Sophia-Antipolis, 17 thNovember 2009

  33. Mycontribution • M.Flammini, G.Monaco, L.Moscardelli, M.Shalom, S.Zaks: Selfishness, Collusion and Power of Local Search for the ADMs Minimization Problem. Computer Networks, 2008 (a preliminaryversionappeared in Wine 2007) - the two cost sharing methods are equivalent and induce games always convergent in polynomialtime. - Price ofanarchyis at most5/3 forgeneralgraph, moreoveritis tight evenforrings Sophia-Antipolis, 17 thNovember 2009

  34. ….. • Collusionof at mostkplayers: - onlythe Egalitariancostsharingyields a well-founded definition of induced game. - The game is still convergent. - Price of collusion 3/2 + 1/k (surprising 3/2+ ε is the best known approximation ratio reached by a centralized algorithm). Sophia-Antipolis, 17 thNovember 2009

  35. Open problems • the determination of new cost sharing methods reaching a compromise between the Shapley and Egalitarian ones may lead in a local search algorithm improving the best known approximation ratio. Sophia-Antipolis, 17 thNovember 2009

  36. Algorithmic Game Theory:IsolationGames • A metricspace (X,d) where: - Xis a set ofpoint - respectingsymmetryandtriangularinequality. • An instanceofIsolation Gameis ((X,d),K) whereKis the set ofplayers. • The strategy set of each player is given by the set X. • the utility of a player is defined as a function of his distances from the otherones. Sophia-Antipolis, 17 thNovember 2009

  37. ….. • l-selectionisolation game: the utililtyof a player isgivenbythe distance from the l-th nearest player • Total-distanceisolation game: the utility of a player isgivenby the sum of the distances from all the other players • l-suffixisolation game: the utility of a player isgivenbythe sum of the distances from the l furthest ones Sophia-Antipolis, 17 thNovember 2009

  38. Previousworks and applications • Isolation games find a natural application in the problem of interference minimization in wireless networks • Zhao and othersintroducedthis game (ISAAC 2008): - 1-selection and total-distance are potential game in anysimmetricspace - Forl-selectionNash equilibriaalwaysexists Sophia-Antipolis, 17 thNovember 2009

  39. Mycontribution • V. Bilò, M. Flammini, G. Monaco, L. Moscardelli: On the performances of Nash Equilibria in Isolation Games. International Computing and Combinatorics Conference (COCOON) 2009. Invited to a special issue in Journal of Combinatorial Optimization. l-selection Sophia-Antipolis, 17 thNovember 2009

  40. ….. Total-distance • l-suffixisolation game are notpotentialgames Sophia-Antipolis, 17 thNovember 2009

  41. Recently • Working on the price ofstability in network design with fair costallocationforundirectedgraph. (Open problem in FOCS 2004) • Modellinga situation in whichselfish mobile userswanttobeconnectedminimizingtheirmovements. Sophia-Antipolis, 17 thNovember 2009

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