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

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Research activities Combinatorial Optimization Problems (from a centralized point of view): - problem complexity - analysis of algorithms Algorithmic Game Theory: - non cooperative game with complete knowledge. - Nash equilibria (convergence, existence and performance). 2 Sophia-Antipolis, 17 th November 2009

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Combinatorial Optimization Problems: WDM optical networks 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). 3 Sophia-Antipolis, 17 th November 2009

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An example b g a dc e fh 4 Sophia-Antipolis, 17 th November 2009

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..… 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. 5 Sophia-Antipolis, 17 th November 2009

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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. 6 Sophia-Antipolis, 17 th November 2009

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ADM grooming g=2 b g a dc e fh 7 Sophia-Antipolis, 17 th November 2009

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OADM grooming g=2 b g a dc e fh 8 Sophia-Antipolis, 17 th November 2009

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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 C ă linescu and others ( IEEE Journal of Selected Area on Communications, 2002 ) for g=1 and general topologies. 9 Sophia-Antipolis, 17 th November 2009

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….. A log g approximation algorithm has been proposed by Flammini and others ( Journal of discrete algorithm ) for every g and ring topologies. Amini and others ( Theoretical Computer Science ) proved that for g>=1 (for ring topologies) and g>=2 (for path topologies) the problem is APX-complete. 10 Sophia-Antipolis, 17 th November 2009

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My contribution M. Flammini, G. Monaco, L. Moscardelli, M. Shalom, e S. Zaks: Approximating the Traffic Grooming Problem in Tree and Star Networks. Journal of Parallel and Distributed Computing, 2008 (a preliminary version appeared 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 ln g+o(ln g) for unbounded directed tree. The main open problem is the determination of an approximation algorithm for general trees. 11 Sophia-Antipolis, 17 th November 2009

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..… M. Flammini, G. Monaco, L. Moscardelli, M. Shalom, e S. Zaks. Approximating the Traffic Grooming Problem with respect to ADMs 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. 12 Sophia-Antipolis, 17 th November 2009

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Combinatorial Optimization Problems: maximum-cover problem Instance: – S: collection of sets – V: ground set of elements – 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. 13 Sophia-Antipolis, 17 th November 2009

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Greedy algorithm for Cover problem Greedy algorithm has approximation ratio The result is tight [Feige 98]) An interesting special case is when the size of the sets of S is small (the inapproximability result does not hold!) k-cover problem, k denotes the maximum size of each set in S 14 Sophia-Antipolis, 17 th November 2009

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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 15 Sophia-Antipolis, 17 th November 2009

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….. Application: to WDM optical networks and in general to Resource allocation scenario. Known approximation algorithm: A 9/7 approximation algorithm for the 3-cover problem has been proposed by Caragiannis (STACS 2007). I. Caragiannis and G. Monaco : A 6/5 approximation algorithm for the maximum 3-cover problem. International Symposium on Mathematical Foundations of Computer Science (MFCS) 2008 (an extended version submitted to Theoretical Computer Science). 16 Sophia-Antipolis, 17 th November 2009

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Combinatorial Optimization Problems: 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. sjsj cjcj Job j 17 Sophia-Antipolis, 17 th November 2009

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….. 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. 18 Sophia-Antipolis, 17 th November 2009

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….. Solution 1: Cost =busy green + busy red + busy blue 19 Sophia-Antipolis, 17 th November 2009

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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) 20 Sophia-Antipolis, 17 th November 2009

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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+ )-approximation algorithm for the case in which all the jobs have in common a point of the time. 21 Sophia-Antipolis, 17 th November 2009

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Combinatorial Optimization Problems: Regenerator Placement Problem in Optical Networks Instance: - Optical Network (a Graph G) - set of requests, i.e. paths in G - integer d 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 22 Sophia-Antipolis, 17 th November 2009

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example Suppose d=4 b g a dc e fh 23 Sophia-Antipolis, 17 th November 2009

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example Suppose d=2 b g a dc e fh 24 Sophia-Antipolis, 17 th November 2009

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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. 25 Sophia-Antipolis, 17 th November 2009

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….. k/rt there is a bound on the number of regenerators at each node ∞/rt k/req there is a bound on the number of regenerators and only requests are given (and part of the solution is also to determine the actual routing) ∞/req 26 Sophia-Antipolis, 17 th November 2009

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My contribution 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. 27 complexityapproximability ∞/rt polynomial for trees, rings NP-hard for general network Θ(logm + log d) k/rt NP-Hard (k=1, d=3) Exp-APX ∞/req NP-Hard (all-to-all case, d=1) Θ(logm) (all-to-all) k/req NP-Hard (k=1, d=1) Exp-APX Sophia-Antipolis, 17 th November 2009

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Open problems The main open problem is solving the RPP/∞/req problem. Considering the objective function 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 th November 2009 28

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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 th November 2009 29

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….. 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 an optimal centralized solution (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 an optimal centralized solution (it gives us information on the minimum loss of performance a non-cooperative system has to suffer). Sophia-Antipolis, 17 th November 2009 30

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Algorithmic Game Theory: ADMs Minimization Problem Instance: - G=(V,E); - P={p 1,p 2,…,p n } n simple paths A wavelength assignment is a function such that w(p i )≠w(p j ) for any pair of paths sharing an edge. Each lightpath needs 2 ADMs, one at each endpoint, If two adjacent lightpaths are assigned the same color, they can share an ADM. 31 Sophia-Antipolis, 17 th November 2009

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….. We assume that every path p i is isuued and handled by 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 (agents using an 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. 32 Sophia-Antipolis, 17 th November 2009

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My contribution 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 preliminary version appeared in Wine 2007) - the two cost sharing methods are equivalent and induce games always convergent in polynomial time. - Price of anarchy is at most 5/3 for general graph, moreover it is tight even for rings 33 Sophia-Antipolis, 17 th November 2009

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….. Collusion of at most k players: - only the Egalitarian cost sharing yields 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). 34 Sophia-Antipolis, 17 th November 2009

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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. 35 Sophia-Antipolis, 17 th November 2009

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Algorithmic Game Theory: Isolation Games A metric space (X,d) where: - X is a set of point - respecting symmetry and triangular inequality. An instance of Isolation Game is ((X,d),K) where K is the set of players. 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 other ones. 36 Sophia-Antipolis, 17 th November 2009

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….. l-selection isolation game: the utililty of a player is given by the distance from the l-th nearest player Total-distance isolation game: the utility of a player is given by the sum of the distances from all the other players l-suffix isolation game: the utility of a player is given by the sum of the distances from the l furthest ones 37 Sophia-Antipolis, 17 th November 2009

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Previous works and applications Isolation games find a natural application in the problem of interference minimization in wireless networks Zhao and others introduced this game (ISAAC 2008): - 1-selection and total-distance are potential game in any simmetric space - For l-selection Nash equilibria always exists 38 Sophia-Antipolis, 17 th November 2009

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My contribution 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 39 Social FunctionPrice of stabilityPrice of anarchy SUM12 MIN∞∞ Sophia-Antipolis, 17 th November 2009

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….. Social FunctionPrice of stabilityPrice of anarchy SUM 1 2 MIN 12 40 Total-distance l-suffix isolation game are not potential games Sophia-Antipolis, 17 th November 2009

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Recently Working on the price of stability in network design with fair cost allocation for undirected graph. (Open problem in FOCS 2004) Modelling a situation in which selfish mobile users want to be connected minimizing their movements. 41 Sophia-Antipolis, 17 th November 2009

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