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Strongly Polynomial-Time Truthful Mechanisms in One Shot Paolo Penna 1, Guido Proietti 2, Peter Widmayer 3 1 Università di Salerno 2 Università de l’Aquila 3 ETH Zurich

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Example: BGP Routing An Autonomous System may report false link status to redirect traffic to another AS AS1 AS2 source destination Link down

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Networks, Protocols, Mechanisms Network Protocol “efficient communication” Private Costs Selfish agent selected not selected ti cost ti cost 0 “bids” bi Payments that incentivize agents to be truthfulTruthtelling is a dominant strategyReporting bi = ti maximizes the utility of agent i, always utility = payment - cost

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Networks, Protocols, Mechanisms Network Protocol “efficient communication” Private Costs Selfish agent selected not selected ti cost ti cost 0 “bids” bi “Efficient protocol” “Incentive compatible efficient protocol” ? “Efficient” Alg “Efficient” truthful mechanism (Alg,Pay) ? Algorithmic Mechanism Design [Nisan&Ronen’99]

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< 3 Truthful Mechanisms 12 x Shortest Path Tree cheap expensive x 3 selectednot selected 3 < Depends on the “1” and “2” agent bid Monotone Algorithm

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Truthful Mechanisms Monotone algorithm Truthful mechanism (Vickrey’61, Myerson’81) Pay 0T cheap expensive x T selectednot selected bid of agent i bids of other agents Alg

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Truthful Mechanisms Monotone algorithm Truthful mechanism (Vickrey’61, Myerson’81) Payments Thresholds

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Truthful Mechanisms (Alg,Pay) Monotonicity Compute the payments Techniques to solve both? Efficient mechanism in one shot Two algorithmic problems naive approach: weakly polynomial-time

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Our (and Prior) Work General technique for obtaining efficient mechanisms is one-shot write Alg as a “combination” of simpler algorithms compute the payments from the simpler algorithms Monotonicity (easy to prove) Payment computations (efficient) Prior work: Monotone “combinations” [Mu’Alem&Nisan’02] Fast payment computations for several “combinations” [Kao&Li&Wang’05] compute payment no min-max problems Limitations

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Our (and Prior) Work General technique for obtaining efficient mechanisms is one-shot write Alg as a “combination” of simpler algorithms compute the payments from the simpler algorithms Mechanism for the Minimum Diameter Spanning Tree max latency

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Our (and Prior) Work General technique for obtaining efficient mechanisms is one-shot write Alg as a “combination” of simpler algorithms compute the payments from the simpler algorithms Mechanism for the Minimum Diameter Spanning Tree first mechanism, strongly polynomial-time (close to “best algorithm”)

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MIN(A1,A2) : run A1 and A2 independently; choose the solution whose cost is smaller. MIN(A1,A2): compute X1 := A1(b) and X2:=A2(b) if cost(X1,b) cost(X2,b) then return X1 else return X2 “Min combinations” Objective function MIN (A1,…,Ak) := MIN(A1,MIN(A2,…,Ak)) [Mu’Alem&Nisan’02] Agents’ bids

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Example: Minimum Radius Spanning Tree Rooted tree minimizing the longest path to the root (locate a server and minimize the maximum latency)

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x 120 1 Cost = height 11 x SPT2 T2 SPT1 MIN(SPT1,SPT2) MRST Example: Minimum Radius Spanning Tree

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11 x x T1 SPT2 T2 120 1 SPT1 MIN(SPT1,SPT2) SPT2SPT1SPT2 MRST MIN(SPT2,SPT1) Order matters!! Different payments! Cost = height Example: Minimum Radius Spanning Tree

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Payment Computations (Idea) x A1 A2 MIN (A1,A2)MIN (A2,A1) A1A2 The order matters T1 R1L1 T T non-decreasing constant

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Ah Aj Payment Computations (Idea) x The order matters RhLh lowest MIN(A1,…, Aj, …Ak) Ah Threshold T = “leftmost ’’

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General Technique Alg is MIN-reducible in time x Aj non-decreasing constant Tj Alg = MIN(A1,…,Ak) LjRj

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General Technique Alg is MIN-reducible in time Truthful (Alg,Pay) running in O(time Alg + ) time Main Application: MDST MDST Truthful mechnism for MDST running in O(n (n,m)time MDST ) time Easy Hard

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Min Diam. Spanning Tree (Idea) MDST = MIN(SPT e1,…,SPT em ) SPT e1 Cost = height (max dist to every other node) x LR [Assin&Tamir’95]

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Min Diam. Spanning Tree (Idea) MDST = MIN(SPT e1,…,SPT em ) x u e1 SPT

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Min Diam. Spanning Tree (Idea) MDST = MIN(SPT e1,…,SPT em ) x u e1 SPT x

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Min Diam. Spanning Tree (Idea) MDST = MIN(SPT e1,…,SPT em ) x u e1 SPT max x

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Min Diam. Spanning Tree (Idea) MDST = MIN(SPT e1,…,SPT em ) x u e1 SPT max x u’ u’’

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Min Diam. Spanning Tree (Idea) MDST = MIN(SPT e1,…,SPT em ) x u e1 SPT max x u’u’’ x LR limit = “at least limit” L R = “larger than limit”

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Extensions Technique: Every “Binary Game” (selected/not selected) Minimum Radius Spanning Tree (better running time) p-center (1-center MRST) Open Approximation: NP-hard problems MIN-reducible APX Alg? More general Agents (e.g. two edges per agent)

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