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

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

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

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]

< 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

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

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

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

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

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

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”)

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

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

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

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

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

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

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

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

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]

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

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

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

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

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”

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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