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Designing Algorithms for Combinatorial Public Projects

This compilation of slides from three talks by Michael Schapira discusses the design of algorithms for environments with selfish agents. It explores computational and economic concerns, algorithmic mechanism design, and the challenges of achieving truthfulness and computational efficiency. The focus is on the paradigmatic problem of combinatorial auctions and the impossibility proofs in the context of Combinatorial Public Projects.

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Designing Algorithms for Combinatorial Public Projects

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  1. Combinatorial Public Projects Slides compiled from 3 talks by Michael Schapira

  2. Designing Algorithms for Environments With Selfish Agents computational efficiency • Computational concerns: • bounded computational resources • exact optimization or good approximation • Economic concerns: • truthfulness incentive-compatibility

  3. Algorithmic Mechanism Design • Can these different desiderata coexist? • The central problem in Algorithmic Mechanism Design[Nisan-Ronen]

  4. Paradigmatic Problem: Combinatorial Auctions • A set of m items on sale {1,…m}. • n bidders {1,…,n}. Each bidder i has valuation functionvi : 2[m] → R≥0. • Goal: find a partition of the items between the bidders S1,…,Sn such that social welfare Si vi(Si) is maximized

  5. What Do We Want? • Quality of the solution:As close to the optimum as possible. • Computationally tractable:Polynomial running time (in n and m). • Truthful:Motivate (via payments) bidders to report their true values. • The utility of each agent is ui = vi(S) – pi • Solution concept: dominant strategies.

  6. Can Truth and Computation Coexist? Computation Hard (Clique) Easy (in APX, e.g., matching) Incentives Easy (social-welfare max. in auctions) Hard(max-min fairness in auctions) NO! [Papadim-Schap.-Singer] easy + easy = easy?

  7. Combinatorial Public Projects • Set of n agents; Set of m resources; • Each agent i has a valuation function: vi : 2[m] → R≥0 • Objective: Given a parameter k, choose a set of resources S* of size k which maximizes the social welfare: S* = argmaxiSi vi(S) S [m], |S|=k

  8. Assumptions Regarding Each Valuation Function • Normalized: v(∅) = 0 • Non-decreasing: v(S) ≤ v(T) S T • Submodular: v(Sυ{j}) − v(S) ≥ v(Tυ{j}) − v(T) S T

  9. Special Cases • Elections for a committee: The agents are voters, resources are potential candidates. • Overlay networks: The agents are source nodes, resources are potential overlay nodes.

  10. Are Combinatorial Public Projects Easy? • Computational Perspective:A 1-1/e approximation ratio is achievable (not truthful!) • A tight lower bound exists [Feige]. • Strategic Perspective:A truthful solution is achievable via VCG payments (but NP-hard to obtain) • What about achieving both simultaneously?

  11. Truth and Computation Don’t Mix • Theorem (Informal): [Papadimitriou-S-Singer]Anyalgorithm for CPPP that is both truthful and computationally-efficient does not have an approximation ratio better than 1/√m • Even for n=2. • Tight! [Schapira-Singer]. • Two models: • Communication complexity. • Computational complexity.

  12. RA all sets of resources of size k Maximal-In-Range Algorithms (= VCG-Based) Definition:A is MIR if there is someRA {|S| = k s.t. S  [m]}s.t. A(v1,…,vn) = argmax S  Rv1(S) + … + vn(S)* we shall refer to RA as A’s range. A

  13. A trivial √m-approximation Algorithm for Subadditive Agents • The algorithm: • If k ≤√m, simply choose the single resource j for which the social-welfare is maximized. • If k > √m, divide the m resources to √m disjoint sets of equal size and choose the one that maximizes the social welfare.

  14. The Algorithm is Truthful • Fact: Maximal-in-range algorithms are truthful (VCG). • The trivial approximation algorithm is (essentially) the best truthful algorithm for the submodular (and the subadditive) case!!!

  15. Upper and Lower Bounds constant non-truthful upper bound Submodular √m truthful upper bound Subadditive ? √m truthful upper bound

  16. Inapproximability of the Subadditive Case • Theorem: Any approximation algorithm for CPPP with subadditive agents which approximates better than O(m1/4) requires exponential communication in m. • Implications: The trivial truthful approximation algorithm is nearly tight even from a purely computational perspective.

  17. Combinatorial Public Projects:The Impossibility Proof mechanism design Complexity theory (what do truthfulalgorithms look like?) (the hardness of truthful algorithms) combinatorics (the combinatorialproperties of truthful algorithms)

  18. Communication Complexity • Theorem:Any truthful algorithm for CPPP that approximates better than 1/√m requires exponential communication.

  19. Proving the Lower Bound • Lemma 1: Any maximal-in-range (MIR) algorithm for CPPP that approximates better than 1/√m requires exponential communication in m. • Lemma 2 (!):An algorithm for the combinatorial public project problem is truthful iff it is MIR

  20. Lower Bound For MIR • Inapproximability Lemma: Any MIRalgorithm for CPPP that approximates better than 1/√m requires exponential communication in m. • Proof in two steps: [Dobzinski-Nisan] • Proposition 1: In order to get an approximation better than 1/√m, the range must be exponentially large (in m). • Even for n=1.Simple (succinctly described) valuations. • Proposition 2: Maximizing over a range RA requires communicating |RA| bits. • Even for n=2. We use the fact that valuations can be exponentially long.

  21. Characterization Lemma • Characterization Lemma:An algorithm for CPPP is truthful iff it is MIR • Theorem (Roberts 79): For unrestricted valuation functions any truthful algorithm is MIR. • Actually, affine maximizer… • We use machinery from simplified proofs of Roberts’ Theorem [Lavi-Mu’alem-Nisan]. • But… our domain is severely restricted! • But… our domain isn’t open!

  22. combinatorial auctions, combinatorial pubic projects, … ? Characterizing Truthfulness (cntd) single-parameterdomains unrestricted valuations Manynon-MIR algorithms(truthfulnessis well-understood) Only MIR(Roberts 1979) Not always MIR[auction environments: Lavi-Mu’alem-Nisan, Bartal-Gonen-Nisan] Truthful = MIRfor CPPP!

  23. Computational Hardness of Truthfulness • To prove our results we had to assume that the “input’’ can be exponential in m. • Realistic? • If users have succinctly described valuations then computational-complexity techniques are required. • No such impossibility results are known.

  24. Our Proof Revisited • Characterization Lemma: an algorithm is truthful iff it is an affine-maximizer. • Observation: The proof only requires succinctly-described valuations. • Inapproximability Lemma: Any affine maximizer which approximates better than √m requires exponential communication. • Proposition 1: In order to get an approximation better than √m, the range must be exponential. • Proposition 2: Maximizing over a range RA requires communicating |RA| bits.

  25. New Proof • Characterization Lemma: an algorithm is truthful iff it is an affine-maximizer. • Inapproximability Lemma: No affine maximizer can approximate better than √m unless [computational assumption] is false. • Proposition 1: In order to get an approximation better than √m, the range must be exponential. • New Challenge: Maximizing over an exponential-size range in polynomial time implies that[computational assumption] is false. • New technique.

  26. All sets of resources of size k RA Computational Complexity Hardness • For many families of succinctly described valuations CPPP is NP-hard. • Special case: MAX-K-COVERAGE[Feige] • So, optimizing over the set of all possible solutions is hard. • What about optimizing over a set of solutions of exponential size? • Intuition - also hard!

  27. So… • Truthulness and computation can clash! • In two complexity models. • APX is not preserved under truthfulness (unlike P and NP).

  28. Combinatorial Public Projects Problem (CPPP) [Papad.-Schap.-Singer] Set of nagents; Set of mresources; Each agent i has a valuation function: vi : 2[m] → R≥0 normalized, non-decreasing. Goal: Given a parameter k, choose a set of resources S* of size k which maximizes the social welfare: S* = argmaxiSi vi(S) S [m], |S|=k

  29. Complement-Free Hierarchy[Lehmann-Lehmann-Nisan] Complement-Free (Subadditive) Fractionally-Subadditive (“XOS”) Submodular Capped Additive (“Budget-Additive”) Gross Substitute Coverage Questions: Where does CPPP cease to be tractable? (VCG!) Where does CPPP cease to be approximable? Multi-Unit-Demand (“OXS”) Unit-Demand (“XS”)

  30. Complement-Free Hierarchy: Tractability Complement-Free (Subadditive) Fractionally-Subadditive (“XOS”) Submodular Capped Additive (“Budget-Additive”) Gross Substitute Coverage even for n=1 Multi-Unit-Demand (“OXS”) combinatorial auctions Unit-Demand (“XS”) CPPP

  31. Complement-Free Hierarchy: Approximability combinatorial auctions Complement-Free (Subadditive) Fractionally-Subadditive (“XOS”) CPPP Submodular Capped Additive (“Budget-Additive”) Gross Substitute Coverage Multi-Unit-Demand (“OXS”) Unit-Demand (“XS”)

  32. Complement-Free Hierarchy: Area of Interest Submodular Capped Additive (“Budget-Additive”) Gross Substitute Coverage Coverage even for n=1 Multi-Unit-Demand (“OXS”) Unit-Demand (“XS”) Unit-Demand (“XS”)

  33. A Simple Environments CPPP with unit-demand agents Each agent only wants one resource!

  34. 2-{0,1}-Unit-Demand user resources 0 0 1 0 1 Each user only wants (value 1) at most two resources and does not want (value 0) all others.

  35. Combinatorial auctions with such valuations are trivial. matching CPPP with such valuations is NP-hard. Vertex Cover But approximable (Solvable for constant n’s) The perfect starting point. What about truthful computation? 2-{0,1}-Unit-Demand

  36. Thm [Schap.-Singer]: There exists a computationally-efficient MIR mechanism for CPPP with complement-free valuations with appx ratio 1/√m. Thm: No computationally-efficient MIR mechanism for CPPP with 2-{0,1}-unit-demand valuations has appx ratio better than 1/√m unless SAT is in P/poly. 2-{0,1}-Unit-Demand

  37. What about general truthful mechanisms? Thm: There exists a computationally-efficient Greedy (non-MIR) mechanism for CPPP with 2-{0,1}-unit-demand valuations that has appx ratio ½. Simply choose the k most demanded resources. 2-{0,1}-Unit-Demand

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