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

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

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

Paradigmatic Problem: Combinatorial Auctions A set of m items on sale {1,…m}. n bidders {1,…,n}. Each bidder i has valuation function vi : 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

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.

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

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* = argmaxi Si vi(S) S [m], |S|=k

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

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.

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?

Truth and Computation Don’t Mix Theorem (Informal): [Papadimitriou-S-Singer] Any algorithm 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.

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

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.

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!!!

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

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.

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

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

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

Lower Bound For MIR Inapproximability Lemma: Any MIR algorithm 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.

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!

Characterizing Truthfulness (cntd) ? combinatorial auctions, combinatorial pubic projects, … single-parameter domains unrestricted valuations Many non-MIR algorithms (truthfulness is well-understood) Only MIR (Roberts 1979) Not always MIR [auction environments: Lavi-Mu’alem-Nisan, Bartal-Gonen-Nisan] Truthful = MIR for CPPP!

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.

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.

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.

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! All sets of resources of size k RA

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

Combinatorial Public Projects Problem (CPPP) [Papad.-Schap.-Singer] Set of n agents; Set of m resources; 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* = argmaxi Si vi(S) S [m], |S|=k 28

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

Complement-Free Hierarchy: Tractability (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 30

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

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”) 32

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

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

2-{0,1}-Unit-Demand 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? 35

2-{0,1}-Unit-Demand 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. 36

2-{0,1}-Unit-Demand 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. 37