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Complexity Issues in Multiagent Resource Allocation

Complexity Issues in Multiagent Resource Allocation. Paul E. Dunne Dept. of Computer Science University of Liverpool United Kingdom. Overview. Modelling resource allocation. Assessing allocations. Complexity considerations Computational complexity properties.

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Complexity Issues in Multiagent Resource Allocation

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  1. Complexity Issues in Multiagent Resource Allocation Paul E. Dunne Dept. of Computer Science University of Liverpool United Kingdom 2nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28th February – 1st March 2005

  2. Overview • Modelling resource allocation. • Assessing allocations. • Complexity considerations • Computational complexity properties. • A Model for negotiating allocations • and its properties. • Open questions and conjectures 2nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28th February – 1st March 2005

  3. Modelling Resource Allocation • A = {a1 , … , an } –set of nagents. • R = {r1 , … , rm }–resource collection. • U = {u1 , … , un } –utility functions. • Utility function – u – maps subsets of R to rational values. • An allocation is apartition of Rinto n sets - P = <P1 ; … ; Pn >- • n,m denotes the set of all allocations. 2nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28th February – 1st March 2005

  4. Assumptions • Exactly one agent owns any resource, i.e. R is non-shareable. • Utility functions have no allocative externality, i.e. for any P, Q n,mwith Pi = Qiit holds that ui(Pi ) = ui(Qi ). 2nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28th February – 1st March 2005

  5. Assessing Allocations • Qualitative measures. Pareto Optimality Envy Freeness • Quantitative measures. Utilitarian Social Welfare Egalitarian Social Welfare 2nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28th February – 1st March 2005

  6. Qualitative Assessment I • An allocation, P, is ParetoOptimal if for every allocation, Q, that differs from it should there be an agent for whom ui(Qi ) > ui(Pi ) then there is another agent for whom ui(Pi ) > ui(Qi ). 2nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28th February – 1st March 2005

  7. Qualitative Assessment II • An allocation, P, is Envy Free if no agent assigns greater utility to the resource set allocated to another agent within P than it attaches to its own allocation under P. 2nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28th February – 1st March 2005

  8. Quantitative Assessment • Utilitarian Social Welfare - u(P) u(P) =  ui(Pi ) • Egalitarian Social Welfare - e(P) e(P) = min {ui(Pi ) } • One aim is to find allocations that maximise these. 2nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28th February – 1st March 2005

  9. Complexity Considerations • Formulating decision problems. • Representing instances of such decision problems. • An important issue being how the collection {u1 , … , un } is described. 2nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28th February – 1st March 2005

  10. Some decision problems I • ENVY-FREE Instance: <A,R,U> Question: Is there an envy-free allocation of R? • PARETO OPTIMAL Instance: <A,R,U> ; P  n,m Question: IsP Pareto Optimal? 2nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28th February – 1st March 2005

  11. Some decision problems II • WELFARE OPTIMISATION Instance: <A,R,U>; K rational value. Question: Is there an allocation with u(P)  K ? • WELFARE IMPROVEMENT Instance: <A,R,U>; P  n,m Question: Is there Q  n,m with u(Q)> u(P)? 2nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28th February – 1st March 2005

  12. Representing Utility Functions • Possible options Enumerate non-zero valued subsets of R (‘bundle’ form) Algorithm that computes u(S) given S (‘program’ form) Suitable algebraic formula, e.g. u(S) = TR : |T|k(T)IS(T) (‘k-additive’ form) 2nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28th February – 1st March 2005

  13. Pros and Cons • Bundle form – ‘easy’ to encode but length of encoding could be exponential in m. • k-additive form – succinct for constant k but not always possible. • Program form – can be succinct; problem Program run-time and termination 2nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28th February – 1st March 2005

  14. ‘Suitable’ Program Form: SLP • Straight-Line Programs – m input bits encode subset S t program lines – vr := vbvd – b, d < r • Can describe as m+t triples <r,b,d>. • Poly-time computable u poly. length SLP • SLP for u can always be defined. 2nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28th February – 1st March 2005

  15. Complexity and Representation • The form chosen to represent U has little effect on the complexity of the decision problems introduced earlier. • Similarly, many results apply even when only two agent settings are used. 2nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28th February – 1st March 2005

  16. Complexity – Qualitative Case • ENVY-FREE is NP-complete with SLP and 2 agents. • PARETO OPTIMAL is coNP-complete with 2 agents in both SLP and 2-additive utility functions 2nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28th February – 1st March 2005

  17. Complexity – Quantitative Case • In 2 agent settings using SLP or 2-additive utility functions: WELFARE OPTIMISATION is NP-complete WELFARE IMPROVEMENT is NP-complete 2nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28th February – 1st March 2005

  18. Negotiation Models • With <A,R,U> there are |A||R| allocations. • For P and Q distinctallocations, the deal =<P,Q> replaces the allocation P with the the allocation Q. • It is not necessary for every agent to be given a new allocation within a deal - A denotes the set of agents whose allocation is changed by implementing the deal. 2nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28th February – 1st March 2005

  19. Reducing the number of deals • It is not feasible to review every deal. • 2 methods to restrict the number of deals in the search space: Structural restrictions Rationality restrictions 2nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28th February – 1st March 2005

  20. Structural Restrictions • Limit deals to those in which the number of participating agents is bounded and/or the number of resources exchanged is bounded, e.g. One resource-at-a-time (O-contract) (at most) k-resources-at-at-time (C(k)-contract) Exchange (or swap) contracts 2nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28th February – 1st March 2005

  21. Rationality Restrictions • Limit deals to those which “improve” an agent’s view of its allocation, e.g. Individual Rationality (IR) deals <P,Q> is said to be IR if u(Q)> u(P) • Thus, each agent places greater value on a ‘new’ allocation or (if it loses value) can be ‘compensated’ for its loss. 2nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28th February – 1st March 2005

  22. Problems with combined restrictions • Assume <P,Q> is IR. • <P,Q> is always realisable by a sequence of O-contracts. • <P,Q> is not always realisable by a sequence of IR O-contracts. • Similarly, replacing O-contracts by C(k)-contract. 2nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28th February – 1st March 2005

  23. Associated decision problems • IRO PATH Instance: <A,R,U> ; IR deal <P,Q> Question: Is there a sequence of IR O-contracts implementing <P,Q>? • IR(k) PATH Instance: <A,R,U> ; IR deal <P,Q> Question: Is there a sequence of IR C(k)-contracts implementing <P,Q>? 2nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28th February – 1st March 2005

  24. Complexity Properties • In SLP model IRO PATH is NP-hard IR(k) PATH is NP-hard k (constant) IR(k) PATH is NP-hard for k=c.|R| with c0.5 • There are difficulties with establishing membership in NP using the “obvious” algorithm, i.e. “guess a path and check its correctness” 2nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28th February – 1st March 2005

  25. Length of IR O-contract paths • Any deal <P,Q> can be implemented by a sequence of at most|R| O-contracts. • There are IR deals <P,Q> that can be implemented by a sequence of IR O-contracts but the shortest such sequence has length (2|R|) – (arbitrary U) (2|R|/2) – (monotone U) 2nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28th February – 1st March 2005

  26. Some Open Questions I • Using 2-additive utility functions: Complexity of ENVY-FREE? Complexity of IRO PATH? • Worst-case length of shortest IR O-contract sequence for k-additive utility functions • Upper bounds on complexity of IRO PATH, noting that IRO PATHNP? is non-trivial. 2nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28th February – 1st March 2005

  27. Some Open Questions II • Suppose the requirement for every deal to be an IR O-contract is relaxed? e.g. by allowing a “small” number of “irrational” deals and/or deals which are not O-contracts. Approximation algorithms Do exponential length paths occur when t irrational deals are allowed, with the same dealhaving poly. length with t+1 irrational deals? 2nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28th February – 1st March 2005

  28. Bibliography • P.E. Dunne, M. Wooldridge & M. Laurence. The Complexity of Contract Negotiation. Artificial Intelligence, 2005 (in press) • P.E. Dunne. Extremal Behaviour in Multiagent Contract Negotiation. Jnl. of Artificial Intelligence Res., 23, (2005), 41-78 Context dependence in mulitagent resource allocation. • Y. Chevaleyre, U. Endriss, S. Estivie, & N. Maudet. Multiagent resource allocation in k-additive domains: preference representation and complexity. 2nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28th February – 1st March 2005

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