Complexity Issues in Multiagent Resource Allocation Paul E

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Complexity Issues in Multiagent Resource Allocation Paul E. Dunne Dept. of Computer Science University

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

Overview 1. 2. 3. 4. 5. 6. 7. Modelling resource allocation. Assessing allocations. Complexity

Overview 1. 2. 3. 4. 5. 6. 7. Modelling resource allocation. Assessing allocations. Complexity considerations Computational complexity properties. A Model for negotiating allocations and its properties. Open questions and conjectures 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 2

Modelling Resource Allocation A = {a 1 , … , an } – set

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

Assumptions • Exactly one agent owns any resource, i. e. R is non-shareable. •

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, m with Pi = Qi it holds that ui(Pi ) = ui(Qi ). 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 4

Assessing Allocations • Qualitative measures. Pareto Optimality Envy Freeness • Quantitative measures. Utilitarian Social

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

Qualitative Assessment I • An allocation, P, is Pareto Optimal if for every allocation,

Qualitative Assessment I • An allocation, P, is Pareto Optimal 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 ). 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 6

Qualitative Assessment II • An allocation, P, is Envy Free if no agent assigns

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. 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 7

Quantitative Assessment • Utilitarian Social Welfare - u(P) = ui(Pi ) • Egalitarian Social

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

Complexity Considerations • Formulating decision problems. • Representing instances of such decision problems. •

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

Some decision problems I • ENVY-FREE Instance: <A, R, U> Question: Is there an

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: Is P Pareto Optimal? 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 10

Some decision problems II • WELFARE OPTIMISATION Instance: <A, R, U>; K rational value.

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)? 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 11

Representing Utility Functions • Possible options Enumerate non-zero valued subsets of R (‘bundle’ form)

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) 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 12

Pros and Cons • Bundle form – ‘easy’ to encode but length of encoding

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 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 13

‘Suitable’ Program Form: SLP • Straight-Line Programs – m input bits encode subset S

‘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. 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 14

Complexity and Representation • The form chosen to represent U has little effect on

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. 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 15

Complexity – Qualitative Case • ENVY-FREE is NP-complete with SLP and 2 agents. •

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

Complexity – Quantitative Case • In 2 agent settings using SLP or 2 additive

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

Negotiation Models • With <A, R, U> there are |A||R| allocations. • For P

Negotiation Models • With <A, R, U> there are |A||R| allocations. • For P and Q distinct allocations, the deal =<P, Q> replaces the allocation P with 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. 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 18

Reducing the number of deals • It is not feasible to review every deal.

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 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 19

Structural Restrictions • Limit deals to those in which the number of participating agents

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 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 20

Rationality Restrictions • Limit deals to those which “improve” an agent’s view of its

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. 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 21

Problems with combined restrictions • Assume <P, Q> is IR. • <P, Q> is

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. 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 22

Associated decision problems • IRO PATH Instance: <A, R, U> ; IR deal <P,

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>? 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 23

Complexity Properties • In SLP model IRO PATH is NP-hard IR(k) PATH is NP-hard

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” 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 24

Length of IR O-contract paths • Any deal <P, Q> can be implemented by

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 Ocontracts but the shortest such sequence has length (2|R|) – (arbitrary U) (2|R|/2) – (monotone U) 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 25

Some Open Questions I • Using 2 -additive utility functions: Complexity of ENVY-FREE? Complexity

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. 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 26

Some Open Questions II • Suppose the requirement for every deal to be an

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 deal having poly. length with t+1 irrational deals? 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 27

Bibliography • P. E. Dunne, M. Wooldridge & M. Laurence. The Complexity of Contract

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. 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 28