We propose models for lobbying in a probabilistic environment, in which an actor (called "The Lob... more We propose models for lobbying in a probabilistic environment, in which an actor (called "The Lobby") seeks to influence voters' preferences of voting for or against multiple issues when the voters' preferences are represented in terms of probabilities. In particular, we provide two evaluation criteria and three bribery methods to formally describe these models, and we consider the resulting forms of lobbying with and without issue weighting. We provide a formal analysis for these problems of lobbying in a stochastic environment, and determine their classical and parameterized complexity depending on the given bribery/evaluation criteria. Specifically, we show that some of these problems can be solved in polynomial time, some are NP-complete but fixed-parameter tractable, and some are W[2]-complete. Finally, we provide approximability and inapproximability results for these problems and several variants. * A preliminary version of this Corollary 19 VB-SM-PLP-WIW and VB-AM-PLP-WIW (parameterized by the budget per issue B/n) are W[2]-hard. Similar modifications show: Corollary 20 Q-VB-SM-PLP-WIW and Q-VB-AM-PLP-WIW (parameterized by the budget and by the objective value O) are W[2]-complete. Corollary 21 Q-VB-SM-PLP-WIW and Q-VB-AM-PLP-WIW (parameterized by the budget) are W[2]-hard.
This work describes a new theoretical framework for uniform storage and management of diverse pro... more This work describes a new theoretical framework for uniform storage and management of diverse probabilistic information.
If the goal of eliciting accurate user preferences over a complex value space is to be realized, ... more If the goal of eliciting accurate user preferences over a complex value space is to be realized, the elicitation process must be through while at the same time not overly burdensome to the user. An ideal preference elicitation tool would maximize the information about preferences acquired from the user while minimizing the number of queries required. To this end, we present POET: the Online Preference Elicitation Tool. POET is a graphical Java applet designed to elicit complex preference structures to form a utility function. Once the elicitation process is complete POET outputs the resulting utility function as an XML document. Internally, POET represents utility for independent and dependent values as a set of dependency trees. Though POET's output and internal representation involves real numbers, from the user's point-of-view, the experience is completely symbolic.
Three Results on the Polynomial Isomorphism of Complete Sets
Abstract This paper proves three results relating to the isomorphism question for NP-complete set... more Abstract This paper proves three results relating to the isomorphism question for NP-complete sets. Result 1: We construct an oracle A such that SATA is ~~ complete for N p A and all ~~-complete sets for N p A are pA_ isomorphic to S ATA Result 2: We construct a time function T(n) ...
In this paper, we present several results about collapsing and non-collapsing degrees ofNP-comple... more In this paper, we present several results about collapsing and non-collapsing degrees ofNP-complete sets. The first, a relativized collapsing result, is interesting because it is the strongest known constructive approximation to a relativization of Berman and Hartmanis' 1977 conjecture that all ≤ m P -complete sets forNP arep-isomorphic. In addition, the collapsing result explores new territory in oracle construction, particularly in combining overlapping and apparently incompatible coding and diagonalizing techniques. We also present non-collapsing results, which are notable for their technical simplicity, and which rely on no unproven assumptions such as one-way functions. The basic technique developed in these non-collapsing constructions is surprisingly robust, and can be combined with many different oracle constructions.
In this paper we study the interplay between three measures of polynomial time behavior in sets: ... more In this paper we study the interplay between three measures of polynomial time behavior in sets: p-cheatability, p-closeness, and membership in P/poly. First we construct (2" -1 for k) -p-cheatable sets that are bi-immune with respect to all recursively enumerable sets. We show that the constructed sets are in P/poly, but can be constructed so that they are not p-close. In fact, they can be constructed so that they are not even recursiuely close. Next, we construct (n for 2)-p-cheatable sets that are bi-immune with respect to arbitrarily difficult deterministic time classes. These sets are also in P/poly, and they also can be constructed so that they are not p-close. Finally, we construct a set that is (n for
We investigate the computational complexity of testing dominance and consistency in CP-nets. Up u... more We investigate the computational complexity of testing dominance and consistency in CP-nets. Up until now, the complexity of dominance has been determined only for restricted classes in which the dependency graph of the CP-net is acyclic. However, there are preferences of interest that define cyclic dependency graphs; these are modeled with general CP-nets. We show here that both dominance and consistency testing for general CP-nets are PSPACE-complete. The reductions used in the proofs are from STRIPS planning, and thus establish strong connections between both areas.
Wp show that the l)robhun of transforming a structured Marker decision process (MDP) into Bounded... more Wp show that the l)robhun of transforming a structured Marker decision process (MDP) into Bounded Interval MDP is coNppr'-hacd. In particular, the test for e-homogvneitv, a ne('essarv part of verifying mlv . ¯.
We show that for several variations of partially observable Markov decision processes, polynomial... more We show that for several variations of partially observable Markov decision processes, polynomial-time algorithms for nding control policies are unlikely to or simply don't have guarantees of nding policies within a constant factor or a constant summand of optimal. Here \unlikely" means \unless some complexity classes collapse," where the collapses considered are P = NP, P = PSPACE, or P = EXP. Until or unless these collapses are shown to hold, any control-policy designer must choose between such performance guarantees and e cient computation.
We examine the computational complexity of testing and nding small plans in probabilistic plannin... more We examine the computational complexity of testing and nding small plans in probabilistic planning domains (both at and succinct). We show that many problems of interest are complete for a variety of complexity classes: PL, P, NP, co-NP, PP, NP PP , co-NP PP , and PSPACE. Of these, the probabilistic classes PP and NP PP are likely to be of special interest in the eld of uncertainty in arti cial intelligence and are deserving of additional study.
We consider the problem of nding good nite-horizon policies for POMDPs under the expected reward ... more We consider the problem of nding good nite-horizon policies for POMDPs under the expected reward metric. The policies considered are free nite-memory policies with limited memory; a policy is a mapping from the space of observation-memory pairs to the space of action-memory pairs (the policy updates the memory as it goes), and the number of possible memory states is a parameter of the input to the policy-nding algorithms. The algorithms considered here are preliminary implementations of three search heuristics: local search, simulated annealing, and genetic algorithms. We compare their outcomes to each other and to the optimal policies for each instance. We compare run times of each policy and of a dynamic programming algorithm for POMDPs developed by Hansen that iteratively improves a nite-state controller | the previous state of the art for nite memory policies. The value of the best policy can only improve as the amount of memory increases, up to the amount needed for an optimal nite-memory policy. Our most surprising nding is that more memory helps in another way: given more memory than is needed for an optimal policy, the algorithms are more likely to converge to optimalvalued policies.
Value Iteration is an inefficient algorithm for Markov decision processes (MDPs) because it puts ... more Value Iteration is an inefficient algorithm for Markov decision processes (MDPs) because it puts the majority of its effort into backing up the entire state space, which turns out to be unnecessary in many cases. In order to overcome this problem, many approaches have been proposed. Among them, LAO*, LRTDP and HDP are state-of-theart ones. All of these use reachability analysis and heuristics to avoid some unnecessary backups. However, none of these approaches fully exploit the graphical features of the MDPs or use these features to yield the best backup sequence of the state space. We introduce an algorithm named Topological Value Iteration (TVI) that can circumvent the problem of unnecessary backups by detecting the structure of MDPs and backing up states based on topological sequences. We prove that the backup sequence TVI applies is optimal. Our experimental results show that TVI outperforms VI, LAO*, LRTDP and HDP on our benchmark MDPs.
We examine the computational complexity of testing and nding small plans in probabilistic plannin... more We examine the computational complexity of testing and nding small plans in probabilistic planning domains with both at and propositional representations. The complexity of plan evaluation and existence varies with the plan type sought; we examine totally ordered plans, acyclic plans, and looping plans, and partially ordered plans under three natural de nitions of plan value. We show that problems of interest are complete for a variety of complexity classes: PL, P, NP, co-NP, PP, NP PP , co-NP PP , and PSPACE. In the process of proving that certain planning problems are complete for NP PP , we introduce a new basic NP PP -complete problem, E-Majsat, which generalizes the standard Boolean satis ability problem to computations involving probabilistic quantities; our results suggest that the development of good heuristics for E-Majsat could be important for the creation of e cient algorithms for a wide variety of problems.
We survey the complexity of computational problems about Markov decision processes: evaluating po... more We survey the complexity of computational problems about Markov decision processes: evaluating policies, finding good and best policies, approximating best policies, and related decision problems
A partially-observable Markov decision process (POMDP) is a generalization of a Markov decision p... more A partially-observable Markov decision process (POMDP) is a generalization of a Markov decision process that allows for incomplete information regarding the state of the system. We consider several flavors of finite-horizon POMDPs. Our results concern the complexity of the policy evaluation and policy existence problems, which are characterized in terms of completeness for complexity classes. We prove a new upper bound for the policy evaluation problem for POMDPs, showing it is complete for Probabilistic Logspace. From this, we prove policy existence problems for several variants of unobservable, succinctly represented MDPs to be complete for NPPP, a class for which not many natural problems are known to be complete.
We propose models for lobbying in a probabilistic environment, in which an actor (called "The Lob... more We propose models for lobbying in a probabilistic environment, in which an actor (called "The Lobby") seeks to influence voters' preferences of voting for or against multiple issues when the voters' preferences are represented in terms of probabilities. In particular, we provide two evaluation criteria and three bribery methods to formally describe these models, and we consider the resulting forms of lobbying with and without issue weighting. We provide a formal analysis for these problems of lobbying in a stochastic environment, and determine their classical and parameterized complexity depending on the given bribery/evaluation criteria. Specifically, we show that some of these problems can be solved in polynomial time, some are NP-complete but fixed-parameter tractable, and some are W[2]-complete. Finally, we provide approximability and inapproximability results for these problems and several variants. * A preliminary version of this Corollary 19 VB-SM-PLP-WIW and VB-AM-PLP-WIW (parameterized by the budget per issue B/n) are W[2]-hard. Similar modifications show: Corollary 20 Q-VB-SM-PLP-WIW and Q-VB-AM-PLP-WIW (parameterized by the budget and by the objective value O) are W[2]-complete. Corollary 21 Q-VB-SM-PLP-WIW and Q-VB-AM-PLP-WIW (parameterized by the budget) are W[2]-hard.
This work describes a new theoretical framework for uniform storage and management of diverse pro... more This work describes a new theoretical framework for uniform storage and management of diverse probabilistic information.
If the goal of eliciting accurate user preferences over a complex value space is to be realized, ... more If the goal of eliciting accurate user preferences over a complex value space is to be realized, the elicitation process must be through while at the same time not overly burdensome to the user. An ideal preference elicitation tool would maximize the information about preferences acquired from the user while minimizing the number of queries required. To this end, we present POET: the Online Preference Elicitation Tool. POET is a graphical Java applet designed to elicit complex preference structures to form a utility function. Once the elicitation process is complete POET outputs the resulting utility function as an XML document. Internally, POET represents utility for independent and dependent values as a set of dependency trees. Though POET's output and internal representation involves real numbers, from the user's point-of-view, the experience is completely symbolic.
Three Results on the Polynomial Isomorphism of Complete Sets
Abstract This paper proves three results relating to the isomorphism question for NP-complete set... more Abstract This paper proves three results relating to the isomorphism question for NP-complete sets. Result 1: We construct an oracle A such that SATA is ~~ complete for N p A and all ~~-complete sets for N p A are pA_ isomorphic to S ATA Result 2: We construct a time function T(n) ...
In this paper, we present several results about collapsing and non-collapsing degrees ofNP-comple... more In this paper, we present several results about collapsing and non-collapsing degrees ofNP-complete sets. The first, a relativized collapsing result, is interesting because it is the strongest known constructive approximation to a relativization of Berman and Hartmanis' 1977 conjecture that all ≤ m P -complete sets forNP arep-isomorphic. In addition, the collapsing result explores new territory in oracle construction, particularly in combining overlapping and apparently incompatible coding and diagonalizing techniques. We also present non-collapsing results, which are notable for their technical simplicity, and which rely on no unproven assumptions such as one-way functions. The basic technique developed in these non-collapsing constructions is surprisingly robust, and can be combined with many different oracle constructions.
In this paper we study the interplay between three measures of polynomial time behavior in sets: ... more In this paper we study the interplay between three measures of polynomial time behavior in sets: p-cheatability, p-closeness, and membership in P/poly. First we construct (2" -1 for k) -p-cheatable sets that are bi-immune with respect to all recursively enumerable sets. We show that the constructed sets are in P/poly, but can be constructed so that they are not p-close. In fact, they can be constructed so that they are not even recursiuely close. Next, we construct (n for 2)-p-cheatable sets that are bi-immune with respect to arbitrarily difficult deterministic time classes. These sets are also in P/poly, and they also can be constructed so that they are not p-close. Finally, we construct a set that is (n for
We investigate the computational complexity of testing dominance and consistency in CP-nets. Up u... more We investigate the computational complexity of testing dominance and consistency in CP-nets. Up until now, the complexity of dominance has been determined only for restricted classes in which the dependency graph of the CP-net is acyclic. However, there are preferences of interest that define cyclic dependency graphs; these are modeled with general CP-nets. We show here that both dominance and consistency testing for general CP-nets are PSPACE-complete. The reductions used in the proofs are from STRIPS planning, and thus establish strong connections between both areas.
Wp show that the l)robhun of transforming a structured Marker decision process (MDP) into Bounded... more Wp show that the l)robhun of transforming a structured Marker decision process (MDP) into Bounded Interval MDP is coNppr'-hacd. In particular, the test for e-homogvneitv, a ne('essarv part of verifying mlv . ¯.
We show that for several variations of partially observable Markov decision processes, polynomial... more We show that for several variations of partially observable Markov decision processes, polynomial-time algorithms for nding control policies are unlikely to or simply don't have guarantees of nding policies within a constant factor or a constant summand of optimal. Here \unlikely" means \unless some complexity classes collapse," where the collapses considered are P = NP, P = PSPACE, or P = EXP. Until or unless these collapses are shown to hold, any control-policy designer must choose between such performance guarantees and e cient computation.
We examine the computational complexity of testing and nding small plans in probabilistic plannin... more We examine the computational complexity of testing and nding small plans in probabilistic planning domains (both at and succinct). We show that many problems of interest are complete for a variety of complexity classes: PL, P, NP, co-NP, PP, NP PP , co-NP PP , and PSPACE. Of these, the probabilistic classes PP and NP PP are likely to be of special interest in the eld of uncertainty in arti cial intelligence and are deserving of additional study.
We consider the problem of nding good nite-horizon policies for POMDPs under the expected reward ... more We consider the problem of nding good nite-horizon policies for POMDPs under the expected reward metric. The policies considered are free nite-memory policies with limited memory; a policy is a mapping from the space of observation-memory pairs to the space of action-memory pairs (the policy updates the memory as it goes), and the number of possible memory states is a parameter of the input to the policy-nding algorithms. The algorithms considered here are preliminary implementations of three search heuristics: local search, simulated annealing, and genetic algorithms. We compare their outcomes to each other and to the optimal policies for each instance. We compare run times of each policy and of a dynamic programming algorithm for POMDPs developed by Hansen that iteratively improves a nite-state controller | the previous state of the art for nite memory policies. The value of the best policy can only improve as the amount of memory increases, up to the amount needed for an optimal nite-memory policy. Our most surprising nding is that more memory helps in another way: given more memory than is needed for an optimal policy, the algorithms are more likely to converge to optimalvalued policies.
Value Iteration is an inefficient algorithm for Markov decision processes (MDPs) because it puts ... more Value Iteration is an inefficient algorithm for Markov decision processes (MDPs) because it puts the majority of its effort into backing up the entire state space, which turns out to be unnecessary in many cases. In order to overcome this problem, many approaches have been proposed. Among them, LAO*, LRTDP and HDP are state-of-theart ones. All of these use reachability analysis and heuristics to avoid some unnecessary backups. However, none of these approaches fully exploit the graphical features of the MDPs or use these features to yield the best backup sequence of the state space. We introduce an algorithm named Topological Value Iteration (TVI) that can circumvent the problem of unnecessary backups by detecting the structure of MDPs and backing up states based on topological sequences. We prove that the backup sequence TVI applies is optimal. Our experimental results show that TVI outperforms VI, LAO*, LRTDP and HDP on our benchmark MDPs.
We examine the computational complexity of testing and nding small plans in probabilistic plannin... more We examine the computational complexity of testing and nding small plans in probabilistic planning domains with both at and propositional representations. The complexity of plan evaluation and existence varies with the plan type sought; we examine totally ordered plans, acyclic plans, and looping plans, and partially ordered plans under three natural de nitions of plan value. We show that problems of interest are complete for a variety of complexity classes: PL, P, NP, co-NP, PP, NP PP , co-NP PP , and PSPACE. In the process of proving that certain planning problems are complete for NP PP , we introduce a new basic NP PP -complete problem, E-Majsat, which generalizes the standard Boolean satis ability problem to computations involving probabilistic quantities; our results suggest that the development of good heuristics for E-Majsat could be important for the creation of e cient algorithms for a wide variety of problems.
We survey the complexity of computational problems about Markov decision processes: evaluating po... more We survey the complexity of computational problems about Markov decision processes: evaluating policies, finding good and best policies, approximating best policies, and related decision problems
A partially-observable Markov decision process (POMDP) is a generalization of a Markov decision p... more A partially-observable Markov decision process (POMDP) is a generalization of a Markov decision process that allows for incomplete information regarding the state of the system. We consider several flavors of finite-horizon POMDPs. Our results concern the complexity of the policy evaluation and policy existence problems, which are characterized in terms of completeness for complexity classes. We prove a new upper bound for the policy evaluation problem for POMDPs, showing it is complete for Probabilistic Logspace. From this, we prove policy existence problems for several variants of unobservable, succinctly represented MDPs to be complete for NPPP, a class for which not many natural problems are known to be complete.
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Papers by Judy Goldsmith