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Integer Programs and Valid Inequalities for Planning Problems
Abstract
Part of the recent work in AI planning is concerned with the development of algorithms that regard planning as a combinatorial search problem. The underlying representation language is basically propositional logic. While this is adequate for many domains, it is not clear if it remains so for problems that involve numerical constraints, or optimization of complex objective functions. Moreover, the propositional representation imposes restrictions on the domain knowledge that can be utilized by these approaches. In order to address these issues, we propose moving to the more expressive language of Integer Programming (IP). We show how capacity constraints can be easily encoded into linear 0-1 inequalities and how rich forms of domain knowledge can be compactly represented and computationally exploited by IP solvers. Then we introduce a novel heuristic search method based on the linear programming relaxation. Finally, we present the results of our experiments with a classical relaxation-based IP solver and a logic-based 0-1 optimizer.
... CPlan [40] is a high performance constraint-based planner that requires a highly tuned model of the domain to operate. Finally, [4] is a planner based on an integer programming formulation that does IP branch-and-bound using a standard IP solver. ...
Branching and lower bounds are two key notions in heuristic search and combinatorial optimization. Branch-ing refers to the way the space of solutions is searched, while lower bounds refer to approximate cost measures used for focusing and pruning the search. In AI Planning, admissible heuristics or lower bounds have received considerable attention and most current optimal planners use them, either explicitly or implicitly (e.g., by relying on a plan graph). Branching, on the other hand, has received less attention and is seldom discussed explicitly. In this paper, we make the notion of branching in planning explicit and relate it to branching schemes used in combinatorial optimization. We analyze from this perspective the relationship between heuristic-search, constraint-based and partial-order planning, and between planning and scheduling. We also introduce a branch-and-bound formulation for planning that handles actions with durations and use unary resources, and consider a range of theories that stand halfway between planning and scheduling. The goals are twofold: to make sense of various techniques that have been found useful in planning and scheduling in a unified framework, and to lay the ground for systems that can effectively combine both capabilities. We have also implemented a planner based on this formulation on top of a constraint programming language and present some preliminary results.
... (Kambhampati, 2000; Rintanen, 1998)) or to model the planning problem itself into a CP substrate (cf. (Van Beek and Chen, 1999; Bockmayr and Dimopoulos, 1999; Kautz and Walser, 1999; Kautz and Selman, 1996 )) and solve it using standard CP techniques . The second approach has the apparent advantage that planning researchers can, to some extent, exploit the advances being made in CP solution techniques and benefit from the high expressiveness. ...
In this paper we describe Constraint-based Attribute and Interval Planning (CAIP), a paradigm for representing and reasoning about plans. The paradigm enables the description of planning domains with time, resources, concurrent activities, mutual exclusions ...
... As a drawback the efficiency of other system has not been gained. Bockmayr and Dimopoulos integrate some domain specific knowledge to the setting of propositional planning [3]. ...
. In this paper we study traditional and enhanced BDDbased exploration procedures capable of handling large planning problems. On the one hand, reachability analysis and model checking have eventually approached AI-Planning. Unfortunately, they typically rely on uninformed blind search. On the other hand, heuristic search and especially lower bound techniques have matured in effectively directing the exploration even for large problem spaces. Therefore, with heuristic symbolic search we address the unexplored middle ground between single state and symbolic planning engines to establish algorithms that can gain from both sides. To this end we implement and evaluate heuristics found in state-of-the-art heuristic single-state search planners. 1 Introduction One currently very successful trend in deterministic fully-automated planning is heuristic single-state space search. The search space incorporates states as lists of instantiated predicates (also called atoms or fluents). The succes...
The Optiplan planning system is the first integer programming-based planner that successfully participated in the international planning competition. This engineering note describes the architecture of Optiplan and provides the integer programming formulation that enabled it to perform reasonably well in the competition. We also touch upon some recent developments that make integer programming encodings significantly more competitive.
The machine and plant automation domain is faced with an ever increasing demand for ensuring the adaptability of manufacturing facilities in context of Industry 4.0. Field level automation software plays a dominant role in strengthening the overall flexibility of manufacturing resources. Classical programming approaches based typically on signal-oriented languages result in disproportionate effort for ensuring necessary flexibility. To address this challenge, a novel approach based on artificial intelligence planning techniques is presented which is able to handle domain specific requirements while facilitating efficient, scalable problem solving. Throughout this article, a discussion of specific requirements on automated planning techniques for field level automation software in the machine and plant automation domain with respect to Industry 4.0 is provided. An intensive study on existing works and their drawbacks towards addressing these requirements is presented. The proposed configurable partial-order planning approach is based upon a combination of an adapted goal-based planning formulation and its reformulation by means of linear programming techniques. It is shown that the proposed approach is able to efficiently solve large planning problems by exhibiting positive scalability characteristics which indicates its applicability for real-size plants.
This paper presents an example of cooperation between AI planning techniques and Constraint Programming or Operations Research. More precisely, it presents a way of boosting forward planning using combinatorial optimization techniques. The idea consists in combining on one hand a dynamic model that represents the Markovian dynamics of the system considered ( i.e. state transitions), and on the other hand a static model that describes the global properties that are required over state trajectories. The dynamic part is represented by so-called constraint-based timed automata , whereas the static part is represented by so-called constraint-based observers . The latter are modeled using standard combinatorial optimization frameworks, such as linear programming, constraint programming, scheduling, or boolean satisfiability. They can be called at any step of the forward search to cut it via inconsistency detection. Experiments show significant improvements on some benchmarks of the International Planning Competition.
The diffusion of domotic and ambient intelligence systems have introduced a new vision in which autonomous deliberative agents operate in environments where reactive responses of devices can be cooperatively exploited to fulfill the agent's goals. In this article a model for automated planning in reactive environments, based on numerical planning, is introduced. A planner system, based on mixed integer linear programming techniques, which implements the model, is also presented. The planner is able to reason about the dynamic features of the environment and to produce solution plans, which take into account reactive devices and their causal relations with agent's goals by exploitation and avoidance techniques, to reach a given goal state. The introduction of reactive domains in planning poses some issues concerning reasoning patterns which are briefly depicted. Experiments of planning in reactive domains are also discussed.
The Optiplan planning system is the first integer programming-based planner that successfully participated in the international planning competition. This engineering note describes the architecture of Optiplan and provides the integer programming formulation that enabled it to perform reasonably well in the competition. We also touch upon some recent developments that make integer programming encodings significantly more competitive.
We represent planning as a set of loosely coupled network flow problems,
where each network corresponds to one of the state variables in the planning
domain. The network nodes correspond to the state variable values and the
network arcs correspond to the value transitions. The planning problem is to
find a path (a sequence of actions) in each network such that, when merged,
they constitute a feasible plan. In this paper we present a number of integer
programming formulations that model these loosely coupled networks with varying
degrees of flexibility. Since merging may introduce exponentially many ordering
constraints we implement a so-called branch-and-cut algorithm, in which these
constraints are dynamically generated and added to the formulation when needed.
Our results are very promising, they improve upon previous planning as integer
programming approaches and lay the foundation for integer programming
approaches for cost optimal planning.
Recently, casting planning as propositional satisfiability (SAT) has been shown to be an efficient technique of plan synthesis. This article is a response to the recently proposed challenge of developing novel propositional encodings that are based on a combination of different types of plan refinements and characterizing the tradeoffs. We refer to these encodings as hybrid encodings. An investigation of these encodings is important, because this can give insights into what kinds of planning problems can be solved faster with hybrid encodings.
Encodings based on partial–order planning and state–space planning have been reported in previous research. We propose a new type of encoding called a unifying encoding that subsumes these two encodings. We also report on several other hybrid encodings. Next, we show how the satisfiability framework can be extended to incremental planning. State–space encoding is attractive because of its lower size and causal encoding is attractive because of its highest flexibility in reordering steps. We show that hybrid encodings have a higher size and a lower flexibility in step reordering and, thus, do not combine the best of these encodings. We discuss in detail several specific planning scenarios where hybrid encodings are likely to be superior to nonhybrid encodings.
Integer and combinatorial optimization deals with problems of maximizing or minimizing a function of many variables subject to (a) inequality and equality constraints and (b) integrality restrictions on some or all of the variables. Because of the robustness of the general model, a remarkably rich variety of problems can be represented by discrete optimization models. This chapter is concerned with the formulation of integer optimization problems, which means how to translate a verbal description of a problem into a mathematical statement of the form linear mixed-integer programming problem ( MIP), linear (pure) integer programming problem ( IP), or combinatorial optimization problem (CP). The chapter presents two important uses of binary variables in the modeling of optimization problems. The first concerns the representation of nonlinear objective functions of the form l>jfj(yj) using linear functions and binary variables. The second concerns the modeling of disjunctive constraints.
Constraint programming, a methodology for solving diffi- cult combinatorial problems by representing them as con- straint satisfaction problems, has shown that a general pur- pose search algorithm based on constraint propagation com- bined with an emphasis on modeling can solve large, prac- tical scheduling problems. Given the success of constraint programming on scheduling problems and the similarity of scheduling to planning, the question arises, would a con- straint programming approach work as well in planning? In thispaper,wepresentevidencethataconstraintprogramming approach to planning does indeed work well and has the ad- vantagein terms of time andspaceefficiencyover the current state-of-the-art planners.
. This paper outlines the basic principles underlying reasoning about resources in IPP, which is a classical planner based on planning graphs originally introduced with the graphplan system. The main idea is to deal with resources in a strictly action-centered way, i.e., one specifies how each action consumes or produces resources, but no explicit temporal model is used. This avoids the computational problems of solving general constraint satisfaction problems by using instead interval arithmetics and propagation of resource requirements over time steps in the planning graph. 1 Actions that provide, produce, and consume Resources The starting point for the language extension is the ADL subset that is available in IPP 3.0 [7]. It offers universally quantified and conditional effects, atomic negation, equality as well as quantified and conditional goals. To reason about resources, an action description is extended in the following way: 1. Following the "ordinary preconditions" (which...
Means-ends analysis is a seemingly well understood search technique, which can be described, using planning terminology, as: keep adding actions that are feasible and achieve pieces of the goal. Unfortunately, it is often the case that no action is both feasible and relevant in this sense. The traditional answer is to make subgoals out of the preconditions of relevant but infeasible actions. These subgoals become part of the search state. An alternative, surprisingly good, idea is to recompute the entire subgoal hierarchy after every action. This hierarchy is represented by a greedy regression-match graph. The actions near the leaves of this graph are feasible and relevant to a sub. . . subgoals of the original goal. Furthermore, each subgoal is assigned an estimate of the number of actions required to achieve it. This number can be shown in practice to be a useful heuristic estimator for domains that are otherwise intractable. Keywords: planning, search, means-ends analysis Reinven...
This paper describes ILP-PLAN, a framework for solving AI planning problems represented as integer linear programs. ILP-PLAN extends the planning as satisfiability framework to handle plans with resources, action costs, and complex objective functions. We show that challenging planning problems can be effectively solved using both traditional branchand -bound IP solvers and efficient new integer local search algorithms. ILP-PLAN can find better quality solutions for a set of hard benchmark logistics planning problems than had been found by any earlier system. 1
. We describe an extension of graphplan to a subset of ADL that allows conditional and universally quantified effects in operators in such a way that almost all interesting properties of the original graphplan algorithm are preserved. 1 Introduction Planning with planning graphs [1] has received considerable attention recently. The impressive performance and in particular the theoretical properties such as soundness, completeness, generation of shortest plans, and termination on unsolvable problems motivated us to use the approach as the kernel algorithm for our own planner IP 2 . But graphplan also has its limitations. First, its performance can decrease dramatically if too much irrelevant information is contained in the specification of a planning task [7]. Second, its simple representation language is restricted to pure STRIPS operators -- no conditional or universally quantified effects are allowed and it was unclear whether the underlying planning algorithm could be extended t...
We propose some domain-independent techniques for bringing well-founded partial-order planners closer to practicality. The first two techniques are aimed at improving search control while keeping overhead costs low. One is based on a simple adjustment to the default A* heuristic used by UCPOP to select plans for refinement. The other is based on preferring ``zero commitment'' (forced) plan refinements whenever possible, and using LIFO prioritization otherwise. A more radical technique is the use of operator parameter domains to prune search. These domains are initially computed from the definitions of the operators and the initial and goal conditions, using a polynomial-time algorithm that propagates sets of constants through the operator graph, starting in the initial conditions. During planning, parameter domains can be used to prune nonviable operator instances and to remove spurious clobbering threats. In experiments based on modifications of UCPOP, our improved plan and goal selection strategies gave speedups by factors ranging from 5 to more than 1000 for a variety of problems that are nontrivial for the unmodified version. Crucially, the hardest problems gave the greatest improvements. The pruning technique based on parameter domains often gave speedups by an order of magnitude or more for difficult problems, both with the default UCPOP search strategy and with our improved strategy. The Lisp code for our techniques and for the test problems is provided in on-line appendices. Comment: See http://www.jair.org/ for an online appendix and other files accompanying this article
A logic view of 0-1 integer programming problems, providing new insights into the structure of problems that can lead the researcher to more effective solution techniques depending on the problem class. Operations research techniques are integrated into a logic programming environment. The first monographic treatment that begins to unify these two methodological approaches.
Logic-based methods for modelling and solving combinatorial problems have recently started to play a significant role in both theory and practice. The application of logic to combinatorial problems has a dual aspect. On one hand, constraint logic programming allows one to declaratively model combinatorial problems over an appropriate constraint domain, the problems then being solved by a corresponding constraint solver. Besides being a high-level declarative interface to the constraint solver, the logic programming language allows one also to implement those subproblems that cannot be naturally expressed with constraints. On the other hand, logic-based methods can be used as a constraint solving technique within a constraint solver for combinatorial problems modelled as 0-1 integer programs.
A logic view of 0-1 integer programming problems, providing new insights into the structure of problems that can lead the researcher to more effective solution techniques depending on the problem class. Operations research techniques are integrated into a logic programming environment. The first monographic treatment that begins to unify these two methodological approaches. Logic-based methods for modelling and solving combinatorial problems have recently started to play a significant role in both theory and practice. The application of logic to combinatorial problems has a dual aspect. On one hand, constraint logic programming allows one to declaratively model combinatorial problems over an appropriate constraint domain, the problems then being solved by a corresponding constraint solver. Besides being a high-level declarative interface to the constraint solver, the logic programming language allows one also to implement those subproblems that cannot be naturally expressed with constraints. On the other hand, logic-based methods can be used as a constraint solving technique within a constraint solver for combinatorial problems modelled as 0-1 integer programs.
Constraint logic programming has become a promising new technology for solving complexcombinatorial problems. In this paper,
we investigate how (constraint) logic programmingcan support the modelling part when solving discrete optimisation problems.
First, we showthat the basic functionality of algebraic modelling languages can be realised very easily ina pure logic programming
system like PROLOG and that, even without using constraints,various additional features are available. Then we focus on the
constraint-solving facilitiesoffered by constraint logic programming systems. In particular, we explain how the constraintsolver
of the constraint logic programming language CLP(PB) can be used in modelling0 - 1 problems.
We introduce a new approach to planning in STRIPS-like domains based on constructing and analyzing a compact structure we call a planning graph. We describe a new planner, Graphplan, that uses this paradigm. Graphplan always returns a shortest possible partial-order plan, or states that no valid plan exists.We provide empirical evidence in favor of this approach, showing that Graphplan outperforms the total-order planner, Prodigy and the partial-order planner, UCPOP, on a variety of interesting natural and artificial planning problems. We also give empirical evidence that the plans produced by Graphplan are quite sensible. Since searches made by this approach are fundamentally different from the searches of other common planning methods, they provide a new perspective on the planning problem.
The ability to plan and react in dynamic environments is central to intelligent behavior yet few algorithms have managed to combine fast planning with a robust execution. In this paper we develop one such algorithm by looking at planning as real time search. For that we develop a variation of Korf's Learning Real Time A* algorithm together with a suitable heuristic function. The resulting algorithm interleaves lookahead with execution and never builds a plan. It is an action selection mechanism that decides at each time point what to do next. Yet it solves hard planning problems faster than any domain independent planning algorithm known to us, including the powerful SAT planner recently introduced by Kautz and Selman. It also works in the presence of perturbations and noise, and can be given a fixed time window to operate. We illustrate each of these features by running the algorithm on a number of benchmark problems.
Pseudo-Boolean constraints are equations or inequalities between integer polynomials in 0-1 variables. On the one hand, they generalize Boolean constraints, on the other hand, they are a restricted form of finite domain constraints. In this paper, we present special constraint solving techniques for the domain {0,1} originating from mathematical programming. The key concepts are the generation of strong valid inequalities for the solution set of a constraint system and the notion of branch-and-cut.
SATPLAN is currently one of the fastest planning systems for domain-independent planning. In nearly all practical applications, however, there exists an abundance of domain-specific knowledge that can be used to improve the performance of a planning system. This knowledge is traditionally encoded as procedures or rules that are tied to the details of the planning engine. We present a way to encode domain knowledge in a purely declarative, algorithm independent manner. We demonstrate that the same heuristic knowledge can be used by completely different search engines, one systematic, the other using greedy local search. This approach greatly enhances the power of SATPLAN: solution times for some problems are reduced from days to seconds. Introduction Planning is a notoriously hard combinatorial search problem. In Kautz and Selman (1996) we considered domain-independent, state-space planning as a computational challenge. This style of planning is a core problem in AI; similar computati...
I introduce a new search heuristic for propositional STRIPS planning that is based on transforming planning instances to linear programming instances. The linear programming heuristic is admissible for finding minimum length plans and can be used by partial-order planning algorithms. This heuristic appears to be the first non-trivial admissible heuristic for partial-order planning. An empirical study compares Lplan, a partial-order planner incorporating the heuristic, to Graphplan, Satplan, and UCPOP on the tower of Hanoi domain, random blocks-world instances, and random planning instances. Graphplan is far faster in the study than the other algorithms. Lplan is often slower because the heuristic is time-consuming, but Lplan shows promise because it often performs a small search. Introduction Planning is the problem of finding a combination of actions that achieves a goal (Allen, Hendler, & Tate 1990; Hendler, Tate, & Drummond 1990). So far, there is limited success in g...
Planning is a notoriously hard combinatorial search problem. In many interesting domains, current planning algorithms fail to scale up gracefully. By combining a general, stochastic search algorithm and appropriate problem encodings based on propositional logic, we are able to solve hard planning problems many times faster than the best current planning systems. Although stochastic methods have been shown to be very effective on a wide range of scheduling problems, this is the first demonstration of its power on truly challenging classical planning instances. This work also provides a new perspective on representational issues in planning. Introduction There is a widespread belief in the AI community that planning is not amenable to general theorem-proving techniques. The origin of this belief can be traced to the early 1970's, when work on plan generation using first-order, resolution theorem-proving (Green 1969) failed to scale up to realistically-sized problems. The re...
Recent new planning paradigms, such as Graphplan and Satplan, have been shown to outperform more traditional domain-independent planners. An interesting aspect of these planners is that they do not incorporate domain specific control knowledge, but instead rely on e#cient graph-based or propositional representations and advanced search techniques. An alternative approach has been proposed in the TLPlan system. TLPlan is an example of a powerful planner incorporating declarative control specified in temporal logic formulas. We show how these control rules can be parsed into Satplan. Our empirical results show up to an order of magnitude speed up. We also provide a detailed comparison with TLPlan, and show how the search strategies in TLPlan lead to e#cient plans in terms of the number of actions but with little or no parallelism. The Satplan and Graphplan formalisms on the other hand do find highly parallel plans, but are less e#ective in sequential domains. Our result...
Recent research has shown the promise of using propositional reasoning and search to solve AI planning problems. In this paper, we further explore this area by applying Integer Programming to solve AI planning problems. The application of Integer Programming to AI planning has a potentially significant advantage, as it allows quite naturally for the incorporation of numerical constraints and objectives into the planning domain. Moreover, the application of Integer Programming to AI planning addresses one of the challenges in propositional reasoning posed by Kautz and Selman, who conjectured that the principal technique used to solve Integer Programs---the linear programming (LP) relaxation---is not useful when applied to propositional search. We discuss various IP formulations for the class of planning problems based on STRIPS-style planning operators. Our main objective is to show that a carefully chosen IP formulation significantly improves the "strength" of the LP relaxation, and that the resultant LPs are useful in solving the IP and the associated planning problems. Our results clearly show the importance of choosing the "right" representation, and more generally the promise of using Integer Programming techniques in the AI planning domain. 1
The Blackbox planning system unifies the planning as satisfiability framework (Kautz and Selman 1992, 1996) with the plan graph approach to STRIPS planning (Blum and Furst 1995). We show that STRIPS problems can be directly translated into SAT and efficiently solved using new randomized systematic solvers. For certain computationally challenging benchmark problems this unified approach outperforms both SATPLAN and Graphplan alone. We also demonstrate that polynomialtime SAT simplification algorithms applied to the encoded problem instances are a powerful complement to the "mutex" propagation algorithm that works directly on the plan graph. 1 Introduction It has often been observed that the classical AI planning problem (that is, planning with complete and certain information) is a form of logical deduction. Because early attempts to use general theorem provers to solve planning problems proved impractical, research became focused on specialized planning algorithms. Somet...