Refining complexity analyses in planning by exploiting the exponential time hypothesis

The use of computational complexity in planning, and in AI in general, has always been a disputed topic. A major problem with ordinary worst-case analyses is that they do not provide any quantitative information: they do not tell us much about the running time of concrete algorithms, nor do they tell us much about the running time of optimal algorithms. We address problems like this by presenting results based on the exponential time hypothesis (ETH), which is a widely accepted hypothesis concerning the time complexity of 3-SAT. By using this approach, we provide, for instance, almost matching upper and lower bounds onthe time complexity of propositional planning.Funding Agencies|National Graduate School in Computer Science (CUGS), Sweden; Swedish Research Council (VR) [621-2014-4086]</p

Computational Complexity of some Optimization Problems in Planning

Automated planning is known to be computationally hard in the general case. Propositional planning is PSPACE-complete and first-order planning is undecidable. One method for analyzing the computational complexity of planning is to study restricted subsets of planning instances, with the aim of differentiating instances with varying complexity. We use this methodology for studying the computational complexity of planning. Finding new tractable (i.e. polynomial-time solvable) problems has been a particularly important goal for researchers in the area. The reason behind this is not only to differentiate between easy and hard planning instances, but also to use polynomial-time solvable instances in order to construct better heuristic functions and improve planners. We identify a new class of tractable cost-optimal planning instances by restricting the causal graph. We study the computational complexity of oversubscription planning (such as the net-benefit problem) under various restrictions and reveal strong connections with classical planning. Inspired by this, we present a method for compiling oversubscription planning problems into the ordinary plan existence problem. We further study the parameterized complexity of cost-optimal and net-benefit planning under the same restrictions and show that the choice of numeric domain for the action costs has a great impact on the parameterized complexity. We finally consider the parameterized complexity of certain problems related to partial-order planning. In some applications, less restricted plans than total-order plans are needed. Therefore, a partial-order plan is being used instead. When dealing with partial-order plans, one important question is how to achieve optimal partial order plans, i.e. having the highest degree of freedom according to some notion of flexibility. We study several optimization problems for partial-order plans, such as finding a minimum deordering or reordering, and finding the minimum parallel execution length

Cost-optimal and Net-benefit Planning : A Parameterised Complexity View

Cost-optimal planning (COP) uses action costs and asks for a minimum-cost plan. It is sometimes assumed that there is no harm in using actions with zero cost or rational cost. Classical complexity analysis does not contradict this assumption; planning is PSPACE-complete regardless of whether action costs are positive or non-negative, integer or rational. We thus apply parameterised complexity analysis to shed more light on this issue. Our main results are the following. COP is W[2]-complete for positive integer costs, i.e. it is no harder than finding a minimum-length plan, but it is para-NPhard if the costs are non-negative integers or positive rationals. This is a very strong indication that the latter cases are substantially harder. Net-benefit planning (NBP) additionally assigns goal utilities and asks for a plan with maximum difference between its utility and its cost. NBP is para-NP-hard even when action costs and utilities are positive integers, suggesting that it is harder than COP. In addition, we also analyse a large number of subclasses, using both the PUBS restrictions and restricting the number of preconditions and effects

A Multi-parameter Complexity Analysis of Cost-optimal and Net-benefit Planning

Aghighi and Bäckström have previously studied cost-optimal planning (COP) and net-benefit planning (NBP) for three action cost domains: the positive integers (Z_+), the non-negative integers (Z_0) and the positive rationals (Q_+). These were indistinguishable under standard complexity analysis for both problems, but separated for COP using parameterised complexity analysis. With the plan cost, k, as parameter, COP was W[2]-complete for Z_+, but para-NP-hard for both Z_0 and Q_+, i.e. presumably much harder. NBP was para-NP-hard for all three domains, thus remaining unseparable. We continue by considering combinations with several additional parameters and also the non-negative rationals (Q_0). Examples of new parameters are the plan length, l, and the largest denominator of the action costs, d. Our findings include: (1) COP remains W[2]-hard for all domains, even if combining all parameters; (2) COP for Z_0 is in W[2] for the combined parameter {k,l}; (3) COP for Q_+ is in W[2] for {k,d} and (4) COP for Q_0 is in W[2] for {k,d,l}. For NBP we consider further additional parameters, where the most crucial one for reducing complexity is the sum of variable utilities. Our results help to understand the previous results, eg. the separation between Z_+ and Q_+ for COP, and to refine the previous connections with empirical findings

Plan Reordering and Parallel Execution — A Parameterized Complexity View

Bäckström has previously studied a number of optimization problems for partial-order plans, like finding a minimum deordering (MCD) or reordering (MCR), and finding the minimum parallel execution length (PPL), which are all NP-complete. We revisit these problems, but applying parameterized complexity analysis rather than standard complexity analysis. We consider various parameters, including both the original and desired size of the plan order, as well as its width and height. Our findings include that MCD and MCR are W[2]-hard and in W[P] when parameterized with the desired order size, and MCD is fixed-parameter tractable (fpt) when parameterized with the original order size. Problem PPL is fpt if parameterized with the size of the non-concurrency relation, but para-NP-hard in most other cases. We also consider this problem when the number (k) of agents, or processors, is restricted, finding that this number is a crucial parameter; this problem is fixed-parameter tractable with the order size, the parallel execution length and k as parameter, but para-NP-hard without k as parameter

A Multi-parameter Complexity Analysis of Cost-optimal and Net-benefit Planning

Aghighi and Bäckström have previously studied cost-optimal planning (COP) and net-benefit planning (NBP) for three action cost domains: the positive integers (Z_+), the non-negative integers (Z_0) and the positive rationals (Q_+). These were indistinguishable under standard complexity analysis for both problems, but separated for COP using parameterised complexity analysis. With the plan cost, k, as parameter, COP was W[2]-complete for Z_+, but para-NP-hard for both Z_0 and Q_+, i.e. presumably much harder. NBP was para-NP-hard for all three domains, thus remaining unseparable. We continue by considering combinations with several additional parameters and also the non-negative rationals (Q_0). Examples of new parameters are the plan length, l, and the largest denominator of the action costs, d. Our findings include: (1) COP remains W[2]-hard for all domains, even if combining all parameters; (2) COP for Z_0 is in W[2] for the combined parameter {k,l}; (3) COP for Q_+ is in W[2] for {k,d} and (4) COP for Q_0 is in W[2] for {k,d,l}. For NBP we consider further additional parameters, where the most crucial one for reducing complexity is the sum of variable utilities. Our results help to understand the previous results, eg. the separation between Z_+ and Q_+ for COP, and to refine the previous connections with empirical findings

Oversubscription Planning: Complexity and Compilability

Many real-world planning problems are oversubscription problems where all goals are not simultaneously achievable and the planner needs to find a feasible subset. We present complexity results for the so-called partial satisfaction and net benefit problems under various restrictions; this extends previous work by van den Briel et al. Our results reveal strong connections between these problems and with classical planning. We also present a method for efficiently compiling oversubscription problems into the ordinary plan existence problem; this can be viewed as a continuation of earlier work by Keyder &amp; Geffner

A Multi-parameter Complexity Analysis of Cost-optimal and Net-benefit Planning

Aghighi and Bäckström have previously studied cost-optimal planning (COP) and net-benefit planning (NBP) for three action cost domains: the positive integers (Z_+), the non-negative integers (Z_0) and the positive rationals (Q_+). These were indistinguishable under standard complexity analysis for both problems, but separated for COP using parameterised complexity analysis. With the plan cost, k, as parameter, COP was W[2]-complete for Z_+, but para-NP-hard for both Z_0 and Q_+, i.e. presumably much harder. NBP was para-NP-hard for all three domains, thus remaining unseparable. We continue by considering combinations with several additional parameters and also the non-negative rationals (Q_0). Examples of new parameters are the plan length, l, and the largest denominator of the action costs, d. Our findings include: (1) COP remains W[2]-hard for all domains, even if combining all parameters; (2) COP for Z_0 is in W[2] for the combined parameter {k,l}; (3) COP for Q_+ is in W[2] for {k,d} and (4) COP for Q_0 is in W[2] for {k,d,l}. For NBP we consider further additional parameters, where the most crucial one for reducing complexity is the sum of variable utilities. Our results help to understand the previous results, eg. the separation between Z_+ and Q_+ for COP, and to refine the previous connections with empirical findings

Tractable Cost-Optimal Planning over Restricted Polytree Causal Graphs

Causal graphs are widely used to analyze the complexity of planning problems. Many tractable classes have been identified with their aid and state-of-the-art heuristics have been derived by exploiting such classes. In particular, Katz and Keyder have studied causal graphs that are hourglasses (which is a generalization of forks and inverted-forks) and shown that the corresponding cost-optimal planning problem is tractable under certain restrictions. We continue this work by studying polytrees (which is a generalization of hourglasses) under similar restrictions. We prove tractability of cost-optimal planning by providing an algorithm based on a novel notion of variable isomorphism. Our algorithm also sheds light on the k-consistency procedure for identifying unsolvable planning instances. We speculate that this may, at least partially, explain why merge-and-shrink heuristics have been successful for recognizing unsolvable instances

Evaluation of Phoma sp. Biomass as an Endophytic Fungus for Synthesis of Extracellular Gold Nanoparticles with Antibacterial and Antifungal Properties

The aim of our study was to examine the different concentrations of AuNPs as a new antimicrobial substance to control the pathogenic activity. The extracellular synthesis of AuNPs performed by using Phoma sp. as an endophytic fungus. Endophytic fungus was isolated from vascular tissue of peach trees (Prunus persica) from Baft, located in Kerman province, Iran. The UltraViolet-Visible Spectroscopy (UV&ndash;Vis spectroscopy) and Fourier transform infrared spectroscopy provided the absorbance peak at 526 nm, while the X-ray diffraction and transmission electron microscopy images released the formation of spherical AuNPs with sizes in the range of 10&ndash;100 nm. The findings of inhibition zone test of Au nanoparticles (AuNPs) showed a desirable antifungal and antibacterial activity against phytopathogens including Rhizoctonia solani AG1-IA (AG1-IA has been identified as the dominant anastomosis group) and Xanthomonas oryzae pv. oryzae. The highest inhibition level against sclerotia formation was 93% for AuNPs at a concentration of 80 &mu;g/mL. Application of endophytic fungus biomass for synthesis of AuNPs is relatively inexpensive, single step and environmentally friendly. In vitro study of the antifungal activity of AuNPs at concentrations of 10, 20, 40 and 80 &mu;g/mL was conducted against rice fungal pathogen R. solani to reduce sclerotia formation. The experimental data revealed that the Inhibition rate (RH) for sclerotia formation was (15, 33, 74 and 93%), respectively, for their corresponding AuNPs concentrations (10, 20, 40 and 80 &mu;g/mL). Our findings obviously indicated that the RH strongly depend on AuNPs rates, and enhance upon an increase in AuNPs rates. The application of endophytic fungi biomass for green synthesis is our future goal