Sector de fabricación de envases y embalajes de metal

We compare the various strengths of 4PH3C+ _{(Section 6.4.2 by limiting the propa-}

gation queue and using the decision tree.

Table 6.9 shows the performance of running the 4PH3C+_{variants at pre-processing,}

while Table 6.10 shows the performance of running the 4PH3C+ _{variants as RFL.}

Table 6.9: 4PH3C+ _{variants as pre-processing with dom/deg}

Algorithm GAC 4DH3C+ _4DPH3C+ _4P3_H3C+ _42DPH3C+ _4PH3C+ #Solved 2,026 1,942 1,946 1,945 1,954 1,947 #Memout 734 756 756 756 756 756 P CPU [sec] >633,595.3 >1,070,904.7 >1,053,561.1 >1,055,859.6 >1,032,102.7 >1,039,284.2 Avg. #NV 1,193,844.1 363,057.2 355,957.0 357,255.8 355,931.5 355,793.4

#Instances 3,780 total, 1,895 by all, 2,081 by at least one

Table 6.10: 4PH3C+ _{variants as RFL with dom/deg}

Algorithm GAC 4DH3C+ _4DPH3C+ _4P3_H3C+ _42DPH3C+ _4PH3C+ #Solved 2,026 1,758 1,773 1,769 1,760 1,769 #Memout 734 756 756 757 758 756 P CPU [sec] >608,395.3 >1,905,784.7 >1,887,383.1 >1,877,515.6 >1,898,658.9 >1,892,971.1 Avg. #NV 1,184,435.0 147,688.6 142,873.1 145,231.5 148,256.6 142,751.5

#Instances 3,780 total, 1,673 by all, 2,074 by at least one

For both pre-processing and RFL, GAC performs better than any of the PH3C variations. This result is predictable given that PH3C is likely too strong for most problems, which we address in our triggering experiments in Section 6.5.5. The statistical rankings, according to a paired t-test, of the considered variations are as follows:

pre-processing: 42DPH3C+  4PH3C+ _{∼ 4DPH3C}+ _4P3_H3C+ _{∼ 4DH3C}+

RFL: 4P3H3C+ ∼ 4DPH3C+ _{∼ 42DPH3C}+ _{∼ 4PH3C}+ _4DH3C+

where A  B denotes that A is statistically better than B and A ∼ B denotes that there are not statistically distinguishable.

6.5.5

4PH3C

_{with PrePeak}

We evaluate the use of PrePeak to trigger 4DH3C+ _{using the decision tree (Sec-}

tion 6.4.4), and using the variation of DPH3C to iterate through the triangles only once (i.e., as a how much strategy). Table 6.11 compares the performance of DPH3C with GAC. 4DPH3C solves more instances than GAC in faster CPU time. Although

Table 6.11: Comparing GAC and PrePeak with 4DPH3C+

GAC 4DPH3C+ #Solved 1,254 1,252 #MemOut 58 85 P CPU [sec] >215,914.2 >213,326.7 Avg. #NV 823,350.7 225,780.4

#Instances 2,126 total, 1,241 by all, 1,265 by at least one

it does have a few more memouts, the approach of using the decision tree and watching for memouts helps avoid them.

Table 6.12 highlights two exemplary benchmark for 4DPH3C: dubois and mug. The dual graph of the dubois benchmark is a ladder graph, thus by Theorem 16 it can

Table 6.12: _{Benchmarks where PrePeak with 4DPH3C}+ _{performs well}

GAC 4DPH3C+

dubois #Instances 13 total, 6 by all, 13 by at least one

#Solved 6 13

#MemOut 0 0

P

CPU [sec] >29,484.0 0.5

Avg. #NV 123,405,942.7 0

mug #Instances 8 total, 4 by all, 8 by at least one

#Solved 6 13

#MemOut 0 0

P

CPU [sec] >14,400.1 721.3

Avg. #NV 94.0 94.0

be solved backtrack free if partial hyper-3 consistency is enforced. This observation

known to be a tractable benchmark by looking at its constraint semantics, meaning the definition of the constraint, as because all of the constraints can be re-written as implication constraints (i.e., ⇔) [Ostrowski et al., 2002]. As for the mug benchmark, DPH3C is not able to solve the instance backtrack free, but it must search. However, the high strength of 4DPH3C was able to shrink the search space to allow all of the instances of mug to be solved.

As for the other benchmarks, 4DPH3C performs similarly to GAC, provided it does not memout. Indeed, PrePeak triggers 4DPH3C few times as it filters relatively few domain-values given the amount of effort it takes to enforce it. This result can be explained by that 4DPH3C not only needs to filter both the equality constraints, and the CSP constraints, before it is able to filter any domain values. The amount of ‘indirection’ between the equality constraints and the CSP variables hinders its ability to filter many values.

Summary

In this chapter we studied a special form of cycles: triangles. Partial Path Consis- tency (PPC) takes advantage of triangles, especially the 4PPC algorithm, thus we empirically evaluate 4PPC as lookahead, which has never been studied. Further, we presented the first algorithm for enforcing Partial Hyper-3 Consistency (PH3P) by adapting 4PPC to 4PH3P and empirically evaluate it as lookahead. We notice that the dubois benchmark can be determined inconsistent at pre-processing by PH3P, and identify the structural property (Theorem 16 of Chapter 5) to explain its tractability.

Chapter 7

Conclusions and Future Work

This chapter concludes the dissertation by summarizing our contributions and giving directions for future research.

7.1

Summary of Contributions

Constraint Satisfaction Problems (CSPs) are usually solved with search. To reduce the size of the search space, backtrack search is typically interleaved with constraint propagation. Stronger consistency algorithms can filter larger portions of the search space at the cost of an increased CPU time.

The research presented in this dissertation addresses the question of enforcing high-level consistency during search. We offer a new perspective that characterizes the various possible approaches in term of when, where, and how much of a higher- level consistency to enforce during search.

Figure 7.1 gives an overview of the different when, where, and how much strategies advocated for in this thesis. In particular, Chapter 4 introduces PrePeak as a strategy for determining where to enforce higher-level consistency (HLC) depending

When? Where? How much? HLC _PREPEAK PREPEAK+ PREPEAK Cycles or Triangles PREPEAK+ Cycles or Triangles Cycles or Triangles

Figure 7.1: The dimensions of enforcing consistency investigated in this dissertation

on the number of backtracks. PrePeak+ _{combines this ‘where strategy’ with a how}

much strategy to interrupt the consistency algorithm after processing a given number

of elements in the algorithm’s propagation queue or after a given CPU time has passed. Chapter 5 and Chapter 6 localize the operations of the consistency algorithms to cycle structures of the CSP and to triangles, respectively. They also combine the resulting new consistencies with PrePeak and PrePeak+_.

In summary, this dissertation introduces a framework for enforcing higher-level consistency on a CSP that adapts its filtering to the problem at hand.