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In this subsection, we discuss a comparison of STR2-C and STR3-C with the related works in Section 5.2 based on the experimental result from the original

Table 5.3: A comparison for STRm-C, STRsl-C, STRsh-C, STR2-C, and STR3-C. The experimental results are from the original papers (time in seconds).

STR1 STR2 STR3 STR∗-C compression

ratio (Cr) STR∗-C/STR2

STRm-C

[GHLR13] search timememory 147M 147M 384M26.3 13.7 50.0 102M30.6 0.69 0.692.19

STRsl-C [GHLR14] search time 298.05 147.73 189.14 115.30 0.66 0.78 STRsh-C [JN13] nodes/second  365.42  639.39 0.53 (365.42/639.39)0.57 STR2-C _{search time  436.1 1628.5} 212.7 0.13 0.49 STR3-C 596.7 1.37 papers (STRm-C [GHLR13], STRsl-C [GHLR14], and STRsh-C [JN13]). We see that STR2-C can be faster than the others, and believe that c-tuples can compress tables the most. To illustrate this, we extract the experimental results for STRm-C [GHLR13], STRsl-C [GHLR14], STRsh-C [JN13] from the original papers. Since MDD-0.5 instances (MDD-half) are the common benchmarks, we only report on those instances' results in Table 5.3.

The three related papers [GHLR13, GHLR14, JN13] present results dierently, so Table 5.3 lists dierent items for each STR variant. Each row gives the results for each variant with the best value in bold. For STRm-C, the results are based on one MDD-half instance. For the others, the results are the mean values (geometric mean for STRsh-C) over the instances that can be solved. `' indicates that the runtime is not given.

To compare the compression, the most fair way is to use compression ratio (Cr). However, the compression ratio is dened in dierent ways in the original papers. We rst give the denitions in the following:

ˆ For STRm-C, only memory usage is reported, thus Cr is the memory used by STRm-C divided by the memory used by STR2;

ˆ For STRsl-C, Cr is the compressed table size divided by the original table size;

ˆ For STRsh-C, Cr is the number of short supports divided by the number of standard tuples5_;

ˆ For STR2-C and STR3-C, Cr is the number of c-tuples divided by the num- ber of standard tuples;

Although the compression ratio is dened dierently, we may still conjecture that 5_{Cr is originally dened as the number of standard tuples divided by the number of short}

supports [JN13]. Thus in Table 5.3, the original Cr for STRsh-C is 1.87(=1/0.53). For a clear comparison, we reverse it.

c-tuple compresses the tables most on these instances, as its compression ratio 0.13 is much smaller than the others.

To compare the runtime, as the time (or nodes/second for STRsh-C) from the related papers are dependent on dierent machines or solvers, we cannot compare the runtime directly. Therefore, we calculate the runtime ratio of STR∗-C/STR2 (column STR∗-C/STR2) based on their own STR times. We nd out that STR2- C has the best speedup, 0.49, over STR2. The second is STRsh-C, then STRsl-C, while STRm-C is 2X slower than STR2. Note that the runtime ratio of STR3- C/STR2 is 1.37. This is because STR3 is slower than STR2 on this benchmark. But, STR3-C is signicantly faster than STR3, and the runtime ratio is 0.37 (STR3-C/STR3).

5.7 Conclusion

Our experiments show that our STR algorithms on compressed tables are com- petitive when the tables are compressed enough. It is quite surprising that such a simple change in representation is eective to speedup state-of-the-art algorithms. In our experiments, as the table compression ratio drops to 25%, the STRx-C algorithms become more ecient. This illustrates that static and dynamic table compression can cooperate well in practice even though the use of c-tuples has the drawbacks that it reduces the amount of table reduction and has some (small) overheads. We also show that the good properties of STR2 and STR3 can be inherited by their compressed versions.

Chapter 6

Factor Encoding for Higher-Order

Consistencies

Generalized arc consistency (GAC) is one of the most fundamental and well studied notions of local consistency for constraint satisfaction problems (CSPs) due to its simplicity and excellent performance of propagation algorithms in practice. Domain reduction interspersed with GAC during backtrack search has become the foremost method for solving a general CSP [SF94]. GAC on positive table constraints, in particular, continues to receive great attention with a number of propagation algorithms proposed [Lec11, MHD14, CY10, LLY12, GJMN07, LS06] during recent years. These advances in turn provide a basis for many algorithms that enforce even stronger consistencies than GAC to build on.

Janssen et al. [JJNV89] shown that a constraint network is pairwise consistent (PWC) if and only if its dual CSP is arc consistent. PWC is a k-wise consis- tency [Gys86] for the case where k = 2. This is one of the earlier works that demonstrates how (G)AC can be used to achieve other types of consistencies. Consistencies of level/order higher than GAC are also the subject of many recent

works [KWR+_{10, PS12, LPS13]. Specically, MaxRPWC, PWC, and FPWC are}

investigated in [BSW08, PS12, LPS13]. Many of the algorithms that enforce these consistencies are based on extensive well-established GAC algorithms. Paparrizou and Stergiou [PS12] extended the GACva algorithm [LS06] to enforce MaxRPWC. Subsequently, STR2 was extended to cope with FPWC, resulting in eSTR [LPS13] algorithm.

As another area of focus in CSPs, researchers have studied how to trans- form non-binary constraint networks into equivalent binary constraint networks so that the algorithms and methods from the binary case can be applied [BCvBW02, SS05]. Two techniques emerge as a result: the hidden transforma-

tion and the dual transformation. Both rely on the dual variable associated with each constraint, whose domain values have a one-to-one correspondence with the constraint's tuples. Nevertheless, the benets of the transformation diminish as ltering algorithms for non-binary CSPs get better.

In this chapter, we study changing a CSP P into a dierent but "equivalent" representation CSP P0 _{. The dierence lies in the constraints and variables. When}

enforcing GAC on P0_{, full pairwise consistency (FPWC) is enforced on P, reach-}

ing both GAC and pairwise consistency (PWC). In this respect, our intention is similar to that of the authors of [MDL14] who proposed another kind of transfor- mation, i.e. reformulating CSPs into k-interleaved CSPs. The distinction is that in [MDL14] the transformation is merely a way to expose PWC to GAC algorithm so that FPWC can be enforced through GAC. Like the transformation from a non- binary to a binary network, the k-interleaved encoding [MDL14] is based on dual variables. But here the dual variable is included into the scope of the constraint that it is associated with. The pruning power comes from the join of tables that uses the dual variables as its scope so that propagation can be transmitted directly to other constraints. However, since k-interleaved encoding uses joins of tables, some problems become prohibitively large and the propagation on the large join tables may be slow.

Our transformation works in a fundamentally dierent way. Instead of forming a dual variable for each constraint, we factor out commonly shared variables among them. These variables form new (compound) variables that will be augmented to only the constraints that are involved. For FPWC, no new constraints are created. We extend this transformation to cover general k-wise consistency, adding new constraints reduced from the join of k tables. Preliminary experiments show that for FPWC our method is faster than both [MDL14] and a specialized FPWC algorithm. For k-wise consistency where k ≥ 3, our transformation can lower the number of nodes visited during search but it is more costly and possibly of limited application unless the search reduction is large.

6.1 Overview of FPWC Algorithms

In this section, we review two related works. The rst one is the state-of- the-art algorithm which enforces FPWC on CSPs directly (instead of reformu- lation) [LPS13]. This algorithm is based on STR principles, so it is called eSTR [LPS13]. The second is the reformulation of k-interleaved CSPs [MDL14]. Enforcing GAC on the k-interleaved (k=2) CSPs guarantees that the original CSPs are FPWC. When k > 2, even higher consistency than FPWC can be achieved. In

addition, a special higher-order consistency based on our factor encoding for k-wise consistency (FKWC) is introduced in section 6.3. In fact, FKWC is not a kind of consistency, but an encoding with new constraints from table joins strictly speak- ing. Enforcing GAC on FKWC gives a consistency level weaker than enforcing kWC on the original CSPs when k > 3, but at least as strong as FPWC.