### Table 3.3 illustrates the strength of the SDP relaxation on several Nugent problems from QAPLIB compared to other bounds in the literature: Gilmore-Lawler bound (GLB) [6, 14], the projection or elimination bound ELI of [8], and the improved eigenvalue bound EVB3 from [26]. We note the high cost of the SDP bound for n=30 in the table and the low relative error for the bounds. A relaxed form of the SDP bound played a major role in the solution to optimality of several hard QAPs, see [3].

2002

### Table 7.3: Nine previously unsolved benchmark instances taken from QAPLIB. The tree-size estimator enabled us to select those instances that could be solved in the available amount of time. After we had published our results, Tschoke et al. implemented a sim- ilar algorithm on their parallel PowerPC computer and solved the nug24 instance. Their result motivated us to develop a dynamic programming al- gorithm based on a new relaxation of the QAP which is stronger than the Gilmore{Lawler bound. By running this algorithm on the Paragon, we could

### Table 1 is a comparison of bounds obtained from MSDR3 and other relaxation methods applied to instances from QAPLIB [6]. The first column OPT denotes the exact optimal value of the problem instance, while the following columns contain the lower bounds from the relaxation methods: GLB , the Gilmore-Lawler bound [10]; KCCEB , the dual linear programming bound [15]; P B , the projected eigenvalue bound [12]; QP B , the convex quadratic programming bound [1]; SDR1 , SDR2 , SDR3 , the vector-lifting semidefi- nite relaxation bounds [27] computed by the bundle method [24]; the last column is our MSDR3 . All output values are rounded up to the nearest integer. To solve QAP , the minimization of trace AXBXT and trace BXAXT are equivalent. But for the relaxation MSDR3 , exchanging the roles of A and B results in two different formulations and bounds. In our tests we use both versions and take the larger output as the bound of MSDR3 . We then keep the better formulation throughout the branch and bound process, so that we do not double the computational work.

2006

"... In PAGE 16: ... We then keep the better formulation throughout the branch and bound process, so that we do not double the computational work. From Table1 , we see that the relative performances between the LP -based bounds GLB , KCCEB are unpredictable. At some instances, both are weaker than even the least expensive P B bounds.... In PAGE 17: ...a. 4887 4965 4621 Nug30 6124 4539 4785 5266 5362 5413 5651 5803 5446 rou12 235528 202272 223543 200024 205461 208685 219018 223680 207445 rou15 354210 298548 323589 296705 303487 306833 320567 333287 303456 rou20 725522 599948 641425 597045 607362 615549 641577 663833 609102 scr12 31410 27858 29538 4727 8223 11117 23844 29321 18803 scr15 51140 44737 48547 10355 12401 17046 41881 48836 39399 scr20 110030 86766 94489 16113 23480 28535 82106 94998 50548 tai12a 224416 195918 220804 193124 199378 203595 215241 222784 202134 tai15a 388214 327501 351938 325019 330205 333437 349179 364761 331956 tai17a 491812 412722 441501 408910 415576 419619 440333 451317 418356 tai20a 703482 580674 616644 575831 584938 591994 617630 637300 587266 tai25a 1167256 962417 1005978 956657 981870 974004 908248 1041337 970788 tai30a 1818146 1504688 1565313 1500407 1517829 1529135 1573580 1652186 1521368 tho30 149936 90578 99855 119254 124286 125972 134368 136059 122778 Table1 : Comparison of bounds for QAPLIB instances... ..."

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### Table 5: Results for rst level in branching tree

"... In PAGE 33: ...rst level of the branching tree, i.e. we want to see how fast the lower bounds increase after branching. Table5 gives the results for the rst level, when one partial assignment is made. As pointed out above, the Nugxx examples posses inherent symmetries due to their distance matrices, e.... In PAGE 33: ... To measure the increasing rate (ir) of the lower bound (lbd) by branching, we de ne the rate in percent as follows. ir := lbdchild ? lbdparent lbdparent 100% In Table5 , the increasing rates are shown by the numbers in the brackets. The results of this table show that the lower bounds given by the SDP relaxations increase much faster than the Gilmore-Lawler bounds in the rst level of the branching tree.... ..."

### Table 1: Comparison of bounds for QAPLIB instances

2006

"... In PAGE 16: ... 3 Numerical Results 3.1 Comparing Bounds for QAPLIB Problems Table1 is a comparison of bounds obtained from MSDR3 and other relaxation methods applied to instances from QAPLIB [6]. The first column OPT denotes the exact optimal value of the problem instance, while the following columns contain the lower bounds from the relaxation methods: GLB , the Gilmore-Lawler bound [10]; KCCEB , the dual linear programming bound [15]; P B , the projected eigenvalue bound [12]; QP B , the convex quadratic programming bound [1]; SDR1 , SDR2 , SDR3 , the vector-lifting semidefi- nite relaxation bounds [27] computed by the bundle method [24]; the last column is our MSDR3 .... In PAGE 16: ... We then keep the better formulation throughout the branch and bound process, so that we do not double the computational work. From Table1 , we see that the relative performances between the LP -based bounds GLB , KCCEB are unpredictable. At some instances, both are weaker than even the least expensive P B bounds.... ..."

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### Table 1: PN (the probability of necessary causation) as a function of assumptions and available data. ERR stands of the excess-risk-ratio 1 ? P (yjx0)=P (yjx) and CERR is given in Eq. (49). The non-entries (|) represent vacuous bounds, that is, 0 P N 1. Assumptions Data Available

2000

"... In PAGE 24: ...6 Summary of results We now summarize the results from Section 4 that should be of value to practicing epidemiologists and policy makers. These results are shown in Table1 , which lists the best estimand of PN under various assumptions and various types of data|the stronger the assumptions, the more informative the estimates. We see that the excess-risk-ratio (ERR), which epidemiologists commonly identify with the probability of causation, is a valid measure of PN only when two assumptions can be ascertained: exogeneity (i.... In PAGE 25: ...concerned with associations between such factors and susceptibility to expo- sure, as is often assumed in the literature [Khoury , 1989, Glymour, 1998]. The last two rows in Table1 correspond to no assumptions about exo- geneity, and they yield vacuous bounds for PN when data come from either experimental or observational study. In contrast, informative bounds (25) or point estimates (49) are obtained when data from experimental and ob- servational studies are combined.... ..."

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### Table 1 The upper and lower bounds of s and standard form II.

### Table4: FractionsofErrorsandSVs,AlongwiththeMarginsofClassSeparation, for the Toy Example Depicted in Figure 11.

### Table 5 Kolmogorov-Smirnov with Lilliefors significance correction

### Table Bounds

1996

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