### Table 5: Approximate dynamic programming results for the cases of Table 4. ~ J denotes the revenue generated by the approximate policy. For Cases 4 and 5 it is computationally intractable to obtain the optimal static and dynamic policies.

2000

"... In PAGE 24: ... The computation of optimal static and dynamic prices becomes computationally prohibitive as the state space grows, thus, we will resort to the approximation methods outlined in Sections 8 and 9. In Table5 we report approximate dynamic programming results for a number of problems including some large-scale ones. We have used the approach outlined in item 2 of Section 9.... In PAGE 25: ...19 35000 3500 2.06 Table 4: The input parameters for the results of Table5 . We considered two-class systems with demand functions of the form i(ui) = 0;i ?ui 1;i and r1 = 4, r2 = 1, 1 = 1, 2 = 2.... In PAGE 26: ...Figure 4: Approximate dynamic prices for Case 3 of Table5 . Figures (a) and (b) depict prices for classes one and two, respectively.... ..."

Cited by 81

### Table 5: Approximate dynamic programming results for the cases of Table 4. ~ J denotes the revenue generated by the approximate policy. For Cases 4 and 5 it is computationally intractable to obtain the optimal static and dynamic policies.

2000

"... In PAGE 24: ... The computation of optimal static and dynamic prices becomes computationally prohibitive as the state space grows, thus, we will resort to the approximation methods outlined in Sections 8 and 9. In Table5 we report approximate dynamic programming results for a number of problems including some large-scale ones. We have used the approach outlined in item 2 of Section 9.... In PAGE 25: ...19 35000 3500 2.06 Table 4: The input parameters for the results of Table5 . We considered two-class systems with demand functions of the form i(ui) = 0;i ?ui 1;i and r1 = 4, r2 = 1, 1 = 1, 2 = 2.... In PAGE 26: ...Figure 4: Approximate dynamic prices for Case 3 of Table5 . Figures (a) and (b) depict prices for classes one and two, respectively.... ..."

Cited by 81

### Table 2. A list of counting constraints that are intractable to propagate with GAC.

2004

"... In PAGE 5: ... We use the basic tools of computational complexity to show their tractability or intractability. Table2 gives some of the intractability results we obtained for counting constraints on integer variables. Proofs are in [3].... ..."

Cited by 7

### Table 2. A list of counting constraints that are intractable to propagate with GAC.

2004

"... In PAGE 5: ... We use the basic tools of computational complexity to show their tractability or intractability. Table2 gives some of the intractability results we obtained for counting constraints on integer variables. Proofs are in [3].... ..."

Cited by 7

### Table 2. A list of counting constraints that are intractable to propagate with GAC.

2004

"... In PAGE 5: ... We use the basic tools of computational complexity to show their tractability or intractability. Table2 gives some of the intractability results we obtained for counting constraints on integer variables. Proofs are in [3].... ..."

Cited by 7

### Table 3: Timing for the di erent methods and examples. abeyond memory limit, bvolume computed incorrectly, cproblem is intractable with this method, d`cdd apos; is faster by a factor of at least 100, 6,9storage performed for 6 and 9 levels resp. Next, to measure potential numerical di culties, Table 4 presents condition numbers for some algorithms. As di erent methods face di erent numerical problems we decided to de ne two di erent condition numbers: For triangulation methods all partial volumes are positive; the only problem lies in summing up volumes of potentially di erent magnitudes. Hence in this case we de ne the condition number as log10 of the biggest simplex volume divided through the smallest nonzero one (zero volumes occur with `qhull apos;, which returns some degenerate simplices). When evaluating Lawrence apos;s formula the main problem is that

"... In PAGE 12: ... Unless otherwise stated the computed volume is identical for all methods for at least six digits and is therefore given only once. The general abbreviations for the codes used in the sequel are given below in the rst column; in the second column the shorter ones used in Table3 and 4 are stated. BND Bnd boundary triangulation using `lrs apos; DEL Del Delaunay triangulation using `qhull apos; C amp;H CH Cohen amp; Hickey apos;s triangulation including our improvements HOT HOT hybrid orthonormalisation technique LAW-nd Lnd Lawrence apos;s formula in the non-degenerate case LAW-d Ld Lawrence apos;s formula in the general case using `lrs apos; for computing all lexicographically feasible cobases LAS original Lasserre r-LAS rL revised Lasserre Sources for the codes are given in the appendix.... In PAGE 14: ... 5.3 Comparing the Main Codes In Table3 the timing in CPU-seconds for all codes and examples is given; because the transformation V!H or H!V can be as demanding as the volume computation itself, and furthermore we need both representations for certain methods, we give in the last two columns the transformation time when using `lrs apos; and `pd apos; with exact arithmetic, see Appendix 6.... In PAGE 16: ... Up to date, only one implementation of these so-called primal-dual algorithms is available, namely `pd apos;, which uses reverse search with exact arithmetic as the oracle. In Table3 the transformation times using `lrs apos; for the `easy apos; and `pd apos; for the `di cult apos; direction are reported. Still, the transformations can be more time consuming than the actual volume computation, so it seems worthwhile to take the representation into account when choosing an algorithm for the volume computation.... ..."

### Table 3: a beyond memory limit, b numerical errors, c problem is intractable with this method, d `cdd apos; is faster by a factor of at least 100, 6,9 storage performed for 6 resp. 9 levels

"... In PAGE 12: ... Unless otherwise stated the computed volume is identical for all methods for at least six digits and is therefore given only once. The general abbreviations for the codes used in the sequel are given below in the rst column; in the second column the shorter ones used in Table3 are stated. BND Bnd boundary triangulation using `lrs apos; DEL Del Delaunay triangulation using `qhull apos; C amp;H original Cohen amp; Hickey triangulation r-C amp;H rCH revised Cohen amp; Hickey triangulation HOT HOT hybrid orthonormalisation technique LAW-nd Lnd Lawrence apos;s formula in the non-degenerate case LAW-d Ld Lawrence apos;s formula in the general case using `lrs apos; for computing all lexicographically feasible cobases LAS original Lasserre r-LAS rL revised Lasserre Sources for the codes are given in the appendix.... In PAGE 13: ... 5.3 Comparing the main codes In Table3 the timing in CPU-seconds for all codes and examples is given; because the transformation V!H or H!V can be as demanding as the volume computation itself, and furthermore we need both representations for certain methods, we give in the last two columns the transformation time when using `lrs apos; and `pd apos; with exact arithmetic.... In PAGE 14: ...Table 3: a beyond memory limit, b numerical errors, c problem is intractable with this method, d `cdd apos; is faster by a factor of at least 100, 6,9 storage performed for 6 resp. 9 levels Table3 reveals huge di erences in the CPU time among the di erent codes for the same polytope. Some problems are even intractable with certain methods whereas they are quite e ciently solved by others, demonstrating the relevance of a good choice.... In PAGE 15: ... Up to date, only one implementation of these so-called primal-dual algorithms is available, namely `pd apos;, which uses reverse search with exact arithmetic as the oracle. In Table3 the transformation times using `lrs apos; for the `easy apos; and `pd apos; for the `di cult apos; direction are reported. Still, the transformations can be more time consuming than the actual volume computation, so it seems worthwhile to take the representation into account when choosing an algorithm for the volume.... ..."

### Table 1: A variety of inference algorithms for graphical models. Note that not every cell is filled in (marked by ?) while some are simply intractable. ME is the maximization-expectation algorithm of [15] and any MCMC means that we can use any MCMC sampler for the parameters once latent variables have been marginalized out.

2007

"... In PAGE 7: ... Over the years a variety of inference algorithms have been proposed based on a combination of {maximize, sample, assume independent, marginalize out} applied to both parameters and latent variables. We conclude by summarizing these algorithms in Table1 , and note that CVB is located in the marginalize out parameters and assume latent variables are independent cell. A Exact Computation of Expectation Terms in (15) We can compute the expectation terms in (15) exactly as follows.... ..."

Cited by 6

### Table 1: A variety of inference algorithms for graphical models. Note that not every cell is filled in (marked by ?) while some are simply intractable. ME is the maximization-expectation algorithm of [15] and any MCMC means that we can use any MCMC sampler for the parameters once latent variables have been marginalized out.

"... In PAGE 6: ... Over the years a variety of inference algorithms has been proposed based on a combination of {maximize, sample, assume independent, marginalize out} applied to both parameters and latent variables. We summarize these algorithms in Table1 , and note that CVB is located in the marginalize out parameters and assume latent variables are independent cell. A Exact Computation of Expectation Terms in (15) We can compute the expectation terms in (15) exactly as follows.... ..."

### Table 1: A variety of inference algorithms for graphical models. Note that not every cell is filled in (marked by ?) while some are simply intractable. ME is the maximization-expectation algorithm of [15] and any MCMC means that we can use any MCMC sampler for the parameters once latent variables have been marginalized out.

"... In PAGE 6: ... Over the years a variety of inference algorithms has been proposed based on a combination of fmaximize, sample, assume independent, marginalize outg applied to both parameters and latent variables. We summarize these algorithms in Table1 , and note that CVB is located in the marginalize out parameters and assume latent variables are independent cell. A Exact Computation of Expectation Terms in (15) We can compute the expectation terms in (15) exactly as follows.... ..."