  Low level algorithms, bit wizardry, combinatorial generation, fast transforms, fast arithmetics and algorithms for finite fields. A real masterpiece of computation !

Great paper! A pleasure to read! - Using the proposed ordering divides the running time by no more than two compared to using other orderings. This is a bit disappointing. - Why no comparison against a random permutation of the nodes? I'm curious how bad a random ordering performs. - "the optimal score $F_w$ cannot be computed due to the NP-hard complexity": such as it is, this statement is not true since only a few real-world datasets are considered. In practice, these real-world datasets could lead to easy instances of the problem. - Is there any publicly available implementation of the proposed algorithm? I find the following paragraph confusing: "In general, the problem of finding the optimal graph ordering by maximizing $F(\phi)$ is the problem of solving maxTSP-$w$ with a window size $w$ as a variant of the maxTSP problem. To solve the graph ordering problem as maxTSP-$w$, we can construct an edge-weighted complete undirected graph $G_w$ from $G$. Here, $V(G_w) =V(G)$, and there is an edge $(v_i, v_j)\in E(G_w)$ for any pair of nodes in $V(G_w)$. The edge-weight for an edge (vi, vj) in $G_w$, denoted as $s_{ij}$, is $S(v_i, v_j)$ computed for the two nodes in the original graph $G$. Then the optimal maxTSP-$w$ over $G$ is the solution of maxTSP over $G$. Note that maxTSP only cares the weight between two adjacent nodes, whereas maxTSP-$w$ cares the sum of weights within a window of size $w$." I think that: - The index $w$ of $G_w$ does not refer to the window size $w$, but stands for "weight". - It should be "the optimal $F(\Phi)$ over $G$ is the solution of maxTSP-$w$ over $G_w$" instead of "the optimal maxTSP-$w$ over $G$ is the solution of maxTSP over $G$". - On the same line, in the paragraph above that one: "The optimal graph ordering, for the window size $w= 1$, by maximizing $F(\Phi)$ is equivalent to the maximum traveling sales-man problem, denoted as maxTSP for short.". It should be clear that maxTSP should be solve on $G_w$, where $w$ stands for weight: complete graph with weight $S(u,v)$ on the edge $u,v$. MINORS: - "The night graph algorithms" and "the night orderings": "night" instead of "nine".
Interesting comments. > - Using the proposed ordering divides the running time by no more than two compared to using other orderings. This is a bit disappointing. Yes, but from Figure 1, we can draw that we cannot expect much more than this... Maybe 3 or 4 at most. > - "the optimal score $F_w$ cannot be computed due to the NP-hard complexity": such as it is, this statement is not true since only a few real-world datasets are considered. In practice, these real-world datasets could lead to easy instances of the problem. Very relevant, still it looks very much like looking for the best solution of a TSPw on a real instance. I understand that they actually found the optimal $F_w$ in some cases (Table 1).