From ddbf92b0daa22bfc5b6b28f3d1fe7f2fdf712c36 Mon Sep 17 00:00:00 2001
From: Chiara Braghin <chiara.braghin@unimi.it>
Date: Wed, 12 Jun 2024 17:35:23 +0200
Subject: [PATCH] added reference

---
 bib_on_BigDataAccessControl.bib | 17 +++++++++++++++++
 metrics.tex                     |  6 +++---
 2 files changed, 20 insertions(+), 3 deletions(-)

diff --git a/bib_on_BigDataAccessControl.bib b/bib_on_BigDataAccessControl.bib
index 4f3b235..c499417 100644
--- a/bib_on_BigDataAccessControl.bib
+++ b/bib_on_BigDataAccessControl.bib
@@ -946,3 +946,20 @@ @inproceedings{ABBJ.ICWS2022
 address = {Barcelona, Spain},
 }
 
+@Inbook{Kellerer2004,
+author="Kellerer, Hans
+and Pferschy, Ulrich
+and Pisinger, David",
+title="Multidimensional Knapsack Problems",
+bookTitle="Knapsack Problems",
+year="2004",
+publisher="Springer Berlin Heidelberg",
+address="Berlin, Heidelberg",
+pages="235--283",
+abstract="In this first chapter of extensions and generalizations of the basic knapsack problem (KP) we will add additional constraints to the single weight constraint (1.2) thus attaining the multidimensional knapsack problem. After the introduction we will deal extensively with relaxations and reductions in Section 9.2. Exact algorithms to compute optimal solutions will be covered in Section 9.3 followed by results on approximation in Section 9.4. A detailed treatment of heuristic methods will be given in Section 9.5. Separate sections are devoted to two special cases, namely the two-dimensional knapsack problem (Section 9.6) and the cardinality constrained knapsack problem (Section 9.7). Finally, we will consider the combination of multiple constraints and multiple-choice selection of items from classes (see Chapter 11 for the one-dimensional case) in Section 9.8.",
+isbn="978-3-540-24777-7",
+doi="10.1007/978-3-540-24777-7_9",
+url="https://doi.org/10.1007/978-3-540-24777-7_9"
+}
+
+
diff --git a/metrics.tex b/metrics.tex
index 0217208..6d9be42 100644
--- a/metrics.tex
+++ b/metrics.tex
@@ -52,8 +52,8 @@ \subsubsection{Pipeline Quality}
 
 \vspace{0.5em}
 
-\begin{definition}[\emph{\quality}]\label{def:quality}
-  Given a metric $M$$\in$$\{M_J,M_{JSD}$\} modeling the data quality, pipeline quality \q$=$$M_{ij}$, with $M_{ij}$ the value of the quality metric retrieved at each vertex \vii{i}$\in$$\V'_S$ of the pipeline instance $G'$ according to service \sii{j}.
+\begin{definition}[\emph{\Quality}]\label{def:quality}
+  Given a metric $M$$\in$$\{M_J,M_{JSD}$\} modeling data quality, the pipeline quality \q is equal to $\sum_{i=1}^{n}M_{ij}$, with $M_{ij}$ the value of the quality metric retrieved at each vertex \vii{i}$\in$$\V'_S$ of the pipeline instance $G'$ according to service \sii{j}.
 \end{definition}
 
 \vspace{0.5em}
@@ -87,7 +87,7 @@ \subsection{NP-Hardness of the Max-Quality Pipeline Instantiation Problem}\label
   The Max-Quality \problem is NP-Hard.
 \end{theorem}
 \emph{Proof: }
-The proof is a reduction from the multiple-choice knapsack problem (MCKP), a classified NP-hard combinatorial optimization problem, which is a generalization of the simple knapsack problem (KP) \cite{}\hl{CITA}. In the MCKP problem, there are $t$ mutually disjoint classes $N_1,N_2,\ldots,N_t$ of items to pack in some knapsack of capacity $C$, class $N_i$ having size $n_i$. Each item $j$$\in$$N_i$ has a profit $p_{ij}$ and a weight $w_{ij}$; the problem is to choose one item from each class such that the profit sum is maximized without having the weight sum to exceed C.
+The proof is a reduction from the multiple-choice knapsack problem (MCKP), a classified NP-hard combinatorial optimization problem, which is a generalization of the simple knapsack problem (KP) \cite{Kellerer2004}. In the MCKP problem, there are $t$ mutually disjoint classes $N_1,N_2,\ldots,N_t$ of items to pack in some knapsack of capacity $C$, class $N_i$ having size $n_i$. Each item $j$$\in$$N_i$ has a profit $p_{ij}$ and a weight $w_{ij}$; the problem is to choose one item from each class such that the profit sum is maximized without having the weight sum to exceed C.
 
 The MCKP can be reduced to the Max quality \problem in polynomial time, with $N_1,N_2,\ldots,N_t$ corresponding to $S^c_{1}, S^c_{1}, \ldots, S^c_{u},$, $t$$=$$u$ and $n_i$ the size of $S^c_{i}$. The profit $p_{ij}$ of item $j$$\in$$N_i$ corresponds to \textit{\q}$_{ij}$ computed for each candidate service $s_j$$\in$$S^c_{i}$, while $w_{ij}$ is uniformly 1 (thus, C is always equal to the cardinality of $V_C$).