From cec9b298f548ce44fe5eaecf71c04d65c95de993 Mon Sep 17 00:00:00 2001
From: Stratis Ioannidis <stratis@stratis-Latitude-E6320.(none)>
Date: Sun, 4 Nov 2012 23:38:08 -0800
Subject: opts

---
 main.tex | 228 ++++++++++++++++++++++++++++++++-------------------------------
 1 file changed, 115 insertions(+), 113 deletions(-)

(limited to 'main.tex')

diff --git a/main.tex b/main.tex
index c79ff88..7faec7e 100644
--- a/main.tex
+++ b/main.tex
@@ -31,7 +31,7 @@ Let $S_G$ be the set computed by the greedy algorithm and define $i^*$:
 \end{displaymath}
 then the following inequality holds:
 \begin{displaymath}
-OPT(V,\mathcal{N},B) \leq \frac{e}{e-1}\big( 3 V(S_G) + 2 V(i^*)\big).
+OPT \leq \frac{e}{e-1}\big( 3 V(S_G) + 2 V(i^*)\big).
 \end{displaymath}
 \end{lemma}
 
@@ -56,8 +56,8 @@ overall polynomial complexity for the allocation algorithm.
 \sloppy
 When the underlying
 full information problem \eqref{eq:non-strategic} can be solved in
-polynomial time, Chen et al. \cite{chen} prove that  $OPT(V,\mathcal{N}\setminus\{i^*\},B)$ can play the role of this quantity: that is, allocating to $i^*$ when
-$V(i^*) \geq C\cdot OPT(V,\mathcal{N}\setminus\{i^*\},B)$ (for some constant $C$)
+polynomial time, Chen et al. \cite{chen} prove that  $OPT_{-i^*}$, the optimal solution to \eqref{eq:non-strategic} when $i^*$ is excluded from set $\mathcal{N}$, can play the role of this quantity: that is, allocating to $i^*$ when
+$V(i^*) \geq C\cdot OPT_{-i^*}$ (for some constant $C$)
 and to $S_G$ otherwise yields a 8.34-approximation mechanism. However, this is not a poly-time mechanism when the underlying problem is NP hard (unless P=NP), as is the case for \EDP.
 
 \fussy
@@ -65,26 +65,26 @@ For  NP-hard
 problems, Chen et al.~\cite{chen} %for \textsc{Knapsack} and Singer
 %\cite{singer-influence} for \textsc{Coverage} instead 
 propose comparing
-$V(i^*)$ to %$OPT(R_{\mathcal{N}\setminus\{i\}}, B)$, where $R_\mathcal{N}$ denotes 
+$V(i^*)$ to %$OPT(R_{\mathcal{N}\setminus\{i\}}, B)$, where $R$ denotes 
 the optimal value of a \emph{fractional relaxation} of the function $V$ over the set
-$\mathcal{N}$. A fuction $R_\mathcal{N}:[0, 1]^{n}\to\reals_+$ defined on the hypercube  $[0, 1]^{n}$ is a fractional relaxation of $V$
-over the set $\mathcal{N}$ if %(a) $R_\mathcal{N}$ is a function defined on the hypercube $[0, 1]^{n}$ and (b) 
-  $R_\mathcal{N}(\mathbf{1}_S) = V(S)$ for all $S\subseteq\mathcal{N}$, where
+$\mathcal{N}$. A fuction $R:[0, 1]^{n}\to\reals_+$ defined on the hypercube  $[0, 1]^{n}$ is a fractional relaxation of $V$
+over the set $\mathcal{N}$ if %(a) $R$ is a function defined on the hypercube $[0, 1]^{n}$ and (b) 
+  $R(\mathbf{1}_S) = V(S)$ for all $S\subseteq\mathcal{N}$, where
 $\mathbf{1}_S$ denotes the indicator vector of $S$. The optimization program
 \eqref{eq:non-strategic} extends naturally to such relaxations:
 \begin{align}
-    OPT(R_\mathcal{N}, B) = \argmax_{\lambda\in[0, 1]^{n}}
-    \left\{R_\mathcal{N}(\lambda) \mid \sum_{i=1}^{n} \lambda_i c_i
+    OPT' = \argmax_{\lambda\in[0, 1]^{n}}
+    \left\{R(\lambda) \mid \sum_{i=1}^{n} \lambda_i c_i
     \leq B\right\}\label{relax}
 \end{align}
-To attain truthful constant approximation mechanism for \textsc{Knapsack},  Chen \emph{et al.}~\cite{chen}  substitute $OPT(R_\mathcal{N},B)$ for $OPT(V,\mathcal{N},B)$ in the above program, where $R_{\mathcal{N}}$ are relaxations of the \textsc{Knapsack} objectives.
+To attain truthful constant approximation mechanism for \textsc{Knapsack},  Chen \emph{et al.}~\cite{chen}  substitute $OPT'_{-i^*}$ for $OPT_{-i^*}$ in the above program, where $R$ is a relaxation of the \textsc{Knapsack} objective.
 Similarly, Singer~\cite{singer-influence} follows the same approach to obtain a mechanism for \textsc{Coverage}.
 
 A relaxation which is commonly used, due to its well-behaved properties in the
 context of maximizing submodular functions, is the \emph{multi-linear}
 extension:
 \begin{equation}\label{eq:multilinear}
-    F_\mathcal{N}(\lambda) 
+    F(\lambda) 
     = \mathbb{E}_{S\sim P_\mathcal{N}^\lambda}\left[V(S)\right]
     = \sum_{S\subseteq\mathcal{N}} P_\mathcal{N}^\lambda(S) V(S)
 \end{equation}
@@ -106,14 +106,14 @@ extension using the \emph{pipage rounding} method of Ageev and Sviridenko
 Here, following these ideas, we introduce a relaxation specifically tailored to the value
 function of \EDP. The multi-linear extension of \eqref{modified} can be written:
 \begin{displaymath}
-    F_\mathcal{N}(\lambda) = \mathbb{E}_{S\sim
+    F(\lambda) = \mathbb{E}_{S\sim
     P_\mathcal{N}^\lambda}\Big[\log\det \big(I_d + \sum_{i\in S} x_i\T{x_i}\big) \Big].
 \end{displaymath}
 As in the case of \textsc{Coverage}, \eqref{relax} is not a convex optimization problem, and is not easy to solve directly. %As in ~\cite{singer-influence},
-We thus introduce an additional relaxation $L_{\mathcal{N}}$. Our relaxation follows naturally by swapping the expectation and the $\log\det$
+We thus introduce an additional relaxation $L$. Our relaxation follows naturally by swapping the expectation and the $\log\det$
 in the above formula:
 \begin{align}\label{eq:concave}
-    L_\mathcal{N}(\lambda) & \defeq \log\det\Big(\mathbb{E}_{S\sim
+    L(\lambda) & \defeq \log\det\Big(\mathbb{E}_{S\sim
     P_\mathcal{N}^\lambda}\big[\log\det (I_d + \sum_{i\in S} x_i\T{x_i})\big]\Big)\notag \\
     &= \log\det\left(I_d + \sum_{i\in\mathcal{N}}
     \lambda_i x_i\T{x_i}\right)
@@ -121,10 +121,10 @@ in the above formula:
 This function is well-known to be concave and even self-concordant (see \emph{e.g.},
 \cite{boyd2004convex}). In this case, the analysis of Newton's method for
 self-concordant functions in \cite{boyd2004convex}, shows that it is possible
-to find the maximum of $L_\mathcal{N}$ to any precision $\varepsilon$ in
+to find the maximum of $L$ to any precision $\varepsilon$ in
 a number of iterations $O(\log\log\varepsilon^{-1})$. The main challenge will
-be to prove that $OPT(L_\mathcal{N}, B)$ is close to $V(S_G)$. To do so, our
-main technical contribution is to prove that $L_\mathcal{N}$ has a bounded
+be to prove that $OPT'$, for the relaxation $R=L$, is close to $V(S_G)$. To do so, our
+main technical contribution is to prove that $L$ has a bounded
 approximation ratio to the value function $V$ (Lemma~\ref{lemma:relaxation}).
 
 \begin{algorithm}[!t]
@@ -132,10 +132,10 @@ approximation ratio to the value function $V$ (Lemma~\ref{lemma:relaxation}).
     \begin{algorithmic}[1]
     \State $\mathcal{N} \gets \mathcal{N}\setminus\{i\in\mathcal{N} : c_i > B\}$
     \State $i^* \gets \argmax_{j\in\mathcal{N}}V(j)$
-    \State $\xi \gets \argmax_{\lambda\in[0,1]^{n}} \{L_{\mathcal{N}\setminus\{i^*\}}(\lambda)
-                                    \mid \sum_{i \in \mathcal{N}\setminus\{i^*\}}c_i\lambda_i\leq B\}$, with precision $\epsilon>0$
+    \State $\xi \gets \argmax_{\lambda\in[0,1]^{n}} \{L(\lambda)
+                                    \mid \lambda_{i^*}=0,\sum_{i \in \mathcal{N}\setminus\{i^*\}}c_i\lambda_i\leq B\}$
         \Statex
-        \If{$L_{\mathcal{N}\setminus\{i^*\}}(\xi) < C \cdot V(i^*)$} \label{c}
+        \If{$L(\xi) < C \cdot V(i^*)$} \label{c}
             \State \textbf{return} $\{i^*\}$ 
         \Else
             \State $i \gets \argmax_{1\leq j\leq n}\frac{V(j)}{c_j}$
@@ -169,7 +169,7 @@ We can now state our main result:
     has  complexity $O\left(\text{poly}(n, d,
     \log\log \varepsilon^{-1})\right)$ and returns a set $S^*$ such that:
     \begin{align*}
-        OPT(V,\mathcal{N}, B)
+        OPT
         & \leq \frac{14e-3 + \sqrt{160e^2-48e + 9}}{2(e-1)} V(S^*)+
         \varepsilon\\
         & \simeq 19.68V(S^*) + \varepsilon
@@ -192,9 +192,9 @@ Suppose, for contradiction, that such a mechanism exists. Consider two experimen
 We now present the proof of Theorem~\ref{thm:main}. Truthfulness and individual
 rationality follows from monotonicity and threshold payments.  Monotonicity and
 budget feasibility follows from the analysis of Chen \emph{et al.} \cite{chen};
-we briefly restate the proofs for the sake of completeness.
+we briefly restate the proofs below for the sake of completeness.
  Our proof of the approximation ratio and uses a bound on our concave relaxation
-$L_\mathcal{N}$ (Lemma~\ref{lemma:relaxation}). This is our main technical
+$L$ (Lemma~\ref{lemma:relaxation}). This is our main technical
 contribution; the proof of this lemma can be found in Section~\ref{sec:relaxation}.
 \begin{lemma}\label{lemma:monotone}
 The mechanism is monotone.
@@ -203,7 +203,7 @@ The mechanism is monotone.
     Consider an agent $i$ with cost $c_i$ that is
     selected by the mechanism, and suppose that she reports
     a cost $c_i'\leq c_i$ while all  other costs stay the same.
-    Suppose that when $i$ reports $c_i$, $L_{\mathcal{N}\setminus\{i^*\}}(\xi) \geq C V(i^*)$; then, as $s_i(c_i,c_{-i})=1$, $i\in S_G$.
+    Suppose that when $i$ reports $c_i$, $L(\xi) \geq C V(i^*)$; then, as $s_i(c_i,c_{-i})=1$, $i\in S_G$.
      By reporting a cost $c_i'\leq c_i$, $i$ may be selected at an earlier iteration of the greedy algorithm.
     %using the submodularity of $V$, we see that $i$ will satisfy the greedy
     %selection rule:
@@ -220,17 +220,17 @@ The mechanism is monotone.
         \frac{B}{2}\frac{V(S_i\cup\{i\})-V(S_i)}{V(S_i\cup\{i\})}
          \leq \frac{B}{2}\frac{V(S_i'\cup\{i\})-V(S_i')}{V(S_i'\cup\{i\})}
     \end{align*}
-    by the monotonicity and submodularity  of $V$. Hence  $i\in S_G'$. As $L_{\mathcal{N}\setminus \{i^*\}}(\xi)$ is the optimal value of \eqref{relax} under relaxation $L_{\mathcal{N}}$, reducing the costs can only increase this value, so under $c'_i\leq c_i$ the greedy set is still allocated and $s_i(c_i',c_{-i}) =1$.
-    Suppose now that when $i$ reports $c_i$, $L_{\mathcal{N}\setminus \{i^*\}}(\xi) < C V(i^*)$. Then $s_i(c_i,c_{-i})=1$ iff $i = i^*$. 
+    by the monotonicity and submodularity  of $V$. Hence  $i\in S_G'$. As $L(\xi)$,  is the optimal value of \eqref{relax} under relaxation $L$ when $i^*$ is excluded from $\mathcal{N}$, reducing the costs can only increase this value, so under $c'_i\leq c_i$ the greedy set is still allocated and $s_i(c_i',c_{-i}) =1$.
+    Suppose now that when $i$ reports $c_i$, $L(\xi) < C V(i^*)$. Then $s_i(c_i,c_{-i})=1$ iff $i = i^*$. 
     Reporting $c_{i^*}'\leq c_{i^*}$ does not change $V(i^*)$ nor
-    $L_{\mathcal{N}\setminus \{i^*\}}(xi) \leq C V(i^*)$; thus $s_{i^*}(c_{i^*}',c_{-i^*})=1$. 
+    $L(\xi) \leq C V(i^*)$; thus $s_{i^*}(c_{i^*}',c_{-i^*})=1$. 
 \end{proof}
 \begin{lemma}\label{lemma:budget-feasibility}
 The mechanism is budget feasible.
 \end{lemma}
 \begin{proof}
-Suppose that $L_{\mathcal{N}\setminus\{i^*\}}(\xi) < C V(i^*)$. Then the mechanism selects $i^*$; as the bid of $i^*$ does not affect the above condition, the threshold payment of $i^*$ is $B$ and the mechanism is budget feasible.
-Suppose thus that $L_{\mathcal{N}\setminus\{i^*\}}(\xi) \geq C V(i^*)$. 
+Suppose that $L(\xi) < C V(i^*)$. Then the mechanism selects $i^*$; as the bid of $i^*$ does not affect the above condition, the threshold payment of $i^*$ is $B$ and the mechanism is budget feasible.
+Suppose thus that $L(\xi) \geq C V(i^*)$. 
 Denote by $S_G$ the set selected by the greedy algorithm, and for $i\in S_G$,  denote by
 $S_i$ the subset of the solution set that was selected by the greedy algorithm just prior to the addition of $i$---both sets determined for the present cost vector $c$. Chen \emph{et al.}~\cite{chen} show that, for any submodular function $V$, and for all $i\in S_G$:
 %the reported cost of an agent selected by the greedy heuristic, and holds for
@@ -274,49 +274,49 @@ Hence, the total payment is bounded by the telescopic sum:
 \end{proof}
 
 Finally, we prove the approximation ratio of the mechanism. We use the
-following lemma, which establishes the optimal fractional relaxation
-$L_\mathcal{N}$ is not too far from $OPT(V,\mathcal{N},B)$.  
+following lemma, which establishes that $OPT'$, the optimal value \eqref{relax} of the fractional relaxation $L$ under the budget constraints
+ is not too far from $OPT$.  
 \begin{lemma}\label{lemma:relaxation}
     %\begin{displaymath}
-     $   OPT(L_\mathcal{N}, B) \leq 4 OPT(V,\mathcal{N},B)
+     $   OPT' \leq 4 OPT
         + 2\max_{i\in\mathcal{N}}V(i)$
     %\end{displaymath}
 \end{lemma}
-Note that the lower bound $OPT(L_\mathcal{N}, B) \geq  OPT(V,\mathcal{N},B)
-$ also holds trivially, as $L_{\mathcal{N}}$ is a fractional relaxation of $V$ over $\mathcal{N}$.
+Note that the lower bound $OPT(L, B) \geq  OPT
+$ also holds trivially, as $L$ is a fractional relaxation of $V$ over $\mathcal{N}$.
 The proof of Lemma~\ref{lemma:relaxation} is our main technical contribution, and can be found in Section \ref{sec:relaxation}.
 Using this lemma, we can complete the proof of Theorem~\ref{thm:main} by showing that, %\begin{lemma}\label{lemma:approx}
     %C_{\textrm{max}} = \max\left(1+C,\frac{3e}{e-1}\left( 1 + \frac{8e}{C
     %(e-1) -10e  +2}\right)\right)
-    for any $\varepsilon > 0$, if $L(x^*)$ has been computed to a precision
+    for any $\varepsilon > 0$, if $OPT_{-i}'$, the optimal value of $L$ when $i^*$ is excluded from $\mathcal{N}$, has been computed to a precision
     $\varepsilon$, then the set $S^*$ allocated by the mechanism is such that:
     \begin{align}
-        OPT(V,\mathcal{N}, B)
+        OPT
          \leq \frac{14e\!-\!3 + \sqrt{160e^2\!-\!48e\!+\!9}}{2(e\!-\!1)} V(S^*)\!+\! \varepsilon \label{approxbound}
     \end{align}
 %\end{lemma}
-    To see this, let $x^*\in [0,1]^{n-1}$ be the true maximizer of $L_{\mathcal{N}\setminus\{i^*\}}$ subject to the budget constraints. Assume that on line 3 of algorithm~\ref{mechanism}, a quantity
-    $\tilde{L}$ such that $\tilde{L}-\varepsilon\leq L_{\mathcal{N}\setminus {i^*}}(x^*) \leq
-    \tilde{L}+\varepsilon$ has been computed (lemma~\ref{lemma:complexity}
+    To see this, let $OPT_{-i^*}'$ be the true maximum value of $L$ subject to $\lambda_{i^*}=0$, $\sum_{i\in \mathcal{N}\setminus{i^*}}c_i\leq B$. Assume that on line 3 of algorithm~\ref{mechanism}, a quantity
+    $\tilde{L}$ such that $\tilde{L}-\varepsilon\leq OPT_{-i^*}' \leq
+    \tilde{L}+\varepsilon$ has been computed (Lemma~\ref{lemma:complexity}
     states that this  is computed in time within our complexity guarantee).
     If the condition on line 3 of the algorithm holds, then:
     \begin{displaymath}
-        V(i^*) \geq \frac{1}{C}L_{N\setminus \{i\}}(x^*)-\frac{\varepsilon}{C} \geq
-        \frac{1}{C}OPT(V,\mathcal{N}\setminus\{i^*\}, B) -\frac{\varepsilon}{C} 
+        V(i^*) \geq \frac{1}{C}OPT_{-i^*}'-\frac{\varepsilon}{C} \geq
+        \frac{1}{C}OPT_{-i^*} -\frac{\varepsilon}{C} 
     \end{displaymath}
-    as $L$ is a fractional relaxation of $V$ on $\mathcal{N}\setminus \{i^*\}$. Also,
+    as $L$ is a fractional relaxation of $V$.  Also,
     \begin{displaymath}
-        OPT(V,\mathcal{N},B) \leq OPT(V,\mathcal{N}\setminus\{i^*\}, B) + V(i^*)
+        OPT \leq OPT_{-i^*} + V(i^*)
     \end{displaymath}
     Hence:
     \begin{equation}\label{eq:bound1}
-        OPT(V,\mathcal{N}, B)\leq (1+C)V(i^*) + \varepsilon
+        OPT\leq (1+C)V(i^*) + \varepsilon
     \end{equation}
-    Note that $OPT(L_{\mathcal{N}\setminus\{i^*\}},B)\leq OPT(L_{\mathcal{N}},B)$. If the condition  does not hold, 
+    Note that $OPT_{-i^*}'\leq OPT'$. If the condition  does not hold, 
     from Lemmas \ref{lemma:relaxation} and \ref{lemma:greedy-bound}:
     \begin{align*}
-        V(i^*) & \stackrel{}\leq \frac{1}{C}L_{\mathcal{N}}(x^*) + \frac{\varepsilon}{C}\leq \frac{1}{C}
-        \big(4 OPT(V,\mathcal{N}, B) + 2 V(i^*)\big) + \frac{\varepsilon}{C}\\
+        V(i^*) & \stackrel{}\leq \frac{1}{C}OPT_{-i^*}'+ \frac{\varepsilon}{C}\leq \frac{1}{C}
+        \big(4 OPT + 2 V(i^*)\big) + \frac{\varepsilon}{C}\\
         & \leq \frac{1}{C}\left(\frac{4e}{e-1}\big(3 V(S_G)
         + 2 V(i^*)\big)
         + 2 V(i^*)\right) + \frac{\varepsilon}{C}
@@ -354,18 +354,18 @@ This solution is:
     gives the approximation ratio in \eqref{approxbound}, and concludes the proof of Theorem~\ref{thm:main}.\hspace*{\stretch{1}}\qed
 %\end{proof}
 \subsection{Proof of Lemma~\ref{lemma:relaxation}}\label{sec:relaxation}
-%Recall that, since $L_\mathcal{N}$ is a fractional relaxation of  $V$, 
+%Recall that, since $L$ is a fractional relaxation of  $V$, 
 %\begin{displaymath}
-%    OPT(V,\mathcal{N},B) \leq OPT(L_\mathcal{N}, B).
+%    OPT \leq OPT(L, B).
 %\end{displaymath}
 %However, for the purpose of proving theorem~\ref{thm:main}, we need to bound
-%$L_\mathcal{N}$ from above (up to a multiplicative factor) by $V$. 
+%$L$ from above (up to a multiplicative factor) by $V$. 
 To prove Lemma~\ref{lemma:relaxation}, we use the
 \emph{pipage rounding} method of Ageev and Sviridenko~\cite{pipage}, where
-$L_\mathcal{N}$ is first bounded from above by the multi-linear relaxation $F_\mathcal{N}$, which is itself
+$L$ is first bounded from above by the multi-linear relaxation $F$, which is itself
 subsequently bounded by $V$. While the latter part is general and can be applied
 to the multi-linear extension of any submodular function, the former part is
-specific to our choice for the function $L_\mathcal{N}$.  %and is our main technical contribution (lemma~\ref{lemma:relaxation-ratio}).
+specific to our choice for the function $L$.  %and is our main technical contribution (lemma~\ref{lemma:relaxation-ratio}).
 
 We first prove a variant of the $\varepsilon$-convexity of the multi-linear
 extension \eqref{eq:multilinear} introduced by Ageev and Sviridenko
@@ -375,7 +375,7 @@ referred to in the literature as \emph{cross-convexity} (see, \emph{e.g.},
 \cite{dughmi}). Formally, %this property states that 
 if we define:
 \begin{displaymath}
-    \tilde{F}_\lambda(\varepsilon) \defeq F_\mathcal{N}\big(\lambda + \varepsilon(e_i
+    \tilde{F}_\lambda(\varepsilon) \defeq F\big(\lambda + \varepsilon(e_i
     - e_j)\big)
 \end{displaymath}
 where $e_i$ and $e_j$ are two vectors of the standard basis of
@@ -388,7 +388,7 @@ or the $j$-th component of $\lambda$ becomes integral.
 
 The lemma below proves that we can similarly trade a fractional component for
 an other until one of them becomes integral \emph{while maintaining the
-feasibility of the point at which $F_\mathcal{N}$ is evaluated}. Here, by feasibility of
+feasibility of the point at which $F$ is evaluated}. Here, by feasibility of
 a point $\lambda$, we mean that it satisfies the budget constraint $\sum_{i=1}^n \lambda_i c_i \leq B$.
 \begin{lemma}[Rounding]\label{lemma:rounding}
     For any feasible $\lambda\in[0,1]^{n}$, there exists a feasible
@@ -404,7 +404,7 @@ a point $\lambda$, we mean that it satisfies the budget constraint $\sum_{i=1}^n
     two fractional components, returns some feasible $\lambda'$ with one less fractional
     component such that:
     \begin{displaymath}
-        F_\mathcal{N}(\lambda) \leq F_\mathcal{N}(\lambda')
+        F(\lambda) \leq F(\lambda')
     \end{displaymath}
     Applying this procedure recursively yields the lemma's result.
     Let us consider such a feasible $\lambda$. Let $i$ and $j$ be two
@@ -448,49 +448,19 @@ a point $\lambda$, we mean that it satisfies the budget constraint $\sum_{i=1}^n
 For all $\lambda\in[0,1]^{n},$
     %\begin{displaymath}
      $           \frac{1}{2}
-        \,L_\mathcal{N}(\lambda)\leq
-        F_\mathcal{N}(\lambda)\leq L_{\mathcal{N}}(\lambda)$.
+        \,L(\lambda)\leq
+        F(\lambda)\leq L(\lambda)$.
     %\end{displaymath}
 \end{lemma}
 \begin{proof}
+    The bound $F_{\mathcal{N}}(\lambda)\leq L_{\mathcal{N}(\lambda)}$ follows by the concavity of the $\log\det$ function.
     We will prove that $\frac{1}{2}$ is a lower bound of the ratio $\partial_i
-    F_\mathcal{N}(\lambda)/\partial_i L_\mathcal{N}(\lambda)$, where
-    $\partial_i\, \cdot$ denotes the partial derivative of a function with respect to the
-    $i$-th variable. This will be enough to conclude, by observing the following cases.
-    First, if the minimum of the ratio
-    $F_\mathcal{N}(\lambda)/L_\mathcal{N}(\lambda)$ is attained at a point interior to the hypercube, then it is
-    a critical point; such a critical point is
-    characterized by:
-    \begin{equation}\label{eq:lhopital}
-        \frac{F_\mathcal{N}(\lambda)}{L_\mathcal{N}(\lambda)}
-        = \frac{\partial_i F_\mathcal{N}(\lambda)}{\partial_i
-        L_\mathcal{N}(\lambda)} \geq \frac{1}{2}
-    \end{equation}
-    Second, if the minimum is attained as 
-    $\lambda$ converges to zero in, \emph{e.g.}, the $l_2$ norm, by the Taylor approximation, one can write:
-    \begin{displaymath}
-        \frac{F_\mathcal{N}(\lambda)}{L_\mathcal{N}(\lambda)}
-        \sim_{\lambda\rightarrow 0}
-        \frac{\sum_{i\in \mathcal{N}}\lambda_i\partial_i F_\mathcal{N}(0)}
-        {\sum_{i\in\mathcal{N}}\lambda_i\partial_i L_\mathcal{N}(0)}
-        \geq \frac{1}{2},
-    \end{displaymath}
-    \emph{i.e.}, the ratio $\frac{F_\mathcal{N}(\lambda)}{L_\mathcal{N}(\lambda)}$ is necessarily bounded from below by 1/2 for small enough $\lambda$. 
-       Finally, if the minimum is attained on a face of the hypercube $[0,1]^n$ (a face is
-    defined as a subset of the hypercube where one of the variable is fixed to
-    0 or 1), without loss of generality, we can assume that the minimum is
-    attained on the face where the $n$-th variable has been fixed
-    to 0 or 1. Then, either the minimum is attained at a point interior to the
-    face or on a boundary of the face. In the first case, relation
-    \eqref{eq:lhopital} still characterizes the minimum for $i< n$.
-    In the second case, by repeating the argument again by induction, we see
-    that all is left to do is to show that the bound holds for the vertices of
-    the cube (the faces of dimension 1).  The vertices are exactly the binary
-    points, for which we know that both relaxations are equal to the value
-    function $V$. Hence, the ratio is equal to 1 on the vertices.
- 
-    Let us start by computing the derivatives of $F_\mathcal{N}$ and
-    $L_\mathcal{N}$ with respect to the $i$-th component.
+    F(\lambda)/\partial_i L(\lambda)$, where
+    $\partial_i\, \cdot$ denotes the partial derivative with respect to the
+    $i$-th variable.  
+
+   Let us start by computing the derivatives of $F$ and
+    $L$ with respect to the $i$-th component.
     For $F$, it suffices to look at the derivative of
     $P_\mathcal{N}^\lambda(S)$:
     \begin{displaymath}
@@ -503,7 +473,7 @@ For all $\lambda\in[0,1]^{n},$
     \end{displaymath}
     Hence:
     \begin{multline*}
-        \partial_i F_\mathcal{N} =
+        \partial_i F =
         \sum_{\substack{S\subseteq\mathcal{N}\\ i\in S}}
         P_{\mathcal{N}\setminus\{i\}}^\lambda(S\setminus\{i\})V(S)
         - \sum_{\substack{S\subseteq\mathcal{N}\\ i\in \mathcal{N}\setminus S}}
@@ -512,7 +482,7 @@ For all $\lambda\in[0,1]^{n},$
     Now, using that every $S$ such that $i\in S$ can be uniquely written as
     $S'\cup\{i\}$, we can write:
     \begin{multline*}
-        \partial_i F_\mathcal{N} =
+        \partial_i F =
         \sum_{\substack{S\subseteq\mathcal{N}\\ i\in\mathcal{N}\setminus S}}
         P_{\mathcal{N}\setminus\{i\}}^\lambda(S)V(S\cup\{i\})
         - \sum_{\substack{S\subseteq\mathcal{N}\\ i\in \mathcal{N}\setminus S}}
@@ -523,15 +493,15 @@ For all $\lambda\in[0,1]^{n},$
   $  V(S\cup\{i\}) - V(S) = \frac{1}{2}\log\left(1 +  \T{x_i} A(S)^{-1}x_i\right)$.
 Using this,
     \begin{displaymath}
-        \partial_i F_\mathcal{N}(\lambda) = \frac{1}{2}
+        \partial_i F(\lambda) = \frac{1}{2}
         \sum_{\substack{S\subseteq\mathcal{N}\\ i\in\mathcal{N}\setminus S}}
         P_{\mathcal{N}\setminus\{i\}}^\lambda(S)
         \log\Big(1 + \T{x_i}A(S)^{-1}x_i\Big)
     \end{displaymath}
-    where $A(S) =I_d+ \T{X_S}X_S$. The computation of the derivative of $L_\mathcal{N}$ uses standard matrix
+    where $A(S) =I_d+ \T{X_S}X_S$. The computation of the derivative of $L$ uses standard matrix
     calculus and gives:
     \begin{displaymath}
-        \partial_i L_\mathcal{N}(\lambda)
+        \partial_i L(\lambda)
         =\frac{1}{2} \T{x_i}\tilde{A}(\lambda)^{-1}x_i
     \end{displaymath}
     where $\tilde{A}(\lambda) = I_d+\sum_{i\in \mathcal{N}}\lambda_ix_i\T{x_i}$. Using the following inequalities (where $A\geq B$ implies $A-B$ is positive semi-definite):
@@ -545,7 +515,7 @@ Using this,
     \end{gather*}
     we get:
     \begin{align*}
-        \partial_i F_\mathcal{N}(\lambda) 
+        \partial_i F(\lambda) 
       %  & = \sum_{\substack{S\subseteq\mathcal{N}\\ i\in\mathcal{N}\setminus S}} P_\mathcal{N}^\lambda(S) \log\Big(1 + \T{x_i}A(S)^{-1}x_i\Big)\\
         & \geq \frac{1}{4}
         \sum_{\substack{S\subseteq\mathcal{N}\\ i\in\mathcal{N}\setminus S}}
@@ -570,43 +540,75 @@ Using this,
     \end{displaymath}
     Hence:
     \begin{displaymath}
-        \partial_i F_\mathcal{N}(\lambda) \geq
+        \partial_i F(\lambda) \geq
         \frac{1}{4}
         \T{x_i}\bigg(\sum_{S\subseteq\mathcal{N}}P_\mathcal{N}^\lambda(S)A(S)^{-1}\bigg)x_i
     \end{displaymath}
     Finally, using that the inverse is a matrix convex function over symmetric
     positive definite matrices:
     \begin{align*}
-        \partial_i F_\mathcal{N}(\lambda) &\geq
+        \partial_i F(\lambda) &\geq
         \frac{1}{4}
         \T{x_i}\bigg(\sum_{S\subseteq\mathcal{N}}P_\mathcal{N}^\lambda(S)A(S)\bigg)^{-1}x_i
          \geq \frac{1}{2}
-        \partial_i L_\mathcal{N}(\lambda)\qed
-    \end{align*}
+        \partial_i L(\lambda)    \end{align*}
+
+This will be enough to conclude, by observing the following cases.
+    First, if the minimum of the ratio
+    $F(\lambda)/L(\lambda)$ is attained at a point interior to the hypercube, then it is
+    a critical point; such a critical point is
+    characterized by:
+    \begin{equation}\label{eq:lhopital}
+        \frac{F(\lambda)}{L(\lambda)}
+        = \frac{\partial_i F(\lambda)}{\partial_i
+        L(\lambda)} \geq \frac{1}{2}
+    \end{equation}
+    Second, if the minimum is attained as 
+    $\lambda$ converges to zero in, \emph{e.g.}, the $l_2$ norm, by the Taylor approximation, one can write:
+    \begin{displaymath}
+        \frac{F(\lambda)}{L(\lambda)}
+        \sim_{\lambda\rightarrow 0}
+        \frac{\sum_{i\in \mathcal{N}}\lambda_i\partial_i F(0)}
+        {\sum_{i\in\mathcal{N}}\lambda_i\partial_i L(0)}
+        \geq \frac{1}{2},
+    \end{displaymath}
+    \emph{i.e.}, the ratio $\frac{F(\lambda)}{L(\lambda)}$ is necessarily bounded from below by 1/2 for small enough $\lambda$. 
+       Finally, if the minimum is attained on a face of the hypercube $[0,1]^n$ (a face is
+    defined as a subset of the hypercube where one of the variable is fixed to
+    0 or 1), without loss of generality, we can assume that the minimum is
+    attained on the face where the $n$-th variable has been fixed
+    to 0 or 1. Then, either the minimum is attained at a point interior to the
+    face or on a boundary of the face. In the first case, relation
+    \eqref{eq:lhopital} still characterizes the minimum for $i< n$.
+    In the second case, by repeating the argument again by induction, we see
+    that all is left to do is to show that the bound holds for the vertices of
+    the cube (the faces of dimension 1).  The vertices are exactly the binary
+    points, for which we know that both relaxations are equal to the value
+    function $V$. Hence, the ratio is equal to 1 on the vertices.
 \end{proof}
- To prove Lemma~\ref{lemma:relaxation},   let us consider a feasible point $\lambda^*\in[0,1]^{n}$ such that $L_\mathcal{N}(\lambda^*)
-    = OPT(L_\mathcal{N}, B)$. By applying Lemma~\ref{lemma:relaxation-ratio}
+ To prove Lemma~\ref{lemma:relaxation},   let us consider a feasible point $\lambda^*\in[0,1]^{n}$ such that $L(\lambda^*)
+    = OPT(L, B)$. By applying Lemma~\ref{lemma:relaxation-ratio}
     and Lemma~\ref{lemma:rounding} we get a feasible point $\bar{\lambda}$ with at most
     one fractional component such that:
     \begin{equation}\label{eq:e1}
-        L_\mathcal{N}(\lambda^*) \leq 2
-        F_\mathcal{N}(\bar{\lambda})
+        L(\lambda^*) \leq 2
+        F(\bar{\lambda})
     \end{equation}
     Let $\lambda_i$ denote the fractional component of $\bar{\lambda}$ and $S$
     denote the set whose indicator vector is $\bar{\lambda} - \lambda_i e_i$.
-    Using the fact that $F_\mathcal{N}$ is linear with respect to the $i$-th
+    Using the fact that $F$ is linear with respect to the $i$-th
     component and is a relaxation of the value function, we get:
     \begin{displaymath}
-        F_\mathcal{N}(\bar{\lambda}) = V(S) +\lambda_i V(S\cup\{i\})
+        F(\bar{\lambda}) = V(S) +\lambda_i V(S\cup\{i\})
     \end{displaymath}
     Using the submodularity of $V$:
     \begin{displaymath}
-        F_\mathcal{N}(\bar{\lambda}) \leq 2 V(S) + V(i)
+        F(\bar{\lambda}) \leq 2 V(S) + V(i)
     \end{displaymath}
     Note that since $\bar{\lambda}$ is feasible, $S$ is also feasible and
-    $V(S)\leq OPT(V,\mathcal{N}, B)$. Hence:
+    $V(S)\leq OPT$. Hence:
     \begin{equation}\label{eq:e2}
-        F_\mathcal{N}(\bar{\lambda}) \leq 2 OPT(V,\mathcal{N}, B)
+        F(\bar{\lambda}) \leq 2 OPT
         + \max_{i\in\mathcal{N}} V(i)
     \end{equation}
     Together, \eqref{eq:e1} and \eqref{eq:e2} imply the lemma. \hspace*{\stretch{1}}\qed
-- 
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