From f64ff3ca389ec33c240bf6d872033a4bc6e4d9c6 Mon Sep 17 00:00:00 2001
From: Thibaut Horel <thibaut.horel@gmail.com>
Date: Sat, 3 Nov 2012 14:42:42 +0100
Subject: Small fixes

---
 main.tex | 41 ++++++++++++-----------------------------
 1 file changed, 12 insertions(+), 29 deletions(-)

(limited to 'main.tex')

diff --git a/main.tex b/main.tex
index e2b8335..178ad1f 100644
--- a/main.tex
+++ b/main.tex
@@ -62,10 +62,10 @@ the allocation function.
 The way this issue has been addressed thus far \cite{chen, singer-influence},
 is to introduce a third quantity: if $V(i^*)$ is larger than this quantity, the
 mechanism allocates to $i^*$, otherwise it allocates to the greedy set $S_G$.
-To keep a bounded approximation ratio, this quantity must be provably close to
-$V(S_G)$ while maintaining the monotonicity of the allocation algorithm.
-Furthermore, it must be computable in polynomial time to keep an overall
-polynomial complexity for the allocation algorithm. If the underlying
+This quantity must be provably close to $V(S_G)$, to keep a bounded
+approximation ratio, while maintaining the monotonicity of the allocation
+algorithm. Furthermore, it must be computable in polynomial time to keep an
+overall polynomial complexity for the allocation algorithm. If the underlying
 non-strategic optimization program \eqref{eq:non-strategic} can be solved in
 polynomial time, Chen et al. \cite{chen} prove that allocating to $i^*$ when
 $V(i^*) \geq C\cdot OPT(V,\mathcal{N}\setminus\{i^*\}$ (for some constant $C$)
@@ -73,26 +73,25 @@ and to $S_G$ otherwise yields a 8.34 approximation mechanism. For specific
 problems, Chen et al. \cite{chen} for knapsack on one hand, Singer
 \cite{singer-influence} for maximum coverage on the other hand, instead compare
 $V(i^*)$ to $OPT(R_{\mathcal{N}\setminus\{i\}}, B)$, where $R_\mathcal{N}$
-denotes a fractional relaxation of the function $V$ over the set
-$\mathcal{N}$.
+denotes a fractional relaxation of the function $V$ over the set $\mathcal{N}$.
 
 We say that $R_\mathcal{N}$ is a fractional relaxation of $V$ over the set
 $\mathcal{N}$, if (a) $R_\mathcal{N}$ is a function defined on the hypercube
 $[0, 1]^{|\mathcal{N}|}$ and (b) for all $S\subseteq\mathcal{N}$, if
 $\mathbf{1}_S$ denotes the indicator vector of $S$ we have
 $R_\mathcal{N}(\mathbf{1}_S) = V(S)$. A relaxation which is commonly used, due
-to its nice properties in the context of maximizing submodular functions, is the
-\emph{multi-linear} extension:
+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}^V(\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}
-where $P_\mathcal{N}^\lambda(S)$ is the probability of choosing the set $S$ if
+$P_\mathcal{N}^\lambda(S)$ is the probability of choosing the set $S$ if
 we decide for each element in $\mathcal{N}$ to pick it with probability
 $\lambda_i$ or to reject it with probability $1-\lambda_i$:
 \begin{displaymath}
-    P_\mathcal{N}^\lambda = \prod_{i\in S} \lambda_i
+    P_\mathcal{N}^\lambda(S) = \prod_{i\in S} \lambda_i
     \prod_{i\in\mathcal{N}\setminus S} 1 - \lambda_i
 \end{displaymath}
 For knapsack, Chen et al. \cite{chen} directly use the multi-linear extension as
@@ -113,8 +112,8 @@ function. Note that in our case, the multi-linear extension can be written:
 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\mathbb{E}_{S\sim
-    P_\mathcal{N}^\lambda} A(S) \\
+    L_\mathcal{N}(\lambda) & \defeq \log\det\left(\mathbb{E}_{S\sim
+    P_\mathcal{N}^\lambda}[A(S)]\right)\notag \\
     &= \log\det\left(I_d + \sum_{i\in\mathcal{N}}
     \lambda_i x_i\T{x_i}\right)
 \end{align}
@@ -158,7 +157,7 @@ We can now state the main result of this section:
     \end{displaymath}
     we get an approximation ratio of:
     \begin{displaymath}
-        1 + C^* = \frac{14e-3 + \sqrt{160e^2-48e + 9}}{2(e-1)}\simeq 18.68
+        1 + C^* = \frac{14e-3 + \sqrt{160e^2-48e + 9}}{2(e-1)}\simeq 19.68
     \end{displaymath}
 \end{theorem}
 
@@ -304,22 +303,6 @@ Our main contribution is lemma~\ref{lemma:relaxation} which gives an
 approximation ratio for $L_\mathcal{N}$ and is key in the proof of
 theorem~\ref{thm:main}.
 
-\begin{lemma}\label{lemma:concave}
-    The \emph{concave relaxation} $L_\mathcal{N}$ is concave\footnote{Hence
-    this relaxation is well-named!
-\stratis{avoid humor}}.
-\end{lemma}
-
-\begin{proof}
-    This follows from the concavity of the $\log\det$ function over symmetric
-    positive semi-definite matrices. More precisely, if $A$ and $B$ are two
-    symmetric positive semi-definite matrices, then:
-    \begin{multline*}
-        \forall\alpha\in [0, 1],\; \log\det\big(\alpha A + (1-\alpha) B\big)\\
-        \geq \alpha\log\det A + (1-\alpha)\log\det B
-    \end{multline*}
-\end{proof}
-
 It has been observed already (see for example \cite{dughmi}) that the
 multi-linear extension presents the cross-convexity property: it is convex along
 any direction $e_i-e_j$ where $e_i$ and $e_j$ are two elements of the standard
-- 
cgit v1.2.3-70-g09d2

