From 385334421230a9a4f7422817cc2fae224fd561da Mon Sep 17 00:00:00 2001
From: Stratis Ioannidis <stratis@stratis-Latitude-E6320.(none)>
Date: Thu, 4 Jul 2013 18:53:24 -0700
Subject: wording, proof, algo

---
 approximation.tex | 13 +++++++------
 1 file changed, 7 insertions(+), 6 deletions(-)

(limited to 'approximation.tex')

diff --git a/approximation.tex b/approximation.tex
index de1fecd..1a4b44b 100644
--- a/approximation.tex
+++ b/approximation.tex
@@ -1,4 +1,5 @@
-Previous approaches towards designing truthful, budget feasible mechanisms for \textsc{Knapsack}~\cite{chen} and \textsc{Coverage}~\cite{singer-influence} build upon polynomial-time algorithms that compute an approximation of $OPT$, the optimal value in the full information case. Crucially, to be used in designing a truthful mechanism, such algorithms need also to be \emph{monotone}, in the sense that decreasing any cost $c_i$ leads to an increase  in the estimation of $OPT$. In the cases of \textsc{Knapsack} and~\textsc{Coverage}, as well as in the case of \EDP{},  monotonicity  precludes using traditional approximation algorithms.
+Previous approaches towards designing truthful, budget feasible mechanisms for \textsc{Knapsack}~\cite{chen} and \textsc{Coverage}~\cite{singer-influence} build upon polynomial-time algorithms that compute an approximation of $OPT$, the optimal value in the full information case. Crucially, to be used in designing a truthful mechanism, such algorithms need also to be \emph{monotone}, in the sense that decreasing any cost $c_i$ leads to an increase  in the estimation of $OPT$; %In the cases of \textsc{Knapsack} and~\textsc{Coverage}, as well as in the case of \EDP{}, 
+the monotonicity property  precludes using traditional approximation algorithms.
 
 In the fist part of this section, we address this issue by designing a convex relaxation of \EDP{}, and showing that its solution can be used to approximate $OPT$. The objective of this relaxation is concave and self-concordant \cite{boyd2004convex} and, as such, there exists an algorithm that solves this relaxed problem with arbitrary accuracy in polynomial time. Unfortunately, the output of this algorithm may not necessarily be monotone. Nevertheless, in the second part of this section, we show how a  solver of the relaxed problem can be used to construct a solver that is ``almost'' monotone: in Section~\ref{sec:main}, we show how this algorithm can be used to design a $\delta$-truthful mechanism for \EDP.
 
@@ -33,7 +34,7 @@ In the fist part of this section, we address this issue by designing a convex re
 A classical way of relaxing combinatorial optimization problems is 
 \emph{relaxing by expectation}, using the so-called \emph{multi-linear}
 extension of the objective function $V$ (see, \emph{e.g.}, \cite{calinescu2007maximizing,vondrak2008optimal,dughmi2011convex}).
-This is because this extension can yield approximation guarantees for a wide class of combinatorial problems through \emph{pipage rounding}, a technique proposed by \citeN{pipage}. Crucially for our purposes, such relaxations preserve monotonicity which, as we discussed, is necessary for our mechanism design.
+This is because this extension can yield approximation guarantees for a wide class of combinatorial problems through \emph{pipage rounding}, a technique proposed by \citeN{pipage}. Crucially for our purposes, such relaxations in general preserve monotonicity which, as discussed, is required in mechanism design.
 
 Formally, let $P_\mathcal{N}^\lambda$ be a probability distribution over $\mathcal{N}$ parametrized by $\lambda\in [0,1]^n$, where a set $S\subseteq \mathcal{N}$ sampled from  $P_\mathcal{N}^\lambda$ is constructed as follows:  each $i\in \mathcal{N}$ is selected to be in $S$ independently with probability $\lambda_i$, \emph{i.e.},
 %\begin{displaymath}
@@ -71,7 +72,7 @@ For all $\lambda\in[0,1]^{n},$
         \,L(\lambda)\leq
         F(\lambda)\leq L(\lambda)$.
 \end{lemma}
-The proof of this lemma can be found in Appendix~\ref{proofofrelaxation-ratio}. In short, we prove this by exploiting the concavity of the $\log\det$ function over the set of positive semi-definite matrices to bound the ratio of all partial derivatives of $F$ and $L$; to conclude the proof, we then show that the bound on the ratio of the derivatives also implies a bound on the ratio $F/L$.
+The proof of this lemma can be found in Appendix~\ref{proofofrelaxation-ratio}. In short,  exploiting the concavity of the $\log\det$ function over the set of positive semi-definite matrices, we first  bound the ratio of all partial derivatives of $F$ and $L$. We then show that the bound on the ratio of the derivatives also implies a bound on the ratio $F/L$.
 
 Armed with this result, we subsequently use  pipage rounding to show that any $\lambda$ that maximizes the multi-linear extension $F$ can be rounded to an ``almost'' integral solution. More specifically, given a set of costs $c\in \reals^n_+$, we say that a $\lambda\in [0,1]^n$ is feasible if it belongs to the set 
 \begin{align}\dom_c =\{\lambda \in [0,1]^n: \sum_{i\in \mathcal{N}} c_i\lambda_i\leq B\}.\label{fdom}\end{align} Then, the following lemma holds:
@@ -116,7 +117,7 @@ approximation $\hat{L}^*_c$ that is $\varepsilon$-accurate, \emph{i.e.}, it sati
 
  Clearly, the optimal value $L^*_c$ of \eqref{eq:primal} is monotone in $c$. Formally, for any two $c,c'\in \reals_+^n$ s.t.~$c\leq c'$ coordinate-wise, $\dom_{c'}\subseteq \dom_c$ and thus $L^*_c\geq L^*_{c'}$. Hence, the map $c\mapsto L^*_c$ is non-increasing. Unfortunately, the same is not true for the output $\hat{L}_c^*$ of the barrier method: there is no guarantee that the $\epsilon$-accurate approximation $\hat{L}^*_c$ exhibits any kind of monotonicity.
 
-Nevertheless, it is possible to use the barrier method to construct an approximation of $L^*_{c}$ that is ``almost'' monotone. More specifically, given $\delta>0$,  we will say that $f:\reals^n\to\reals$ is
+Nevertheless, we prove that it is possible to use the barrier method to construct an approximation of $L^*_{c}$ that is ``almost'' monotone. More specifically, given $\delta>0$,  we will say that $f:\reals^n\to\reals$ is
 \emph{$\delta$-decreasing}  if
 \begin{equation}\label{eq:dd}
   f(x) \geq  f(x+\mu e_i), \qquad
@@ -182,14 +183,14 @@ Note that $\dom_c=\dom_{c,0}.$ Consider the following perturbation of the concav
 \begin{algorithm}[t]
     \caption{}\label{alg:monotone}
     \begin{algorithmic}[1]
-	\Require{ $B\in \reals_+$,$c\in\reals^n_+$, $\delta\in (0,1]$, $\epsilon\in (0,1]$ }
+	\Require{ $B\in \reals_+$,$c\in[0,B]^n$, $\delta\in (0,1]$, $\epsilon\in (0,1]$ }
         \State $\alpha \gets \min\left(\varepsilon B^{-1}(\delta+n^2)^{-1},\frac{1}{nB}\right) $ 
         \State Use the barrier method to solve \eqref{eq:perturbed-primal} with accuracy $\varepsilon'=\frac{1}{2^{n+1}}\alpha\delta b$; denote the output by $\hat{L}^*_{c,\alpha}$
 	\State \textbf{return} $\hat{L}^*_{c,\alpha}$
     \end{algorithmic}
 \end{algorithm}
 
-Our construction of a $\delta$-decreasing, $\varepsilon$-accurate approximator of $\hat{L}_c^*$ proceeds as follows: first, it computes an appropriately selected lower bound $\alpha$; using this bound, it proceeds by solving the perturbed problem \eqref{eq:perturbed-primal} using the barrier method, also at an appropriately selected accuracy $\varepsilon'$, obtaining thusly a $\varepsilon'$-accurate approximation of $L^*_{c,\alpha}\defeq \max_{\lambda\in \dom_{c,\alpha}} L(\lambda)$ . The corresponding output is returned as an approximation of $L^*_c$. A summary of algorithm and the specific choices of $\alpha$ and $\varepsilon'$ are given in Algorithm~\ref{alg:monotone}. The following proposition, which is proved in Appendix~\ref{proofofpropmonotonicity}, establishes that this algorithm has the desirable properties:
+Our construction of a $\delta$-decreasing, $\varepsilon$-accurate approximator of $\hat{L}_c^*$ proceeds as follows: first, it computes an appropriately selected lower bound $\alpha$; using this bound, it proceeds by solving the perturbed problem \eqref{eq:perturbed-primal} using the barrier method, also at an appropriately selected accuracy $\varepsilon'$, obtaining thusly a $\varepsilon'$-accurate approximation of $L^*_{c,\alpha}\defeq \max_{\lambda\in \dom_{c,\alpha}} L(\lambda)$ . The corresponding output is returned as an approximation of $L^*_c$. A summary of the algorithm and the specific choices of $\alpha$ and $\varepsilon'$ are given in Algorithm~\ref{alg:monotone}. The following proposition, which is proved in Appendix~\ref{proofofpropmonotonicity}, establishes that this algorithm has both properties we desire:
 \begin{proposition}\label{prop:monotonicity}
     For any $\delta\in(0,1]$ and any $\varepsilon\in(0,1]$,
     Algorithm~\ref{alg:monotone} computes a $\delta$-decreasing,
-- 
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