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Diffstat (limited to 'main.tex')
| -rw-r--r-- | main.tex | 44 |
1 files changed, 21 insertions, 23 deletions
@@ -45,7 +45,7 @@ approximation ratio \paragraph{Strategic case} We could design the allocation function of our mechanism by following the full information approach: allocating to agent $i^*$ when $V(i^*)\geq V(S_G)$ and to -$S_G$ otherwise. However, Singer~\cite{singer-influence} notes that this +$S_G$ otherwise. However, \citeN{singer-influence} notes that this allocation function is not monotonous, and thus breaks incentive compatibility by Myerson's theorem (Theorem~\ref{thm:myerson}). @@ -55,14 +55,13 @@ full-information case after removing $i^*$ from the set $\mathcal{N}$: OPT_{-i^*} \defeq \max_{S\subseteq\mathcal{N}\setminus\{i^*\}} \Big\{V(S) \;\Big| \; \sum_{i\in S}c_i\leq B\Big\} \end{equation} -Singer~\cite{singer-mechanisms} and Chen \emph{et al.}~\cite{chen} prove that -allocating to $i^*$ when $V(i^*)\geq C.OPT_{-i^*}$ (for some constant $C$) and -to $S_G$ otherwise yields a 8.34-approximation mechanism. However, -$OPT_{-i^*}$, the solution of the full-information case, cannot be computed in -poly-time when the underlying problem is NP-hard (unless P=NP), as is the case -for \EDP{}. +\citeN{singer-mechanisms} and \citeN{chen} prove that allocating to $i^*$ when +$V(i^*)\geq C.OPT_{-i^*}$ (for some constant $C$) and to $S_G$ otherwise yields +a 8.34-approximation mechanism. However, $OPT_{-i^*}$, the solution of the +full-information case, cannot be computed in poly-time when the underlying +problem is NP-hard (unless P=NP), as is the case for \EDP{}. -Instead, Singer~\cite{singer-mechanisms} and Chen \emph{et al.}~\cite{chen} +Instead, \citeN{singer-mechanisms} and Chen \emph{et al.}~\cite{chen} suggest to replace $OPT_{-i^*}$ by a quantity $L^*$ satisfying the following properties: \begin{itemize} @@ -94,11 +93,10 @@ Section~\ref{sec:relaxation}) yields an constant-approximation mechanism for In \cite{singer-influence}, the optimization program \eqref{relax} cannot be solved efficiently when $R$ is the multilinear extension of $V$. Consequently, Singer introduces a second relaxation which he relates to the -multilinear extension through the \emph{pipage rounding} framework of Ageev and -Sviridenko~\cite{pipage}. +multilinear extension through the \emph{pipage rounding} framework of \citeN{pipage}. \paragraph{Our approach} -Following Chen \emph{et al.}~\cite{chen} and Singer~\cite{singer-influence}, +Following Chen \citeN{chen} and \citeN{singer-influence}, we in turn introduce a relaxation $L$ specifically tailored to the value function of \EDP: \begin{displaymath} @@ -114,7 +112,7 @@ $O(\log\log\varepsilon^{-1})$ iterations. Being the solution to a maximization problem, $L^*$ satisfies the required monotonicity property. The main challenge will be to prove that $OPT'_{-i^*}$, for the relaxation $R=L$, is close to $OPT$. As in~\cite{singer-influence} this will be done by using the -\emph{pipage rounding} framework of~\cite{pipage} and forms the technical bulk +\emph{pipage rounding} framework of~\citeN{pipage} and forms the technical bulk of the paper. The resulting mechanism for \EDP{} is composed of @@ -140,10 +138,10 @@ set can be found in~\cite{singer-mechanisms}. \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(\lambda) + \State $L^* \gets \argmax_{\lambda\in[0,1]^{n}} \{L(\lambda) \mid \lambda_{i^*}=0,\sum_{i \in \mathcal{N}\setminus\{i^*\}}c_i\lambda_i\leq B\}$ \Statex - \If{$L(\xi) < C \cdot V(i^*)$} \label{c} + \If{$L^* < C \cdot V(i^*)$} \label{c} \State \textbf{return} $\{i^*\}$ \Else \State $i \gets \argmax_{1\leq j\leq n}\frac{V(j)}{c_j}$ @@ -229,18 +227,18 @@ in the above formula: \lambda_i x_i\T{x_i}\right) \end{align} \end{comment} + \subsection{Proof of Theorem~\ref{thm:main}}\label{sec:proofofmainthm} %\stratis{individual rationality???} %The proof of the properties of the mechanism is broken down into lemmas. We now present the proof of Theorem~\ref{thm:main}. Truthfulness and individual rationality follows from monotonicity and threshold payments. Monotonicity and -budget feasibility follow the same steps as the analysis of Chen \emph{et al.} \cite{chen}; +budget feasibility follow the same steps as the analysis of \citeN{chen}; for the sake of completeness, we restate their proof in the Appendix. - Our proof of the approximation ratio uses a bound on our concave relaxation -$L$ (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:complexity} + +The complexity of the mechanism is given by the following lemma. +\begin{lemma}[Complexity]\label{lemma:complexity} For any $\varepsilon > 0$, the complexity of the mechanism is $O(\text{poly}(n, d, \log\log \varepsilon^{-1}))$. \end{lemma} @@ -266,7 +264,7 @@ contribution; the proof of this lemma can be found in Section~\ref{sec:relaxatio Finally, we prove the approximation ratio of the mechanism. We use the 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{lemma}[Approximation]\label{lemma:relaxation} %\begin{displaymath} $ OPT' \leq 2 OPT + 2\max_{i\in\mathcal{N}}V(i)$ @@ -319,11 +317,11 @@ Using Lemma~\ref{lemma:relaxation} we can complete the proof of Theorem~\ref{thm + \frac{(e-1)\varepsilon}{C(e-1)- 6e + 2} \end{align*} Finally, using again Lemma~\ref{lemma:greedy-bound}, we get: - \begin{multline}\label{eq:bound2} + \begin{equation}\label{eq:bound2} OPT(V, \mathcal{N}, B) \leq \frac{3e}{e-1}\left( 1 + \frac{4e}{C - (e-1) -6e +2}\right) V(S_G)\\ + (e-1) -6e +2}\right) V(S_G) +\frac{2e\varepsilon}{C(e-1)- 6e + 2} - \end{multline} + \end{equation} To minimize the coefficients of $V_{i^*}$ and $V(S_G)$ in \eqref{eq:bound1} and \eqref{eq:bound2} respectively, we wish to chose for $C=C^*$ such that: \begin{displaymath} |