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Diffstat (limited to 'main.tex')
| -rw-r--r-- | main.tex | 22 |
1 files changed, 11 insertions, 11 deletions
@@ -138,10 +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 $i \gets \argmax_{1\leq j\leq n}\frac{V(j)}{c_j}$ @@ -227,18 +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} @@ -264,7 +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)$ @@ -317,11 +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} |