The challenging aspect of the adaptive seeding problem expressed in
Equation~\ref{eq:problem} is its adaptivity: a seed set must be selected during
the first stage such that in expectation a high influence value can be reached
when adaptively selecting nodes on the second stage.  A standard approach in 
stochastic optimization for overcoming this challenge is to use sampling to
estimate the expectation of the influence value reachable on the second stage.
However, as will be discussed in Section~\ref{sec:experiments}, this approach
quickly becomes infeasible even with modest size graphs.

In this section we develop an approach which avoids sampling and allows
designing adaptive seeding algorithms that can be applied to large graphs.  We
show that for additive influence functions one can optimize a relaxation of the
problem which we refer to as the \emph{non-adaptive} version of the problem.
After defining the non-adaptive version, we show in sections~\ref{sec:gap} that
the optimal solution for the non-adaptive version is an upper bound on the
optimal solution of the adaptive seeding problem.  We then argue in
Section~\ref{sec:round} that any solution to the non-adaptive version of the
problem can be converted to an adaptive solution, losing an arbitrarily small
factor in the approximation ratio.  Together, this implies that one can design
algorithms for the non-adaptive problem instead, as we do in
Section~\ref{sec:algorithms}.\newline 
      
%
%, namely \emph{non-adaptive}. As we will discuss in
%Sections~\ref{sec:gap} and~\ref{sec:round}, these non-adaptive policies can be
%used as tool to construct adaptive solutions.
%

\noindent \textbf{Non-adaptive policies.}  We say that a policy is \emph{non-adaptive} if it selects a set of nodes $S
\subseteq X$ to be seeded in the first stage and a vector of probabilities
$\mathbf{q}\in[0,1]^n$, such that each neighbor $u$ of $S$ which realizes is
included in the solution independently with probability $q_u$.  The constraint
will now be that the budget is only respected in expectation, \emph{i.e.}  $|S|
+ \textbf{p}^T\textbf{q} \leq k$. Formally the optimization problem for
non-adaptive policies can be written as:
\begin{equation}\label{eq:relaxed}
    \begin{split}
    %&\max_{}
        \max_{\substack{S\subseteq X\\\textbf{q}\in[0,1]^n} }& \;
    \sum_{R\subseteq\neigh{X}} \Big (\prod_{u\in R} p_uq_u\prod_{u\in\neigh{X}\setminus
    R}(1-p_uq_u) \Big )
 f(R)\\
    \text{s.t. } & \; |S|+\textbf{p}^T\textbf{q} \leq k,\;
q_u \leq \mathbf{1}\{u\in\neigh{S}\}
\end{split}
\end{equation}
%$\textbf{p}\odot \textbf{q}$ denotes componentwise multiplication and $\textbf{q}_R$ denotes the positive entries of $\textbf{q}$ on nodes in $R$: 
%
%\begin{displaymath}
%    \textbf{p}\circ \textbf{q}_R = \prod_{u\in R} p_uq_u\prod_{u\in\neigh{X}\setminus
%    R}(1-p_uq_u)
%\end{displaymath}
%
%\begin{displaymath}
%    \Pr[R \ |\ \textbf{p}\odot \textbf{q} ] = \prod_{u\in R} p_uq_u\prod_{u\in\neigh{X}\setminus
%    R}(1-p_uq_u)
%\end{displaymath}
where we denote by $\mathbf{1}\{E\}$ the indicator variable of the event $E$.
Note that because of the condition $q_u\leq \mathbf{1}\{u\in\neigh{S}\}$,
the summand associated with $R$ in \eqref{eq:relaxed} vanishes whenever $R$
contains $u\in\neigh{X}\setminus\neigh{S}$. Hence, the summation is restricted
to $R\subseteq\neigh{S}$ as in \eqref{eq:problem}.

%The relaxed non-adaptive optimization \eqref{eq:relaxed} problem can be written
%more concisely using the \emph{multi-linear extension} of $f$:
%\begin{displaymath}
%    \forall p\in[0,1]^m,\; F(p)
%    \defeq\mathbb{E}_{p_R}\big[f(R)\big]
%    =\sum_{R\subseteq\neigh{X}} p_R f(R)
%\end{displaymath}
%This \emph{extension by expectation} is known to present cross-convecity
%properties which can exploited in maximization problems when $f$ is
%a submodular function \cite{dughmi, vondrak}. Using this definition, \eqref{eq:relaxed}
%can be re-written as:
%\begin{equation}\label{eq:relaxed-multi}
%    \begin{split}
%    &\max_{S\subseteq X} \max_{q\in[0,1]^n}F(p\circ q)\\
%    &\text{s.t. }|S|+\sum_{u\in\neigh{X}}p_uq_u \leq k,\;
%q_u \leq \mathbf{1}_{\{u\in\neigh{S}\}}
%\end{split}
%\end{equation}
%
%When $f$ is an ìnfluence function from the triggering model, it has been proved
%in \cite{singer} that adaptive seeding has a bounded \emph{adaptivity gap}:
%denoting by $OPT$ the optimal solution of \eqref{eq:problem} and by $OPT_{NA}$
%the optimal solution of \eqref{eq:relaxed}, we have $OPT\leq\alpha OPT_{NA}$.
%This inequality justifies using non-adaptive policies to approximate solutions
%to the adaptive seeding problem.

\subsection{Adaptivity Gap}\label{sec:gap}

We will now justify the use of non-adaptive strategies by showing that the
optimal solution for this form of non-adaptive strategies yields a higher value
than adaptive ones.  For brevity, given a probability vector $\pi\in[0,1]^m$ we
write:
\begin{equation}\label{eq:multi}
        F(\pi) \defeq
        \sum_{R\subseteq\neigh{X}}\left(\prod_{u\in
        R}\pi_u\prod_{u\in\neigh{X}\setminus R}(1-\pi_u)\right)
    f(R)
\end{equation}
as well as $\textbf{p}\otimes \textbf{q}$ to denote the component-wise
multiplication between vectors $\textbf{p}$ and $\textbf{q}$.  Finally, we write
$\mathcal{F}_{A} \defeq \{S \subseteq X : |S|\leq k\}$, and $\mathcal{F}_{NA}
\defeq\{(S,\textbf{q}), |S|+\textbf{p}^T\textbf{q} \leq k, q_u \leq
\mathbf{1}_{\{u\in\neigh{S}\}}\}$ to denote the feasible regions of the
adaptive and non-adaptive problems, respectively.

%this form of non-adap
%We already know that the adaptivity gap is bounded for a general class of
%influence function. For adaptive functions, we get a stronger result:
%\emph{relaxed-non adaptive policies are stronger than non-adaptive policies}.
%
%
\begin{proposition}\label{prop:gap}
For additive functions given by \eqref{eq:voter}, the value of the optimal
adaptive policy is upper bounded by the optimal non-adaptive policy:
    \begin{displaymath}
        \begin{aligned}[t]
            &\max_{S\subseteq X} \sum_{R\subseteq\neigh{S}} p_R
            \max_{\substack{T\subseteq R\\|T|\leq k-|S|}}f(T)\\
            &\text{s.t. }S \in \mathcal{F}_{A}
        \end{aligned}
        \leq
        \begin{aligned}[t]
            &\max_{\substack{S\subseteq X\\\textbf{q}\in[0,1]^n}}
            F(\mathbf{p}\otimes\mathbf{q})\\
            &\text{s.t. } (S,\textbf{q}) \in \mathcal{F}_{NA}
                    \end{aligned}
    \end{displaymath}
\end{proposition}

%\begin{proposition}\label{prop:gap}
%Let $\mathcal{F}_{A} := \{T \subseteq X : |T|\leq k\}$, $\mathcal{F}_{NA} :=\{(T,\textbf{q}), |S|+\textbf{p}^T\textbf{q} \leq k, q_u \leq \mathbf{1}_{\{u\in\neigh{S}\}}\}$.
%For additive functions of the form given by \eqref{eq:voter}, non-adaptive
%    policies are stronger than adaptive policies:
%    \begin{displaymath}
%        \begin{aligned}[t]
%            &\max_{S\subseteq X} \sum_{R\subseteq\neigh{S}} p_R
%            \max_{\substack{T\subseteq R\\|T|\leq k-|S|}}f(T)\\
%            &\text{s.t. }|S|\leq k
%        \end{aligned}
%        \leq
%        \begin{aligned}[t]
%            &\max_{S\subseteq X} \max_{q\in[0,1]^n}F(p\circ q)\\
%            &\text{s.t. }|S|+\sum_{u\in\neigh{X}}p_uq_u \leq k,\;
%            q_u \leq \mathbf{1}_{\{u\in\neigh{S}\}}
%        \end{aligned}
%    \end{displaymath}
%\end{proposition}

The proof of this proposition can be found in Appendix~\ref{sec:ad-proofs} and
relies on the following fact: the optimal adaptive policy can be written as
a feasible non-adaptive policy, hence it provides a lower bound on the value of
the optimal non-adaptive policy.

\subsection{From Non-Adaptive to Adaptive Solutions}\label{sec:round}

From the above proposition we now know that optimal non-adaptive solutions have
higher values than adaptive solutions. Given a non-adaptive solution
$(S,\mathbf{q})$, a possible scheme would be to use $S$ as an adaptive
solution.  But since $(S, \mathbf{q})$ is a solution to the non-adaptive
problem, Proposition~\ref{prop:gap} does not provide any guarantee on how well
$S$ performs as an adaptive solution.

However, we show that from a non-adaptive solution $(S,\mathbf{q})$, we can
obtain a lower bound on the adaptive value of $S$, that is, the expected
influence attainable in expectation over all possible arrivals of neighbors of
$S$. Starting from $S$, in every realization of neighbors $R$, sample every
node $u \in R \cap \mathcal{N}(S)$ with probability $q_{u}$, to obtain a random
set of nodes $I_R \subseteq R \cap S$. $(S, \mathbf{q})$ being a non-adaptive
solution, it could be that selecting $I_R$ exceeds our budget. Indeed, the only
guarantee that we have is that $|S| + \mathbb{E}\big[|I_R|\big]\leq k$. As
a consequence, an adaptive solution starting from $S$ might not be able to
select $I_R$ on the second stage.

%execute an adaptive policy by selecting a set $S$ in the first stage, and then in the second stage, while there is still budget remaining, select every neighbor $u$ of $S$ that realized with probability $q_{u}$


%To do so, one needs to prove that in expectation over all the realizations, the average values obtainable in the second stage, are close to the value of the non-adaptive objective evaluated over $(S,\mathbf{q})$.
 
%the objective function of the adaptive problem \eqref{eq:relaxed}. However, the set resulting from the sampling might exceed the budget on the second stage, hence preventing us from directly obtaining a feasible adaptive solution.    
%Given a a non-adaptive policy $(S,\mathbf{q})$ one can execute an adaptive policy by selecting a set $S$ in the first stage, and then in the second stage, while there is still budget remaining, select every neighbor $u$ of $S$ that realized with probability $q_{u}$.  
%In order to use the above lemma, one needs to show that in almost all realizations   
%non-adaptive policy

%Fortunately, by using contention resolution schemes~\cite{vondrak} given a feasible solution $\mathbf{q}$ to the non-adaptive version, one can compute a \emph{feasible} random set $I$, such that:
%\begin{equation}\label{eq:cr}
%\mathbb{E}\big[f(I)\big] 
%\geq (1-\varepsilon) F(\mathbf{q}) 
%\end{equation}

Fortunately, the probability of exceeding the budget is small enough and with
high probability $I_R$ will be feasible. This is exploited in \cite{vondrak} to
design a randomized rounding method with approximation guarantees. These
rounding methods are called \emph{contention resolution schemes}.
%More precisely, we note that once the set $S$ is fixed, the feasibility constraint of problem~\eqref{eq:relaxed} becomes a single Knapsack constraint, for which \cite{vondrak} constructs a $(1-\varepsilon, 1-\varepsilon)$-balanced contention resolution scheme.
Theorem~1.3 of this paper gives us a contention resolution scheme which will
compute from $\mathbf{q}$ and for any realization $R$ a \emph{feasible} set
$\tilde{I}_R$, such that:
\begin{equation}
    \label{eq:cr}
    \mathbb{E}_R\big[f(\tilde{I}_R)\big] \geq (1-\varepsilon) F(\mathbf{q})
\end{equation}
What this means is that starting from a non-adaptive solution $(S,\mathbf{q})$,
there is a way to construct a random \emph{feasible} subset on the second stage
such that in expectation, this set attains almost the same influence value as
the non-adaptive solution. Since the adaptive solution starting from $S$ will
select optimally from the realizations $R\subseteq\neigh{S}$,
$\mathbb{E}_R[f(\tilde{I}_R)]$ provides a lower bound on the adaptive value of
$S$ that we denote by $A(S)$.

More precisely, denoting by $\text{OPT}_A$ the optimal value of the adaptive
problem~\eqref{eq:problem}, we have the following proposition whose proof can
be found in Appendix~\ref{sec:ad-proofs}.
\begin{proposition}\label{prop:cr}
    Let $(S,\textbf{q})$ be an $\alpha$-approximate solution to the
    non-adaptive problem \eqref{eq:relaxed}, then $\mathrm{A}(S) \geq \alpha
    \mathrm{OPT}_A$.
\end{proposition}