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\maketitle +% +\section{Submodular functions and Lov\'{a}sz extension} + +\begin{definition}[Submodular functions] + Let $N = \{1, \dots, n\}$, and $2^N$ denote the power set of $N$. + Then a function $f : 2^N \rightarrow \mathbb{R}$ is \emph{submodular} if: + \begin{align*} + f(A \cup \{ j \} ) - f(A) \geq f(B \cup \{ j \}) - f(B) + \end{align*} + for all $A \subseteq B \subseteq N$ and for all $j \in N$. + + This is equivalent to: + \begin{align*} + f(A \cap B) + f(A \cup B) \leq f(A) + f(B) + \end{align*} + for all $\forall A, B \subseteq N$. + +\end{definition} + +Some examples of submodular functions: +\begin{itemize} + \item Let $A$ be a subset of nodes of a bipartite graph. Then $f(A) = | \text{neighborhood}(A) |$ is submodular. + + \item Let $X = \{ X_1, \dots X_n \}$ be a set of random variables, and $A \subseteq X$. Then $f(A) = H(A)$, where $H$ denotes entropy, is submodular. + + \item Let $E, V$ be the edges and nodes of a graph, and $A \subseteq V$. Then the size of the graph cut, $f(A) = \big| \{ (u, v) \in E : u \in A, v \in V \setminus A \} \big| $, is submodular. +\end{itemize} + +There is an equivalence between submodular and boolean functions: a submodular function $f: 2^N \rightarrow \mathbb{R}$ can be expressed as a boolean function $f: \{0, 1\}^n \rightarrow \mathbb{R}$. + +\begin{definition}[Lov\'{a}sz extension] +Let $f(x)$ be a submodular function. Then its Lov\'{a}sz extension, $\hat f(z): [0, 1]^n \rightarrow \mathbb{R}$ is given by: +\begin{align*} +\hat f(z) = \mathbb{E}_{\lambda \sim \text{U}(0, 1)} \bigg[ f(\{i : z_i \geq \lambda \}) \bigg] +\end{align*} +where $\text{U}(0, 1)$ denotes the uniform distribution on $[0, 1]$. +\end{definition} + +Some observations about $\hat f(z)$: +\begin{itemize} + \item For any $x \in \{0, 1\}^n$, $\hat f(x) = f(x)$. That is, $f$ and $\hat f$ agree on the domain of $f$. + + \item For any $z \in [0, 1]^n$, there exists an $x \in \{0, 1\}^n$, such that $f(x) \leq \hat f(z)$. Also, $\min_{x \in \{0, 1\}^n} f(x) = \min_{z \in [0,1]^n} \hat f(z)$. + +\end{itemize} + + +\begin{theorem}[Submodularity and convexity] +\theoremlabel{submod_cvx} +Consider a function $f(x) : 2^N \rightarrow \mathbb{R}$ and its Lov\'{a}sz extension $\hat f(z) : 2^N \rightarrow \mathbb{R}$. Then $f(x)$ is submodular iff $\hat f(z)$ is convex. + +\end{theorem} +Proof. We only show the forward direction ($f$ is submodular $\Rightarrow$ $\hat f$ is convex), the reverse direction is left as an exercise. + +Without loss of generality, assume $z$ is order: $z_1 \geq z_2 \geq \dots \geq z_n$. + +Also, let $S_i = \{1, 2, \dots, i\}$. We also take $f(\emptyset) = 0$. + +Then: +\begin{align*} +\hat f(z) +&= \sum_{i=1}^{n-1} P(z_{i+1} \leq \lambda \leq z_i) \cdot f(S_i) + z_n \cdot f(S_n) \\ +&= \sum_{i=1}^{n-1} (z_{i+1} - z_i) \cdot f(S_i) + z_n \cdot f(S_n) \\ +\end{align*} + +Now, consider the following lemma: +\begin{lemma} +\lemmalabel{lovaszLP} +The Lov\'{a}sz extension $\hat f(z)$ of submodular $f(x)$ can be expressed an LP: +\begin{align*} +\hat f(z) = \max_x x^T z : x(S) \leq f(S), x(N) = f(N), \forall S \subseteq N +\end{align*} +\end{lemma} +where $x(S) = \sum_{i \in S} x_i$. Let $F$ denote the feasible region. + +Given \lemmaref{lovaszLP}, the rest of the proof follows easily: +\begin{align*} +\hat f(\lambda z + (1 - \lambda) z') +&= \max_{x \in F} x^T (\lambda z + (1-\lambda)z') \\ +&\leq \lambda \max_{x \in F} x^T z + (1 - \lambda) \max_{x \in F} x^T z' \\ +&= \lambda \hat f(z) + (1 - \lambda) \hat f(z') +\end{align*} + + + +Now we just have to prove \lemmaref{lovaszLP}. + +Consider the dual problem to the above LP: +\begin{align*} +\min_y & \sum_{S \subseteq N} y_S \cdot f(S) : \sum_{S \subseteq N} y_S e_S = z, y_S \geq 0 \\ +& \text{where } (e_S)_i = \begin{cases} i & \text{ if $i \in S$ } \\ 0 & \text{ else} \end{cases} +\end{align*} + +Our goal is to find a primal-dual feasible solution $x^*, y^*$ that satisfies: +\begin{align*} +\hat f(z) = c^T x^* = \sum_{S \subseteq N} y_S^* \cdot f(S) +\end{align*} + +Consider the following $x^*, y^*$: +\begin{align*} +x_i^* &= f(S_i) - f(S_{i-1}) \\ +y_S^* &= \begin{cases} +z_i - z_{i+1} & \text{if } S = S_i \\ +z_n & \text{if } S = N \\ +0 & \text{else} +\end{cases} +\end{align*} + +First, we show that $x^*$ is primal feasible. + + For the constraint $x(N) = f(N)$, we see that: +\begin{align*} +x^*(N) = \sum_{i=1}^{n} x_i^* +&= \sum_{i=1}^{n} f(S_i) - f(S_{i-1}) \\ +&= f(N) - f(\emptyset) = f(N) +\end{align*} + +For the other constraint, $x(S) \leq f(S)$, we proceed by induction on $|S|$ (the size of $S$). + +As a base case, we take $S = \emptyset$. + +Suppose that the constraint is satisfied for all sets of size $i - 1$. + +Then consider an arbitrary $S$, whose largest element is $i$: +\begin{align*} +f(S) + f(S_{i-1}) +&\geq f(S \cup S_{i-1}) + f(S \cup S_{i-1}) \\ +&= f(S_i) + f(S \setminus \{i\}) \\ +&\geq f(S_i) + x^*(S \setminus \{i\}) \\ +\Rightarrow f(S) +&\geq f(S_i) - f(S_{i-1}) + x^*(S \setminus \{i\}) \\ +&= x_i^* + x^*(S \setminus \{i\}) \\ +&= x^*(S) +\end{align*} +where we used fact that $f$ is submodular and that $i$ is the largest element in $S$. + +It follows that $x^*(S) \leq f(S)$, for all $S \subseteq N$, and thus $x^*$ is primal feasible. + +Next, we show that $y^*$ is dual feasible. +\begin{align*} +\sum_{S \subseteq N} y_S^* \cdot f(S) +&= \sum_{j = 1}^{n-1} (z_j - z_{j+1}) e_{S_j} + z_n e_n \\ +\bigg( \sum_{S \subseteq N} y_S^* \cdot f(S) \bigg)_i +&= \bigg( \sum_{j = 1}^{n-1} (z_j - z_{j+1}) e_{S_j} + z_n e_n \bigg)_i & \text{ (consider it elementwise) } \\ +&= \sum_{j = i}^{n} (z_j - z_{j+1}) + z_n \\ +&= z_i +\end{align*} +Since this equality holds elementwise for each $i$, we see that $\sum_{S \subseteq N} y_S^* \cdot f(S) = z$. + +We also observe that $y_S^* \geq 0$ due to the nonincreasing ordering of $z_1, \dots, z_n$. + +Thus, it follows that $y_S^*$ is dual feasible. + +Finally, we show that $x^*, y^*$ are optimal, achieving a dualty gap of zero: +\begin{align*} +z^T x^* +&= \sum_{i=1}^{n} z_i \big( f(S_i) - f(S_{i-1}) \big) \\ +&= \sum_{i=1}^{n} (z_i - z_{i+1}) f(S_i) + z_n f(S_n) \\ +&= \hat f(z) \\ +\hat f(z) +&= \sum_{i=1}^{n} (z_i - z_{i+1}) f(S_i) + z_n f(S_n) \\ +&= \sum_{S \subseteq N} y_S^* f(S) +\end{align*} + +This concludes the proof of \lemmaref{lovaszLP}, and that of \theoremref{submod_cvx}. + +% \end{document} \ No newline at end of file