SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} \Pr \left[ w _ {G} \left(C _ {u}\right) <   w _ {G} \left(C _ {v}\right) \wedge w _ {G} \left(C _ {u}\right) <   w _ {G} \left(C _ {u, v}\right) \wedge w _ {G} \left(C _ {u}\right) <   w _ {G} \left(C _ {\emptyset}\right) \right] \\ = \Pr [ \Delta_ {v} + X _ {s, v} + X _ {t, u} <   X _ {t, v} + X _ {s, u} \\ \wedge \Delta_ {u, v} + X _ {s, v} + X _ {t, u} + \tau <   X _ {t, v} + X _ {t, u} \wedge \Delta_ {\emptyset} + X _ {s, v} + X _ {t, u} + \tau <   X _ {s, v} + X _ {s, u} ] \\ = \Pr \left[ \Delta_ {v} + X _ {s, v} + X _ {t, u} <   X _ {t, v} + X _ {s, u} \right. \\ \wedge \Delta_ {u, v} + X _ {s, v} + \tau <   X _ {t, v} \wedge \Delta_ {\emptyset} + X _ {t, u} + \tau <   X _ {s, u} ]. \tag {2} \\ \end{array}
$$
Now, we replace  $X_{s,v} - X_{t,v}$  by  $x$  and  $X_{t,u} - X_{s,u}$  by  $y$ . Then, Eq. (2) can be rewritten as
$$
\begin{array}{l} \Pr \left[ w _ {G} \left(C _ {u}\right) <   w _ {G} \left(C _ {v}\right) \wedge w _ {G} \left(C _ {u}\right) <   w _ {G} \left(C _ {u, v}\right) \wedge w _ {G} \left(C _ {u}\right) <   w _ {G} \left(C _ {\emptyset}\right) \right] \\ = \Pr \left[ x + y <   - \Delta_ {v} \wedge x <   - \Delta_ {u, v} - \tau \wedge y <   - \Delta_ {\emptyset} - \tau \right]. \\ \end{array}
$$
Applying the same analysis for  $G'$  we derive
$$
\begin{array}{l} \Pr \left[ w _ {G ^ {\prime}} \left(C _ {u}\right) <   w _ {G ^ {\prime}} \left(C _ {v}\right) \wedge w _ {G ^ {\prime}} \left(C _ {u}\right) <   w _ {G ^ {\prime}} \left(C _ {u, v}\right) \wedge w _ {G ^ {\prime}} \left(C _ {u}\right) <   w _ {G ^ {\prime}} \left(C _ {\emptyset}\right) \right] \\ = \Pr \left[ \Delta_ {v} + X _ {s, v} + X _ {t, u} <   X _ {t, v} + X _ {s, u} \right. \\ \wedge \Delta_ {u, v} + X _ {s, v} + X _ {t, u} <   X _ {t, v} + X _ {t, u} \wedge \Delta_ {\emptyset} + X _ {s, v} + X _ {t, u} <   X _ {s, v} + X _ {s, u} ] \\ = \Pr \left[ \Delta_ {v} + X _ {s, v} + X _ {t, u} <   X _ {t, v} + X _ {s, u} \wedge \Delta_ {u, v} + X _ {s, v} <   X _ {t, v} \wedge \Delta_ {\emptyset} + X _ {t, u} <   X _ {s, u} \right]. \\ \end{array}
$$
Replacing  $X_{s,v} - X_{t,v}$  by  $x$  and  $X_{t,u} - X_{s,u}$  by  $y$  yields
$$
\begin{array}{l} \Pr \left[ w _ {G ^ {\prime}} \left(C _ {u}\right) <   w _ {G ^ {\prime}} \left(C _ {v}\right) \wedge w _ {G ^ {\prime}} \left(C _ {u}\right) <   w _ {G ^ {\prime}} \left(C _ {u, v}\right) \wedge w _ {G ^ {\prime}} \left(C _ {u}\right) <   w _ {G ^ {\prime}} \left(C _ {\emptyset}\right) \right] \\ = \Pr \left[ x + y <   - \Delta_ {v} \wedge x <   - \Delta_ {u, v} \wedge y <   - \Delta_ {\emptyset} \right]. \\ \end{array}
$$
Note that by Fact 2.3 we have that  $x$  and  $y$  follow  $\mathrm{Lap}(1 / \epsilon)$ . Moreover, observe that the random variables  $x$  and  $y$  are independent by definition. Therefore, by invoking Lemma 3.1 with  $\alpha = -\Delta_{u,v} - \tau$ ,  $\beta = -\Delta_{\emptyset} - \tau$ , and  $\gamma = -\Delta_v$  we derive
$$
\frac {\operatorname * {P r} \left[ w _ {G ^ {\prime}} (C _ {u}) <   w _ {G ^ {\prime}} (C _ {v}) \wedge w _ {G ^ {\prime}} (C _ {u}) <   w _ {G ^ {\prime}} (C _ {u , v}) \wedge w _ {G ^ {\prime}} (C _ {u}) <   w _ {G ^ {\prime}} (C _ {\emptyset}) \right]}{\operatorname * {P r} \left[ w _ {G} (C _ {u}) <   w _ {G} (C _ {v}) \wedge w _ {G} (C _ {u}) <   w _ {G} (C _ {u , v}) \wedge w _ {G} (C _ {u}) <   w _ {G} (C _ {\emptyset}) \right]} \leq e ^ {4 \tau \epsilon}.
$$
Considering the second case  $w_{G}(e) + \tau = w_{G^{\prime}}(e)$ , for some  $\tau > 0$ , we derive that the ratio Eq. (1) is upper-bounded by  $1 \leq e^{4\tau \epsilon}$ .
The remaining cases. The proof for the remaining case, i.e., analysis when  $C_v, C_{u,v}$  and  $C_{\emptyset}$  is the minimum, follows the same steps as the case we have just analyzed.
Finalizing the proof. It remains to discuss two properties that are needed to complete the proof. First, our analysis above is applied for all but 4 random variables  $X$  fixed. Nevertheless, our analysis does not depend on how those random variables are fixed. Therefore, for any fixed cut  $C$ , summing/integrating over all the possible outcomes of those variables yields
$$
\frac {\operatorname* {P r} \left[ \text {A l g o r i t h m 1 o u t p u t s C g i v e n G} \right]}{\operatorname* {P r} \left[ \text {A l g o r i t h m 1 o u t p u t s C g i v e n G ^ {\prime}} \right]} \leq e ^ {4 \tau \epsilon}.
$$
Second, our proof shows the ratio of a cut  $C$  being the minimum one in  $G$  and  $G'$ . However, the DP definition applies to any set of multiple cuts. It is folklore that in the case of pure DP, i.e., when  $\delta = 0$ , these two cases are equivalent. This completes the analysis.
3.2 Approximation Analysis
Observe that showing that Algorithm 1 has  $O(n / \epsilon \cdot \log n)$  additive error is straightforward as a random variable drawn from  $\mathrm{Exp}(b)$  is upper-bounded by  $O(\log n / b)$  whp. Lemma 3.3 proves the  $O(n / \epsilon)$  bound. On a very high level, our proof relies on the fact that for a sufficiently large constant  $c$ , it holds that only a small fraction of random variables  $X_{i,j}$  exceeds  $c / \epsilon$ ; this fraction is  $e^{-c}$ .
Lemma 3.3. With probability at least  $1 - n^{-2}$ , Algorithm 1 outputs a min  $s$ -t cut with additive error  $O(n / \epsilon)$ .
Proof. Let  $G$  be an input graph to Algorithm 1 and let  $\widetilde{G}$  be the graph after the edges on Line 2 are added.  $\widetilde{G}$  contains  $2(n - 1)$  more edges than  $G$ . This proof shows that with probability at least  $1 - n^{-2}$ , the total sum of weights of all these edges is  $O(n / \epsilon)$ .
We first provide a brief intuition. For the sake of it, assume  $\epsilon = 1$ . Whp, each  $X_{i,j}$  weighs at most  $5\log n$ . Also, consider only those  $X_{i,j}$  such that  $X_{i,j} \geq 2$ ; the total sum of those random variables having weight less than 2 is  $O(n)$ . It is instructive to think of the interval  $[2, 5\log n]$  being partitioned into buckets of the form  $[2^i, 2^{i+1})$ . Then, the value of each edge added by Algorithm 1 falls into one of the buckets. Now, the task becomes upper-bounding the number  $c_i$  of edges in bucket  $i$ . That is, we let  $Y_{s,u}^i = 1$  iff  $X_{s,u} \in [2^i, 2^{i+1})$ , which results in  $c_i = \sum_{u \in V} (Y_{s,u}^i + Y_{t,u}^i)$ . Hence,  $c_i$  is a sum of  $0/1$  independent random variables, and we can use Chernoff bound to argue about its concentration.
There are two cases. If  $\mathbb{E}[c_i]$  is more than  $O(\log n)$ , then  $c_{i}\in O(\mathbb{E}[c_{i}])$  by the Chernoff bound. If  $\mathbb{E}[c_i]\in o(\log n)$ , e.g.,  $\mathbb{E}[c_i] = O(1)$ , we can not say that with high probability  $c_{i}\in O(\mathbb{E}[c_{i}])$ . Nevertheless, it still holds  $c_{i}\in O(\log n)$  whp.
Edges in  $E(\widetilde{G}) \setminus E(G)$  with weights more than  $5\frac{\log n}{\epsilon}$ . Let  $Y \sim \mathrm{Exp}(\epsilon)$ . By Fact 2.1,  $\operatorname*{Pr}\left[Y > 5\frac{\log n}{\epsilon}\right] = \exp (-5\log n) = n^{-5}$ . Since each  $X_{s,v}$  and  $X_{t,v}$  is drawn from  $\mathrm{Exp}(\epsilon)$ , we have that for  $n \geq 2$  with probability at least  $1 - n^{-3}$  each of the edges in  $E(\widetilde{G}) \setminus E(G)$  has weight at most  $5\frac{\log n}{\epsilon}$ .
Edges in  $E(\widetilde{G}) \setminus E(G)$  with weights at most  $5\frac{\log n}{\epsilon}$ . Observe that the sum of the weights of all the edges in  $E(\widetilde{G}) \setminus E(G)$  having weight at most  $2 / \epsilon$  is  $O(n / \epsilon)$ . Hence, we focus on the edge weights in the interval  $\left[\frac{2}{\epsilon}, 5\frac{\log n}{\epsilon}\right]$ . We partition this interval into  $O(\log \log n)$  subintervals where each, except potentially the last one, of the form  $[2^i / \epsilon, 2^{i+1} / \epsilon)$ .
Let  $Y\sim \mathrm{Exp}(\epsilon)$  .We have
$$
\Pr \left[ Y \in \left[ 2 ^ {i} / \epsilon , 2 ^ {i + 1} / \epsilon\right) \right] \leq \Pr \left[ Y \geq 2 ^ {i} / \epsilon \right] = e ^ {- 2 ^ {i}}.
$$
Let  $c_{i}$  be the number of random variables among  $X_{s,v}$  and  $X_{t,v}$  whose values belong to  $[2^{i} / \epsilon, 2^{i + 1} / \epsilon)$ . Then, we derive
$$
\mathbb {E} \left[ c _ {i} \right] \leq \frac {2 (n - 1)}{e ^ {2 ^ {i}}} \leq \frac {2 n}{2 ^ {2 ^ {i}}} \leq \frac {2 n}{2 ^ {2 ^ {i}}},
$$
where to obtain the inequalities, we use that  $i \geq 1$ . By the Chernoff bound, for appropriately set constant  $b > 0$ , it holds that  $c_{i} \leq b \cdot \max \left( \log n, \frac{2n}{2^{2i}} \right)$  with probability at least  $1 - n^{-5}$ . By the union bound, this claim holds for all the  $O(\log \log n)$  partitions simultaneously and with probability at least  $1 - n^{-4}$ . Hence, with probability at least  $1 - n^{-4}$ , the sum of the edge-weights in  $E(\widetilde{G}) \setminus E(G)$  across all the  $O(\log \log n)$  partitions is at most
$$
\sum_ {i = 1} ^ {\log (5 \log n)} b \cdot \max  \left(\log n, \frac {2 n}{2 ^ {2 i}}\right) \cdot \frac {2 ^ {i + 1}}{\epsilon} \leq \frac {O (\operatorname {p o l y} \log n)}{\epsilon} + 2 b \sum_ {i \geq 0} \frac {2 n}{2 ^ {i} \cdot \epsilon} = \operatorname {p o l y} \log n + 8 b \frac {n}{\epsilon} \in O \left(\frac {n}{\epsilon}\right),
$$
where we used  $\max (\log n,2n / 2^{2i})\leq \log n + 2n / 2^{2i}$ . By taking the union bound over both cases, we have that with probability at least  $1 - n^{-2}$  it holds that the sum of the weights of all edges added to  $G$  is  $O(n / \epsilon)$ . Hence, with probability  $1 - n^{-2}$  at least, the min  $s$ -t cut in  $\widetilde{G}$  has weight at most the min  $s$ -t cut in  $G$  plus  $O(n / \epsilon)$ . This completes our analysis.
4 DP algorithm for multiway cut
In this section we show our approach for proving Theorem 1.3 restated below.
We first develop an algorithm for the non-private multiway  $k$ -cut problem and then use it to prove Theorem 1.3. The algorithm we design invokes a min  $s - t$  cut procedure  $O(\log k)$  times.
4.1 Solving Multiway Cut in  $\log k$  Rounds of Min  $s$ - $t$  Cut
Our new myultiway  $k$ -cut algorithm is presented in Algorithm 2. Algorithm 2 first finds a cut that separates  $s_1, \ldots, s_{k'}$  from  $s_{k' + 1}, \ldots, s_k$ . This separation is obtained by contracting  $s_1, \ldots, s_{k'}$  into a single node called  $s$ , contracting  $s_{k' + 1}, \ldots, s_k$  into a single node called  $t$ , and then running min  $s - t$  cut on this new graph. Afterward, each of the two partitions is processed separately by recursion, and the algorithm outputs the union of the outputs on each of the two partitions.
Algorithm 2: Given a weighted graph  $G$ , this algorithm outputs a multiway  $k$ -cut.
Input :A weighted graph  $G = (V,E,w)$  Terminals  $s_1,\ldots ,s_k$  A min  $s - t$  cut algorithm  $\mathcal{A}$  that with probability  $1 - \alpha$  is  $(1,e(n))$  -approximate   
if  $k = 1$  then   
return  $G$  as one partition   
else   
Let  $k^{\prime} = \left\lfloor \frac{k}{2}\right\rfloor$    
Let  $\widetilde{G}$  be the graph obtained by contracting  $s_1,\dots ,s_{k'}$  into  $s$  and contracting  $s_{k^{\prime} + 1},\ldots ,s_{k}$  into  $t$  Run algorithm  $\mathcal{A}$  on  $\widetilde{G}$  with terminals  $s,t$  to obtain cut  $C$  with partitions  $\widetilde{G}_1$  and  $\widetilde{G}_2$  For  $i = 1,2$  , let  $G_{i}$  be the graph obtained from  $\widetilde{G}_i$  by reversing the contraction of terminal nodes.   
Let  $C_1$  be the output of Algorithm 2 on  $G_{1}$  and  $C_2$  be the output of Algorithm 2 on  $G_{2}$  return  $C_1\cup C_2\cup C.$
We first show that in each recursion level of the algorithm, we can run the min  $s - t$  cut step (Line 6) of all the instances in that level together as one min  $s - t$  cut, so that we only run  $O(\log k)$  many min  $s - t$  cuts.
Lemma 4.1. Algorithm 2 is a reduction of multiway  $k$ -cut on  $n$ -node graphs to  $O(\log k)$  many instances of min  $s-t$  cut on  $O(n)$ -node graphs. Moreover, if  $T(\mathcal{A}, n, m)$  is the running time of  $\mathcal{A}$  on an  $n$ -node  $m$ -edge graphs, Algorithm 2 runs in  $O(\log k \cdot [T(\mathcal{A}, n) + m])$ .
Proof. Let  $G$  be the input graph and suppose that it has  $n$  nodes. If we consider the recursion tree of the algorithm on  $G$ , we can perform the min  $s-t$  cut step (Line 6) of all of the subproblems on one level of the recursion tree by a single min  $s-t$  cut invocation. To see this, assume that  $G_1^{r'}, \ldots, G_r^{r'}$  are in level  $r'$  of the recursion tree for  $r = 2^{r' - 1}$ , and for sake of simplicity, assume that  $k$  is a power of 2. Let  $s_i, t_i$  be the two terminals in  $\widetilde{G}_i^{r'}$  (defined in Line 5) for  $i \in \{1, \ldots, r\}$ . We can think of the collection of graphs  $\widetilde{G}_1^{r'}, \ldots, \widetilde{G}_r^{r'}$  as one graph  $G^{r'}$ . Contract  $s_1, \ldots, s_r$  into  $s$ , and contract  $t_1, \ldots, t_r$  into  $t$ , to obtain the graph  $\widetilde{G}^{r'}$  from  $G^{r'}$ . Then performing a min  $s-t$  cut algorithm on  $\widetilde{G}^{r'}$  is equivalent to performing min  $s-t$  cut on each of  $G_i^{r'}$ , as there is no edge between  $G_i^{r'}$  and  $G_j^{r'}$  for any  $i, j$ . Note that since  $G_1^{r'}, \ldots, G_r^{r'}$  are disjoint and are subgraphs of  $G$ ,  $\widetilde{G}^{r'}$  has at most  $n$  nodes. Moreover, the node contraction processes in each recursion level takes at most  $O(m)$  as each edge is scanned at most once.
To prove that Algorithm 2 outputs a multiway  $k$ -cut that is a 2-approximation of the optimal multiway  $k$ -cut, we first present a few definitions. Given graph  $G$  with node set  $V$ , a partial multiway  $k$ -cut is a set of disjoint node subsets  $V_{1},\ldots ,V_{k}$  such that for each  $i$ , terminal  $s_i\in V_i$ . Note that  $V_{1}\cup \dots \cup V_{k}$  is not necessarily the whole  $V$ . For a set of nodes  $S\subseteq V$ , let  $\delta_G(S)$  be the sum of the weights of the edges with exactly one endpoint in  $S$ , i.e., the sum of the weights of the edges in  $E(S,V\setminus S)$ . Let  $w(E(S))$  be the sum of the weights of the edges with both endpoints in  $S$ . Theorem 4.1 is the main result in this section and we state the proof of it the Supplementary material
Theorem 4.1. Let  $e(n)$  be a convex function and  $G = (V, E, w)$  be a weighted graph. Let  $\mathcal{A}$  be an algorithm that with probability  $1 - \alpha$  outputs a  $(1, e(n))$ -approximate min  $s$ -t cut for  $e(n) = cn / \epsilon$  for constants  $\epsilon > 0, c \geq 0$ . Then, for any graph with terminals  $s_1, \ldots, s_k$  and any partial multiway  $k$ -cut  $V_1, \ldots, V_k$  of it, Algorithm 2 outputs a multiway cut that with probability  $1 - \alpha O(\log k)$  has a value at most  $\sum_{i=1}^{k} \delta(V_i) + w(E(\overline{V})) + O(\log (k)e(n))$  where  $\overline{V} = V \setminus [\cup_{i=1}^k V_i]$ .
A direct Corollary of Theorem 4.1 is a 2-approximation algorithm of the optimal multiway  $k$ -cut. Recall that the novelty of this algorithm is in using  $O(\log k)$  many min  $s-t$  cut runs as opposed to  $O(k)$  which is depicted in Lemma 4.1.
Corollary 4.1. Given a graph  $G$ , integer  $k \geq 2$  and any exact min  $s-t$  cut algorithm, Algorithm 2 returns a multiway  $k$ -cut that is a 2-approximation of the optimal multiway  $k$ -cut.
Proof. Let  $\mathcal{A}$  be an exact  $s$ -t-cut algorithm that is part of the input to Algorithm 2. So we have  $e(n) = 0$ . Let  $C_{ALG}$  be the output of Algorithm 2. By Theorem 4.1,  $C_{ALG}$  is a multiway  $k$ -cut. Let  $C_{OPT}$  be the optimal multiway  $k$ -cut with partitions  $V_1, \ldots, V_k$ . Note that there are no nodes that are not partitioned, and hence  $w(E(\overline{V})) = 0$ . So by Theorem 4.1, we have that  $w(C_{ALG}) \leq \sum_{i=1}^{k} \delta_G(V_i) = 2w(C_{OPT})$ .
4.2 Differentially Private Multiway Cut
Our Differentially Private Multiway cut algorithm is essentially Algorithm 2 where we use a differentially private min  $s - t$  cut algorithm (such as Algorithm 1) in Line 6 to compute a differentially private multiway  $k$ -cut. This concludes the approach for Theorem 1.3 which is a direct corollary of Theorem 4.2 below.
Theorem 4.2. Let  $\epsilon >0$  be a privacy parameter,  $G$  an  $n$ -node graph, and  $\mathcal{A}$  an  $\epsilon$ -DP algorithm that with probability at least  $1 - 1 / n^2$  computes a min  $s$ - $t$  cut with additive error  $O(e(n) / \epsilon)$  on any  $n$ -node graph. Then, Algorithm 2 is a  $(\epsilon \log k)$ -DP algorithm that with probability at least  $1 - O(\log k)\cdot n^{-2}$  finds a multiway  $k$ -cut with multiplicative error 2 and additive error  $O(\log k\cdot e(n) / \epsilon)$ .
Proof. The error guarantees are a direct result of Theorem 4.1 and Lemma 3.3. Hence here we prove the privacy guarantee. Consider the recursion tree of Algorithm 2 which has depth  $O(\log k)$ . As proved by Lemma 4.1, each level of the recursion tree consists of running algorithm  $\mathcal{A}$  on disjoint graphs, and so each level is  $\epsilon$ -DP. By basic composition (see Definition 2.5), since Algorithm 2 is a composition of  $O(\log k)$  many  $\epsilon$ -DP mechanisms, it is  $(\epsilon \log k)$ -DP.
We perform an empirical evaluation on the additive error of Algorithm 1 and validate our theoretical bounds.
Set-up. As our base graph, we use email-Eu-core network [22] which is an undirected unweighted 1,005-node 25,571-edge graph. This graph represents email exchanges, where two nodes  $u$  and  $v$  are adjacent if the person representing  $u$  has sent at least one email to the person representing  $v$  or vice-versa. However, this graph does not account for multiple emails sent between two individuals. To make the graph more realistic, we add random weights to the edges from the exponential distribution with a mean of 40 (rounded to an integer) to denote the number of emails sent between two nodes. We take 10 percent of the nodes and contract them into a terminal node  $s$ , and take another 10 percent of the nodes and contract them into another terminal node  $t$ . We make different instances of the problem by choosing the nodes that contract into  $s$  or  $t$  uniformly at random. Note that node contraction into terminals is a standard practice in real-world min  $s - t$  cut instances, as there are often nodes with
Figure 1: (left) Terminal cut and private cut with error bars over 50 graph instances, for  $\epsilon = 0.5$ . For each instance, Algorithm 1 was run 100 times. (right) Average terminal cut value and private cut value over different values of  $\epsilon$ . The  $x$  axis is  $\frac{1}{\epsilon}$ . For each value of  $\epsilon$ , the set up is the same as in the left figure.
predetermined partitions and to make a  $s-t$  cut (or multiway  $k$ -cut) instance one needs to contract these nodes into one terminal node for each partition. We take the floor of the values  $X_{s,u}$  and  $X_{t,u}$  to obtain integral weights.
**Baseline.** Let  $C_s$  be the cut where one partition consists of only  $s$ , and let  $C_t$  be the cut where one partition consists of only  $t$ . We compare the cut value output by Algorithm 1 against  $\min(w(C_s), w(C_t))$ . Particularly, if  $C_0$  is the min  $s-t$  cut of an instance and Algorithm 1 outputs  $C_{alg}$ , then we compare relative errors  $\frac{w(C_{alg}) - w(C_0)}{w(C_0)}$  and  $\frac{\min(w(C_s), w(C_t)) - w(C_0)^*}{w(C_0)}$ . We refer to the former value as the private cut relative error, and to the later value as the terminal cut relative error.
Results. We first set  $\epsilon = 0.5$ . We then consider 50 graph instances, and for each of them, we evaluate the average private error over 50 rounds of randomness used in Algorithm 1, see Figure 1(left) for the results. For almost all instances, the private cut error (including the error bars) is less than the terminal cut error. Moreover, the private cut error is quite stable, and far below the theoretical bound of  $n / \epsilon$ . We next change  $\epsilon$  and repeat the set-up above for each  $\epsilon \in \{\frac{1}{15}, \frac{1}{14}, \dots, \frac{1}{2}, 1\}$ . For each  $\epsilon$  in this set we produce 50 graph instances, measure the average private error over 100 rounds of randomness for each, and then average the terminal cut error and mean private cut error for each graph instance. In this way, we obtain an average value for terminal cut and private cut errors for each of the  $\epsilon$  values; we refer to Figure 1(right) for the results. The linear relationship between the private cut error and  $\frac{1}{\epsilon}$  can be observed in Figure 1(right).
6 Conclusion And Future Work
In this work, we investigate the differential privacy of the min  $s-t$  cut and of, its generalization, the multiway  $k$ -cut problem. For the first problem, we show almost tight lower- and upper-bounds in terms of the error guarantee. Moreover, our DP algorithm is as fast as the fastest non-private one. For the multiway  $k$ -cut problem, we develop a novel approach that yields only  $O(n/\epsilon \cdot \log k)$  additive error, as opposed to  $O(n/\epsilon \cdot k)$  that standard algorithms lead to, in order to achieve DP. This algorithm might be of independent interest. In particular, the total running time of our multiway  $k$ -cut algorithm is only  $O(\log k)$  more than that of the fastest non-private min  $s-t$  cut. This is in stark contrast with the prior non-private 2-approximate approach, which is  $O(k)$  times slower than the fastest non-private min  $s-t$  cut algorithm.
Comparing our lower- and upper-bounds, it remains to close the gap of  $1 / \epsilon$  multiplicative factor in the additive error of the min  $s - t$  cut problem. It also remains open whether  $O(\log k)$  factor in the additive error for the multiway  $k$ -cut problem is needed. We believe that techniques relying on solving LPs might be able to avoid this dependence on  $O(\log k)$ . Finally, it would be interesting to explore whether smaller additive error is possible for special but interesting classes of graphs.
*As another standard heuristic one could consider a random  $s - t$  cut. We do not include this baseline here as it often has a very high error and the terminal cut does much better than a random cut.
We are grateful to anonymous reviewers for the valuable feedback. S. Mitrovic was supported by the Google Research Scholar Program.