SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Mina Dalirrooyfard*
mina.dalirrooyfard@morganstanley.com
Slobodan Mitrovic*
smitrovic@ucdavis.edu
Yuriy Nevmyvaka
yuriy.nevmvaka@morganstanley.com
Finding min  $s-t$  cuts in graphs is a basic algorithmic tool, with applications in image segmentation, community detection, reinforcement learning, and data clustering. In this problem, we are given two nodes as terminals and the goal is to remove the smallest number of edges from the graph so that these two terminals are disconnected. We study the complexity of differential privacy for the min  $s-t$  cut problem and show nearly tight lower and upper bounds where we achieve privacy at no cost for running time efficiency. We also develop a differentially private algorithm for the multiway  $k$ -cut problem, in which we are given  $k$  nodes as terminals that we would like to disconnect. As a function of  $k$ , we obtain privacy guarantees that are exponentially more efficient than applying the advanced composition theorem to known algorithms for multiway  $k$ -cut. Finally, we empirically evaluate the approximation of our differentially private min  $s-t$  cut algorithm and show that it almost matches the quality of the output of non-private ones.
Min  $s-t$  cut, or more generally Multiway  $k$ -cut, is a fundamental problem in graph theory and occupies a central place in combinatorial optimization. Given a weighted graph and  $k$  terminals, the multiway  $k$ -cut problem asks to divide the nodes of the graph into  $k$  partitions such that (1) each partition has exactly one terminal, and (2) the sum of the weights of edges between partitions, known as the cut value, is minimized [9]. Multiway  $k$ -cut is a clustering method used in a large variety of applications, including energy minimization [21] and image segmentation [27, 32] in vision, reinforcement learning [25], community detection [33, 29] and many other learning and clustering tasks [4, 5, 30, 23].
The above applications are carried on large data sets, and more and more frequently those applications are executed on sensitive data. Hence, designing algorithms that preserve the privacy of data has been given substantial attention recently. A widely used and conservative standard of data privacy is differential privacy (DP) developed by Dwork [12] which indicates that an algorithm is differentially private if, given two neighboring databases, it produces statistically indistinguishable outputs. When DP is applied to graph data, two variants of DP were introduced [19]: in edge DP, two neighboring graphs differ in only one edge, and in node DP, two graphs are neighboring if one graph is obtained
from removing one node and all the edges incident to that node from the other graph. In this work, we focus on the edge DP. In a large fraction of cases, the nodes of a network are public but the edge attributes - which are the relationships between nodes - are private.
For many clustering algorithms such as  $k$ -means,  $k$ -median, and correlation clustering, tight or near-tight differentially private algorithms have already been developed [31, 17, 20, 8]. In particular, tight algorithms for differentially private min-cut has been long known [18], where the min-cut problem asks to divide the graph into two partitions such that the cut value is minimized; in the min-cut problem, there is no restriction that two nodes must be on different sides of the cut. Even though min-cut and min  $s-t$  cut might seem similar, the algorithm techniques for min-cut do not extend to min  $s-t$  cut. A glimpse of this difference can be seen in the fact that there exist polynomially many min-cuts in any graph, but there can be exponentially many min  $s-t$  cuts for fixed terminals  $s$  and  $t^*$ .
Our Results and Technical Overview In this paper, we provide the edge-differentially private algorithm for min  $s-t$  cut and prove that it is almost tight. To the best of our knowledge, this is the first DP algorithm for the min  $s-t$  cut problem.
Our first result is an  $\epsilon$ -private algorithm for min  $s-t$  cut with  $O\left(\frac{n}{\epsilon}\right)$  additive error.
Theorem 1.1. For any  $\epsilon >0$  and for weighted undirected graphs, there is an algorithm for min  $s - t$  cut that is  $(\epsilon ,0)$  -private and with high probability returns a solution with additive error  $O\left(\frac{n}{\epsilon}\right)$ . Moreover, the running time of this private algorithm is the same as the running time of the fastest non-private one.
Moreover, our proof of Theorem 1.1 extends to when an edge-weight changes by at most  $\tau$  in two neighboring graphs. In that case, our algorithm is  $(\epsilon, 0)$ -private with  $O(n \cdot \tau / \epsilon)$  additive error.
Furthermore, our approach uses the existing min  $s-t$  cut algorithms in a black-box way. More precisely, our method changes the input graph in a specific way, i.e., it adds  $2(n-1)$  edges with special weight, and the rest of the processing is done by an existing algorithm. In other words, our approach automatically transfers to any computation setting – centralized, distributed, parallel, streaming – as long as there is a non-private min  $s-t$  cut for that specific setting. The main challenge with this approach is showing that it actually yields privacy. Perhaps surprisingly, our approach is almost optimal in terms of the approximation it achieves. Specifically, we complement our upper-bound by the following lower-bound result.
Theorem 1.2. Any  $(\epsilon, \delta)$ -differential private algorithm for min  $s$ - $t$  cut on  $n$ -node graphs requires expected additive error of at least  $n/20$  for any  $\epsilon \leq 1$  and  $\delta \leq 0.1$ .
Note that Theorem 1.2 holds regardless of the graph being weighted or unweighted. Given an unweighted graph  $G$ , let the cut  $C_s(G)$  be the  $s-t$  cut where node  $s$  is in one partition and all the other nodes are in the other partition. The algorithm that always outputs  $C_s(G)$  is private, and has the additive error of  $O(n)$  since the value of  $C_s(G)$  is at most  $n-1$  in unweighted graphs. Thus one might conclude that we do not need Theorem 1.1 for a private algorithm, as one cannot hope to do better than  $O(n)$  additive error by Theorem 1.2. However, for weighted graphs, this argument fails when  $C_s(G)$  has a very large value due to weighted edge incident to  $s$ , or when a graph has parallel unweighted edges. In Section 5, we show an application in which one computes a weighted min  $s-t$  cut while as the input receiving an unweighted graph. We further evaluate our theoretical results on private min  $s-t$  cut in Section 5 and show that despite our additive approximation, our approach outputs a cut with the value fairly close to the min  $s-t$  cut. Moreover, we show that it is in fact more accurate than more natural heuristics (such as outputting  $C_s(G)$ ) for a wide range of the privacy parameter  $\epsilon$ .
Theorem 1.1 and Theorem 1.2 depict an interesting comparison to differentially private min cut algorithms. Gupta et al [18] provide an  $\epsilon$ -private algorithm with  $O(\log n / \epsilon)$  additive error for min cut and shows that any  $\epsilon$ -private algorithm requires  $\Omega (\log n)$  additive error. The large gap between
*Take  $n - 2$  nodes  $v_{1},\ldots ,v_{n - 2}$  in addition to  $s$  and  $t$ , and add  $v_{i}$  to both  $s$  and  $t$  for all  $i$ . Suppose the weight of all edges is 1. There are  $2^{n - 2} \min s - t$  cuts, as each min s-t cut should remove exactly one edge from each  $s - v_{i} - t$  path; either  $(s,v_{i})$  or  $(v_{i},t)$  is fine. In this example, there are  $n - 2$  min cuts: for each  $i$ , remove edges  $(v_{i},s)$  and  $(v_{i},t)$  and this would be a cut of size 2. Moreover, one can show that the number of min-cuts is polynomial for any graph, there are at most  $\binom{n}{2}$ ; please see [11].
$\epsilon$ -private algorithms for the min cut and min  $s-t$  cut could provide another reason one cannot easily extend algorithms for one to the other problem.
While finding (non-private) Min  $s-t$  cut is polynomially solvable using max-flow algorithms [1, 7], the (non-private) Multiway  $k$ -cut problem is NP-hard for  $k \geq 3$  [10] and finding the best approximation algorithm for it has been an active area of research [10, 6, 2, 24]. The best approximation factor for the Multiway cut problem is 1.2965 [28] and the largest lower bound is 1.20016 [3, 24]. From a more practical point of view, there are a few simple 2-approximation algorithms for multiway  $k$ -cut such as greedy splitting [34] which splits one partition into two partitions in a sequence of  $k-1$  steps. Another simple 2-approximation algorithm by [10] is the following: (1) if the terminals of the graph are  $s_1, \ldots, s_k$ , for each  $i$ , contract all the terminals except  $s_i$  into one node,  $t_i$ ; (2) then run min  $s-t$  cut on this graph with terminals  $s_i, t_i$ ; (3) finally output the union of all these  $k$  cuts. This algorithm reduces multiway  $k$ -cut into  $k$  instances of min  $s-t$  cut. Hence it is easy to see that if we use a  $\epsilon$ -private min  $s-t$  cut with additive error  $r$ , we get a  $k\epsilon$ -private multiway  $k$ -cut algorithm with additive error  $kr$  and multiplicative error 2. In a similar way one can make the greedy splitting algorithm private with same error guarantees. Using advanced composition [15, 16] can further reduce this dependency polynomially at the expense of private parameters. We design an algorithm that reduces the dependency on  $k$  of the private multiway cut algorithm to  $\log k$ , which is exponentially smaller dependence than applying the advanced composition theorem. We obtain this result by developing a novel 2-approximation algorithm for multiway  $k$ -cut that reduces this problem to  $\log k$  instances of min  $s-t$  cut. Our result is represented in Theorem 1.3 and we defer the proof to supplementary materials. We note that the running time of the Algorithm of Theorem 1.3 is at most  $O(\log k)$  times the private min  $s-t$  cut algorithm of Theorem 1.1.
Theorem 1.3. For any  $\epsilon >0$ , there exists an  $(\epsilon ,0)$ -private algorithm for multiway  $k$ -cut on weighted undirected graphs that with probability  $1 - 1 / n$  at least returns a solution with value  $2\cdot OPT + O(n\log k / \epsilon)$ , where OPT is the value of the optimal non-private multiway  $k$ -cut.
Finally, we emphasize that all our private algorithms output cuts as opposed to only their value. For outputting a number  $x$  privately, the standard approach is to add Laplace noise  $\mathrm{Lap}(1 / \epsilon)$  to get an  $\epsilon$ -private algorithm with additive error at most  $O(1 / \epsilon)$  with high probability.
Additional implications of our results. First, the algorithm for Theorem 1.1, i.e., Algorithm 1, uses a non-private min  $s - t$  cut algorithm in a black-box manner. Hence, Algorithm 1 essentially translates to any computation setting without asymptotically increasing complexities compared to non-private algorithms. Second, Algorithm 2 creates (recursive) graph partitions such that each vertex and edge appears in  $O(\log k)$  partitions. To see how it differs from other methods, observe that the standard splitting algorithm performs  $k - 1$  splits, where some vertices appear in each of the split computations. There are also LP-based algorithms for the multiway  $k$ -cut problem, but to the best of our knowledge, they are computationally more demanding than the 2-approximate greedy splitting approach. As a consequence, in the centralized and parallel settings such as PRAM, in terms of total work, Algorithm 2 depends only logarithmically on  $k$  while the popular greedy splitting algorithm has a linear dependence on  $k$ . To the best of our knowledge, no existing method matches these guarantees of Algorithm 2.
In this section, we provide definitions used in the paper. We say that an algorithm  $\mathcal{A}$  outputs an  $(\alpha, \beta)$ -approximation to a minimization problem  $P$  for  $\alpha \geq 1, \beta \geq 0$ , if on any input instance  $I$ , we have  $OPT(I) \leq \alpha \mathcal{A}(I) + \beta$  where  $\mathcal{A}(I)$  is the output of  $\mathcal{A}$  on input  $I$  and  $OPT(I)$  is the solution to problem  $P$  on input  $I$ . We refer to  $\alpha$  and  $\beta$  as the multiplicative and additive errors respectively.
Graph Cuts We use  $uv$  to refer to the edge between nodes  $u$  and  $v$ . Let  $G = (V, E, w_G)$  be a weighted graph with node set  $V$ , edge set  $E$  and  $w_G : V^2 \to \mathbb{R}_{\geq 0}$ , where for all edges  $uv \in E$ ,  $w_G(uv) > 0$ , and for all  $uv \notin E$ ,  $w_G(uv) = 0$ . For any set of edges  $C$ , let  $w_G(C) = \sum_{e \in C} w_G(e)$ . If removing  $C$  disconnects the graph, we refer to  $C$  as a cut set. When  $C$  is a cut set, we refer to  $w_G(C)$  as the weight or value of  $C$ . We drop the subscript  $G$  when it is clear from the context. For subsets  $A, B \subseteq V$ , let  $E(A, B)$  be the set of edges  $uv = e \in E$  such that  $u \in A$  and  $v \in B$ .
Definition 2.1 (Multiway  $k$ -cut). Given  $k$  terminals  $s_1, \ldots, s_k$ , a multiway  $k$ -cut is a set of edges  $C \subseteq E$  such that removing the edges in  $C$  disconnects all the terminals. More formally, there
exists  $k$ -disjoint node subsets  $V_{1}, \ldots, V_{k} \in V$  such that  $s_{i} \in V_{i}$  for all  $i$ ,  $\cup_{i=1}^{k} V_{i} = V$ , and  $C = \cup_{i,j \in \{1, \ldots, k\}} E(V_{i}, V_{j})$ . The Multiway  $k$ -cut problem asks for a Multiway  $k$ -cut with the lowest value.
In our algorithms, we use the notion of node contractions which we formally define here.
Definition 2.2 (Node contractions). Let  $Z \subseteq V$  be a subset of nodes of the graph  $G = (V, E, w_G)$ . By contracting the nodes in  $Z$  into node  $t^{\star}$ , we add a node  $t^{\star}$  to the graph and remove all the nodes in  $Z$  from the graph. For every  $v \in V \setminus Z$ , the weight of the edge  $t^{\star}v$  is equal to  $\sum_{z \in Z} w_G(vz)$ . Note that if none of the nodes in  $Z$  has an edge to  $v$ , then there is no edge from  $t^{\star}$  to  $v$ .
Differential Privacy We formally define neighboring graphs and differential private algorithms.
Definition 2.3 (Neighboring Graphs). Graphs  $G = (V, E, w)$  and  $G' = (V, E', w')$  are called neighboring if there is  $uv \in V^2$  such that  $|w_G(uv) - w_{G'}(uv)| \leq 1$  and for all  $u'v' \neq uv$ ,  $u'v' \in V^2$ , we have  $w_G(u'v') = w_{G'}(u'v')$ .
Definition 2.4 (Differential Privacy[12]). A (randomized) algorithm  $\mathcal{A}$  is  $(\epsilon, \delta)$ -private (or  $(\epsilon, \delta)$ -DP) if for any neighboring graphs  $G$  and  $G'$  and any set of outcomes  $O \subset \text{Range}(\mathcal{A})$  it holds
$$
\Pr \left[ \mathcal {A} (G) \in O \right] \leq e ^ {\epsilon} \Pr \left[ \mathcal {A} \left(G ^ {\prime}\right) \in O \right] + \delta .
$$
When  $\delta = 0$ , algorithm  $\mathcal{A}$  is pure differentially private, or  $\epsilon$ -private.
Definition 2.5 (Basic composition [14, 13]). Let  $\epsilon_1,\ldots ,\epsilon_t > 0$  and  $\delta_1,\dots ,\delta_t\geq 0$ . If we run  $t$  (possibly adaptive) algorithms where the  $i$ -th algorithm is  $(\epsilon_i,\delta_i)$ -DP, then the entire algorithm is  $(\epsilon_{1} + \ldots +\epsilon_{t},\delta_{1} + \ldots +\delta_{t})$ -DP.
Exponential distribution. For  $\lambda > 0$ , we use  $X \sim \mathrm{Exp}(\lambda)$  to denote that a random variable  $X$  is drawn from the exponential distribution with parameter  $\lambda$ . In our proofs, we use the following fact.
Fact 2.1. Let  $X \sim \operatorname{Exp}(\lambda)$  and  $z \geq 0$ . Then  $\Pr [X \geq z] = \exp(-\lambda z)$ .
Laplace distribution. We use  $X \sim \mathrm{Lap}(b)$  to denote that  $X$  is a random variable is sampled from the Laplace distribution with parameter  $b$ .
Fact 2.2. Let  $X \sim \operatorname{Lap}(b)$  and  $z > 0$ . Then
$$
\Pr \left[ X > z \right] = \frac {1}{2} \exp \left(- \frac {z}{b}\right) a n d \Pr \left[ | X | > z \right] = \exp \left(- \frac {z}{b}\right).
$$
Fact 2.3. Let  $X, Y \sim \mathrm{Exp}(\epsilon)$ . Then, the distribution  $X - Y$  follows  $\mathrm{Lap}(1 / \epsilon)$ .
We use  $f_{\mathrm{Lap}}^b$  to denote the cumulative distribution function of  $\operatorname{Lap}(b)$ , i.e.,  $f_{\mathrm{Lap}}^b(x) = \frac{1}{2b} \exp \left( -\frac{|x|}{b} \right)$ . We use  $F_{\mathrm{Lap}}^b$  to denote the cumulative distribution function of  $\operatorname{Lap}(b)$ , i.e.,
$$
F _ {\mathrm {L a p}} ^ {b} (x) = \left\{ \begin{array}{l l} \frac {1}{2} \exp \left(\frac {x}{b}\right) & \text {i f} x \leq 0 \\ 1 - \frac {1}{2} \exp \left(- \frac {x}{b}\right) & \text {o t h e r w i s e} \end{array} \right.
$$
When it is clear from the context what  $b$  is, we will drop it from the superscript.
Claim 1. For any  $t \in \mathbb{R}$  and  $\tau \geq 0$  it holds
$$
\frac {F _ {\mathrm {L a p}} ^ {1 / \epsilon} (t + \tau)}{F _ {\mathrm {L a p}} ^ {1 / \epsilon} (t)} \leq e ^ {\tau \epsilon} a n d \frac {f _ {\mathrm {L a p}} ^ {1 / \epsilon} (t + \tau)}{f _ {\mathrm {L a p}} ^ {1 / \epsilon} (t)} \leq e ^ {\tau \epsilon}.
$$
The proof of this claim is standard and, for completeness, appears in the Supplementary material.
3 DP Algorithm for Min  $s - t$  Cut
In this section, we prove Theorem 1.1. Our DP min  $s-t$  cut algorithm is quite simple and is provided as Algorithm 1. The approach simply adds an edge between  $s$  and every other node, and an edge between  $t$  and every other node. These edges have weight drawn from  $\mathrm{Exp}(\epsilon)$ . The challenging part is showing that this simple algorithm actually preserves privacy. Moreover, when we couple the
Algorithm 1: Given a weighted graph  $G$ , this algorithm outputs a DP min  $s - t$  cut.
Input :A weighted graph  $G = (V,E,w)$  Terminals s and t Privacy Parameter  $\epsilon$    
for  $u\in V\setminus \{s,t\}$  do Sample  $X_{s,u}\sim \mathrm{Exp}(\epsilon)$  and  $X_{t,u}\sim \mathrm{Exp}(\epsilon)$  In  $G$  add an edge between  $s$  and  $u$  with weight  $X_{s,u}$  , and an edge between  $t$  and  $u$  with weight  $X_{t,u}$    
return the min  $s - t$  cut in (the modified)  $G$
approximation guarantee of Algorithm 1 with our lower-bound shown in the Supplementary material, then up to a  $1 / \epsilon$  factor Algorithm 1 yields optimal approximation guarantee.
Remark: Technically, there might be multiple min  $s - t$  cuts that can be returned on Line 4 of Algorithm 1. We discuss that, without loss of generality, it can be assumed that there is a unique one. We elaborate on this in the Supplementary material.
3.1 Differential Privacy Analysis
Before we dive into DP guarantees of Algorithm 1, we state the following claim that we use in our analysis. Its proof is given in the Supplementary material.
Lemma 3.1. Suppose  $x$  and  $y$  are two independent random variables drawn from  $\mathrm{Lap}(1 / \epsilon)$ . Let  $\alpha, \beta, \gamma$  be three fixed real numbers, and let  $\tau \geq 0$ . Define
$$
P (\alpha , \beta , \gamma) \stackrel {\text {d e f}} {=} \Pr [ x <   \alpha , y <   \beta , x + y <   \gamma ].
$$
$$
1 \leq \frac {P (\alpha + \tau , \beta + \tau , \gamma)}{P (\alpha , \beta , \gamma)} \leq e ^ {4 \tau \epsilon}.
$$
Lemma 3.2. Let  $\tau$  be the edge-weight difference between two neighboring graphs. Algorithm 1 is  $(4\tau \epsilon, 0)$ -DP.
Proof. Let  $G$  and  $G'$  be two neighboring graphs. Let edge  $e = uv$  be the one for which  $w_e$  in  $G$  and  $G'$  differ; recall that it differs by at most  $\tau$ . To analyze the probability with which Algorithm 1 outputs the same cut for input  $G$  as it outputs for input  $G'$ , we first sample all the  $X_{s,x}$  and  $X_{t,x}$  for  $x \in V \setminus \{s, t, u, v\}$ . Intuitively, the outcomes of those random variables are not crucial for the difference between outputs of Algorithm 1 invoked on  $G$  and  $G'$ . We now elaborate on that.
Consider all the  $s-t$  cuts - not only the minimum ones - in  $G$  before  $X_{s,u}$ ,  $X_{s,v}$ ,  $X_{t,u}$  and  $X_{t,v}$  are sampled; for instance, assume for a moment that those four random variables equal 0. Let  $\mathcal{C}_{\mathrm{all-st-cuts}}(G)$  be all those cuts sorted in a non-decreasing order with respect to their weight. Observe that  $\mathcal{C}_{\mathrm{all-st-cuts}}(G) = 2^{n-2}$ . As we will show next, although there are exponentially many cuts, we will point to only four of them as crucial for our analysis. This will come in very handy in the rest of this proof. We now partition  $\mathcal{C}_{\mathrm{all-st-cuts}}(G)$  into four groups based on which  $u$  and  $v$  are on the same side of the cut as  $s$ . Let  $\mathcal{C}_u(G)$  be the subset of  $\mathcal{C}_{\mathrm{all-st-cuts}}(G)$  for which  $u$  is on the same side of a cut as  $s$  while  $v$  is on the same side of the cut as  $t$ . Analogously we define  $\mathcal{C}_v(G)$ . We use  $\mathcal{C}_{u,v}(G)$  to represent the subset of  $\mathcal{C}_{\mathrm{all-st-cuts}}(G)$  for which  $s$ ,  $u$ , and  $v$  are all on the same side. Finally, by  $\mathcal{C}_{\emptyset}(G)$  we refer to the subset of  $\mathcal{C}_{\mathrm{all-st-cuts}}(G)$  for which  $t$ ,  $u$ , and  $v$  are all on the same side.
Let  $C_u, C_v, C_{u,v}$ , and  $C_\emptyset$  be the min cuts in  $\mathcal{C}_u(G), \mathcal{C}_v(G), \mathcal{C}_{u,v}(G)$ , and  $\mathcal{C}_\emptyset (G)$ , respectively, before  $X_{s,u}, X_{s,v}, X_{t,u}$  and  $X_{t,v}$  are sampled. It is an easy observation that sampling  $X_{s,u}, X_{s,v}, X_{t,u}$  and  $X_{t,v}$  and altering the weight of the edge  $e$  changes the weight of all the cuts in  $\mathcal{C}_u(G)$  by the same amount. This same observation also holds for  $\mathcal{C}_v(G), \mathcal{C}_{u,v}(G)$  and  $\mathcal{C}_\emptyset (G)$ . This further implies that the min  $s-t$  cut of  $G$  after  $X_{s,u}, X_{s,v}, X_{t,u}$  and  $X_{t,v}$  are sampled will be among  $C_u, C_v, C_{u,v}$ , and  $C_\emptyset$ . Observe that these four cuts are also the minimum-weight cuts in  $\mathcal{C}_u(G'), \mathcal{C}_v(G'), \mathcal{C}_{u,v}(G')$ , and  $\mathcal{C}_\emptyset (G')$ , respectively.
In other words, the min  $s$ -t cuts in  $G$  and in  $G'$  are among  $C_u, C_v, C_{u,v}$  and  $C_{\emptyset}$ , but not necessarily the same; this holds both before and after sampling  $X_{s,u}, X_{s,v}, X_{t,u}$  and  $X_{t,v}$ . In the rest of this proof, we show that sampling  $X_{s,u}, X_{s,v}, X_{t,u}$  and  $X_{t,v}$  makes it likely that the min  $s$ -t cuts in  $G$  and  $G'$  are the same cut.
The ratio of probabilities in  $G'$  and  $G$  that  $C_u$  is the min  $s-t$  cut. Simply, our goal is to compute
$$
\frac {\Pr \left[ w _ {G ^ {\prime}} \left(C _ {u}\right) <   w _ {G ^ {\prime}} \left(C _ {v}\right) \wedge w _ {G ^ {\prime}} \left(C _ {u}\right) <   w _ {G ^ {\prime}} \left(C _ {u , v}\right) \wedge w _ {G ^ {\prime}} \left(C _ {u}\right) <   w _ {G ^ {\prime}} \left(C _ {\emptyset}\right) \right]}{\Pr \left[ w _ {G} \left(C _ {u}\right) <   w _ {G} \left(C _ {v}\right) \wedge w _ {G} \left(C _ {u}\right) <   w _ {G} \left(C _ {u , v}\right) \wedge w _ {G} \left(C _ {u}\right) <   w _ {G} \left(C _ {\emptyset}\right) \right]}. \tag {1}
$$
First consider the case  $w_{G}(e) = w_{G^{\prime}}(e) + \tau$ , for some  $\tau > 0$ . For the ease of calculation, for  $x \in \{\emptyset, \{u, v\}\}$ , define  $\Delta_{x} = w_{G}(C_{u}) - w_{G}(C_{x}) - \tau$  and define  $\Delta_{v} = w_{G}(C_{u}) - w_{G}(C_{v})$  before  $X_{s,u}, X_{s,v}, X_{t,u}$  and  $X_{t,v}$  are sampled. In the rest of this proof, we are analyzing the behavior of cuts when  $X_{s,u}, X_{s,v}, X_{t,u}$  and  $X_{t,v}$  are sampled. Then, we have