# Parameterized Optimization in Uncertain Graphs—A Survey and Some Results

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Uncertain Graphs-Definition and Semantics

#### 1.2. Survey of Optimization Problems in Uncertain Graphs

**Coverage in Uncertain Graphs.**Apart from the themes of network flows and connectedness, coverage problems are very practical when the uncertainty is on the survival of the edges. In this framework, the uncertainty is on whether an edge will survive a disaster and the goal is to place facilities in the network such that the expected coverage over the possible worlds is maximized. Each possible world can be thought of as the set of edges which survive a disaster. Formally, an uncertain graph $\mathcal{G}=(V,E,p)$ is a succinct description of the set of possible worlds that can arise due to a disaster in which an edge e is known to survive with probability $p(e)$. The nature of dependence among the edge samples is used to model the nature of edge failures in the event of a disaster. As mentioned earlier, a fixed nature of dependence among the edge samples uniquely defines a probability distribution on the possible worlds. From this point, in this paper, we consider uncertain graphs where the uncertainty is on the survival of an edge (recall, the other possibility is on the uncertain edge weight). Further, all the results we present are for a fixed dependence among edge samples and such a fixed dependence among the edge samples is called a distribution model. Given the motivation of disasters, the distribution model is specified based on the dependence among the edge failures (recall, the failure probability of an edge e is $1-p(e)$). In this framework, for a fixed distribution model, the function to be optimized is the coverage function.

**Definition**

**1**

**.**The input consists of an uncertain graph $\mathcal{G}=(V,E,p)$ and an integer k. The goal is to compute a k-sized vertex set S which maximizes the expected total weight of the vertices which are at distance at most r from S. The expectation is over the possible worlds represented by the uncertain graph.

**Coverage in Social Networks.**Coverage problems also have a natural interpretation in social networks. The dynamics of a social network based on the word-of-mouth effect have been of significant interest in marketing and consumer research [30], where a social network is referred to as an interpersonal network. The work by Brown and Reingen [30] state the different hypotheses for estimating the amount of uncertainty in a relationship between two persons in an interpersonal network. With the advent of social networks on the internet, the works of Domingos and Richardson [4,31] formalized the questions relating to the effective marketing of a product based on interpersonal relationships in a social network. In 1969, the work of Bass [32] had modeled the adoption of products as a diffusion process as a global phenomenon, independent of the interpersonal relationships between people in a society. Kempe, Kleinberg and Tardos (KKT) [14] brought together the earlier works on adoption of products in an interpersonal network and posed the question of selecting the most influential nodes in a social network with an aim to influence the maximum number of people to adopt a certain product or opinion. They considered the uncertainty in the social network to be the influence exerted by one individual on another individual and this is naturally modeled as an uncertain graph $\mathcal{G}=(V,E,p)$. The propagation of influence is modeled as a diffusion process which is a function of time as in Bass [32]. The influence maximization problems in KKT are considered under distribution models, which are described as diffusion phenomena, called the Independent Cascade model and the Linear Threshold Model. Among these two models, the Independent Cascade model is defined as a sampling procedure whose outcome is an edge subgraph of a given uncertain graph. Thus, this model is more relevant for our study of uncertain graphs and the IC model is defined in Section 2. In the influence maximization problems considered in KKT [14], the aim is to select k influential people S such that the expected number of people connected to S is maximized. The expected number of vertices, over the distribution model, connected to a set S is called the influence of a set S or the expected coverage of S and is denoted by $\sigma (S)$. Thus, the influence maximization problem is to find the set $\underset{S\subseteq V,\left|S\right|=k}{arg\; max}\sigma (S)$. In Section 2, we discuss the computational complexity of $\sigma (S)$ for different distribution models. Thus, the influence maximization problem in Reference [14] is essentially a facility location problem where the distribution model is specified by a diffusion phenomenon, $r=\infty $ and each input instance consists of an uncertain network and a budget k.

**Coverage and Facility Location.**The survey due to Snyder [33] in 2006 collects the vast body of results on the facility location problem in uncertain graphs into a single research article. The earliest result known to us, due to Daskin [19] who formulated the maximum expected coverage problem where vertices are uncertain, is different from our case where the uncertainty is on the edges and the vertices are known. To the best of our knowledge, it was Daskin’s work [19] that considered the general setting of dependence among vertex failures. In this case, the probability distribution would have been defined uniquely on the possible worlds which would have been the set of induced subgraphs. Subsequently, many variants of the facility location problem for uncertain graphs have been studied for different distribution models where the uncertainty is on the survival of edges.

**Finding Communities in Uncertain Graphs.**Finding communities is a significant problem in social network and in bioinformatics. A natural graph theoretic model for a community is a dense subgraph. A well-known dense subgraph is the core. Given an integer d, a graph $G=(V,E)$ is said to be d-core if degree of every vertex $v\in V$ is at least d. The way of obtaining the unique maximal induced subgraph of a graph G which is a d-core is to repeatedly discard vertices of degree less than d. If the procedure terminates with a non-empty graph, then the graph is a d-core of the graph G. A vertex v is said to be in a d-core if there is a d-core which contains v. This idea is generalized to the uncertain graph framework as follows: Given an uncertain graph $\mathcal{G}=(V,E,p)$ and the distribution model is the RF model, the d-core probability of a vertex $v\in V$, denoted by ${q}_{d}(v)$, is the probability that v is in the d-core of a possible world. In other words, ${q}_{d}(v)=\sum _{H\u2291\mathcal{G}}P(H)I(H,d,v)$, where $I(H,d,v)$ is an indicator function that takes value one if and only if there is a d-core of H that contains v, and $P(H)$ is probability of the possible world H. In this setting, we consider the $(d,\theta )$-core problem defined by Peng et al. [16]. We refer to this problem as the Individual Core problem and it is defined as follows.

**Definition**

**2**

**.**The input consists of an uncertain graph $\mathcal{G}=(V,E,p)$, an integer d and a probability threshold $\theta \in [0,1]$. The objective is to compute an $S\subseteq V$ such that for each $v\in S$, ${q}_{d}(v)\ge \theta $.

**Definition**

**3**

**.**Given an uncertain graph $\mathcal{G}=(V,E,p)$, an integer d and a probability threshold $\theta \in (0,1]$, then the aim of the Probabilistic-Core problem is to find a set $K\subseteq V$ such that the $Pr(K\phantom{\rule{4.pt}{0ex}}\mathit{is}\phantom{\rule{4.pt}{0ex}}a\phantom{\rule{4.pt}{0ex}}\mathit{d}-\mathit{core}\phantom{\rule{4.pt}{0ex}}\mathit{in}\phantom{\rule{4.pt}{0ex}}\mathcal{G})$ is at least θ.

#### 1.3. Our Questions and Results

## 2. Distribution Models for Uncertain Graphs

#### 2.1. Random Failure Model

#### 2.2. Independent Cascade Model

**Observation**

**1.**

#### 2.3. Set-Based Dependency (SBD) Model

**Observation**

**2.**

#### 2.4. Linear Reliable Ordering (LRO) Model

**Lemma**

**1**

**.**For $0\le i\le m$, the probability of the possible world ${G}_{i}$ is given by

## 3. Definitions Related to Graphs

**Definition**

**4**

**.**A tree decomposition of a graph G is a pair $(X,H)$ such that H is a tree and $X=\{{X}_{i}\subseteq V:i\in H\}$. For each node $i\in H$, ${X}_{i}$ is referred to as bag of i. The following three conditions hold for a tree decomposition $(X,H)$ of the graph G.

- (a)
- For each vertex $v\in V$, there is a node $i\in H$ such that $v\in {X}_{i}$.
- (b)
- For each edge $uv\in E$, there is a node $i\in H$ such that $u,v\in {X}_{i}$.
- (c)
- For each vertex $v\in V$, the induced subtree of the nodes in H that contains v is connected.

**Definition**

**5**

**.**A nice tree decomposition is a tree decomposition, rooted by a node r with ${X}_{r}=\varnothing $ and each node in the tree decomposition is one of the following four type of nodes.

- 1.
**Leaf node.**A node $i\in H$ with no child and ${X}_{i}=\varnothing $.- 2.
**Introduce node.**A node $i\in H$ with one child j such that ${X}_{i}={X}_{j}\cup \left\{v\right\}$ for some $v\notin {X}_{j}$.- 3.
**Forget node.**A node $i\in H$ with one child j such that ${X}_{i}={X}_{j}\backslash \left\{v\right\}$ for some $v\in {X}_{j}$.- 4.
**Join node.**A node $i\in H$ with two children j and g such that ${X}_{i}={X}_{j}={X}_{g}$.

## 4. Max-Exp-Cover-1-RF Problem is FPT by $(\mathrm{Treewidth}\xb7\Delta $)

**Definition**

**6.**

**Lemma**

**2.**

**Proof.**

#### 4.1. Recursive Formulation of the Value of a Solution

**Note:**f is dependent on S and i and wherever f is used, it must be used as per the definition at the corresponding node in the tree decomposition.

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

#### 4.2. Bottom-Up Computation of an Optimal Set

**Leaf node.**Let $i\in H$ be a leaf node with bag ${X}_{i}=\varnothing $. The only possible five-way partition of an empty set is the set with five empty sets and the budget is $b=0$. The only valid SN-function is $f:\varnothing \to \varnothing $. Therefore, the value of the corresponding row in the table is

**Introduce node.**Let i be an introduce node with child j such that ${X}_{i}={X}_{j}\cup \left\{v\right\}$ for some $v\notin {X}_{j}$. Let $0\le b\le k$ be an integer, $\mathtt{P}=(A,C,L,R,B)$ be a five-way partition of ${X}_{i}$ and f be the SN-function defined on ${X}_{i}^{+}$. The computation of the table entry is split into two cases, depending on whether the vertex v belongs to the set A in the partition $\mathtt{P}$ or not.

**Case $v\in A$.**Define ${C}_{v}=\{u\in {X}_{i}\backslash A\mid v\in N(u)\cap f(u)\}$. Let ${\mathtt{P}}_{j}$ denote the partition of ${X}_{j}$ obtained by removing vertex v from the set A of the partition $\mathtt{P}$. Let ${f}_{j}:{X}_{j}\to {2}^{V}$ be the SN-function defined as follows:$$\begin{array}{c}\hfill {f}_{j}(u)=\left(\right)open="\{"\; close>\begin{array}{cc}f(u)\backslash \left\{v\right\}\hfill & \phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}u\in {C}_{v}\hfill \\ f(u)\hfill & \phantom{\rule{4.pt}{0ex}}\mathrm{otherwise}.\hfill \end{array}\end{array}$$$$\begin{array}{ccc}\hfill {T}_{i}[b,\mathtt{P},f].\mathtt{Solution}& =& {T}_{j}[b-1,{\mathtt{P}}_{j},{f}_{j}].\mathtt{Solution}\cup \left\{v\right\}\hfill \\ \hfill {T}_{i}[b,\mathtt{P},f].\mathtt{Value}& =& {T}_{j}[b-1,{\mathtt{P}}_{j},{f}_{j}].\mathtt{Value}+w(v)+{\mathcal{C}}_{f}({C}_{v},v)\hfill \end{array}$$**Case $v\notin A$.**Since v is in ${X}_{i}$ but not in ${X}_{j}$ it follows that $N(v)\cap {X}_{i}^{+}\subseteq {X}_{i}$. Therefore, the coverage of the vertex v by the vertices that occur only in ${X}_{j}^{+}\backslash {X}_{j}$ is zero. Let ${\mathtt{P}}_{j}$ be the partition of ${X}_{j}$ obtained by removing the vertex v from the appropriate set in the partition $\mathtt{P}$. The SN-function ${f}_{j}$ is defined as follows on the set ${X}_{j}^{+}$: For $u\in {X}_{i}\backslash \left\{v\right\}$, ${f}_{j}(u)=f(u)$.$$\begin{array}{ccc}\hfill {T}_{i}[b,\mathtt{P},f].\mathtt{Solution}& =& {T}_{j}[b,{\mathtt{P}}_{j},{f}_{j}].\mathtt{Solution}\hfill \\ \hfill {T}_{i}[b,\mathtt{P},f].\mathtt{Value}& =& {T}_{j}[b,{\mathtt{P}}_{j},{f}_{j}].\mathtt{Value}+{\mathcal{C}}_{f}(v,A)\hfill \end{array}$$

**Forget node.**Let i be a forget node with child j such that ${X}_{i}={X}_{j}\backslash \left\{v\right\}$ for some $v\in {X}_{j}$. Let $0\le b\le k$ be a budget, $\mathtt{P}=(A,C,L,R,B)$ be a five-way partition of ${X}_{i}$ and f be an SN-function. We consider the following five-way partitions of ${X}_{j}$.

- ${\mathtt{P}}_{1}=(A\cup \left\{v\right\},C,L,R,B)$
- ${\mathtt{P}}_{2}=(A,C\cup \left\{v\right\},L,R,B)$
- ${\mathtt{P}}_{3}=(A,C,L\cup \left\{v\right\},R,B)$
- ${\mathtt{P}}_{4}=(A,C,L,R\cup \left\{v\right\},B)$
- ${\mathtt{P}}_{5}=(A,C,L,R,B\cup \left\{v\right\})$

**Join node.**Let i be a join node with children j and g such that ${X}_{i}={X}_{j}={X}_{g}$. Let $0\le b\le k$ be a budget, $\mathtt{P}=(A,C,L,R,B)$ be a five-way partition of ${X}_{i}$ and f be an SN-function. We consider the sets ${\mathcal{U}}_{j}$ and ${\mathcal{U}}_{g}$ of all SN-functions defined on ${X}_{j}^{+}$ and ${X}_{g}^{+}$, respectively, satisfying the following conditions:

- For each $v\in L$, ${f}_{j}(v)=f(v)$ and ${f}_{g}(v)=\varnothing $.
- For each $v\in R$, ${f}_{j}(v)=\varnothing $ and ${f}_{g}(v)=f(v)$.
- For each $v\in B$, ${f}_{j}(v)$ and ${f}_{g}(v)$ are defined as follows. For each partition ${f}_{1}(v)\cup {f}_{2}(v)=f(v)\cap {X}_{i}^{+}$, ${f}_{j}(v)={f}_{1}(v)\cup (f(v)\backslash {X}_{i}^{+})$ and ${f}_{g}(v)={f}_{2}(v)\cup (f(v)\backslash {X}_{i}^{+})$.

**Lemma**

**5.**

**Proof.**

**Case when i is an introduce node.**Let j be the child of i and ${X}_{i}={X}_{j}\cup \left\{v\right\}$ for some $v\notin {X}_{j}$. We now consider two cases depending on whether v belongs to A or not.

**Case $v\in A$.**Let ${\mathtt{P}}_{j}$ be the partition of ${X}_{j}$ obtained from $\mathtt{P}$ by removing v from A. Let ${f}_{j}$ be the SN-function on ${X}_{j}^{+}$ such that ${f}_{j}(u)=f(u)\backslash v$, if $u\in {C}_{v}$ and ${f}_{j}(u)=f(u)$, otherwise. We know from our claimed optimality of ${S}^{\prime}$, and that ${S}^{\prime}\cap {X}_{i}=S\cap {X}_{i}=A$, and the value of ${T}_{i}[b,\mathtt{P},f]$, that$$\begin{array}{ccc}\hfill {\mathcal{C}}_{f}({X}_{i}^{+},{S}^{\prime})& =& {\mathcal{C}}_{{f}_{j}}({X}_{j}^{+},{S}^{\prime}\backslash \left\{v\right\})+{\mathcal{C}}_{f}({C}_{v},v)\hfill \\ & >& {\mathcal{C}}_{f}({X}_{i}^{+},S)={\mathcal{C}}_{{f}_{j}}({X}_{j}^{+},S\backslash \left\{v\right\})+{\mathcal{C}}_{f}({C}_{v},v).\hfill \end{array}$$**Case $v\notin A$.**Let ${\mathtt{P}}_{j}$ be the partition of ${X}_{j}$ obtained by removing v from the appropriate set in the partition $\mathtt{P}$. Let ${f}_{j}$ be the SN-function on ${X}_{j}^{+}$ such that ${f}_{j}(u)=f(u)$ for each $u\in {X}_{j}^{+}$. We know from our claimed optimality of ${S}^{\prime}$, and that ${S}^{\prime}\cap {X}_{i}=S\cap {X}_{i}=A$, and the value of ${T}_{i}[b,\mathtt{P},f]$ that ${\mathcal{C}}_{f}({X}_{i}^{+},{S}^{\prime})={\mathcal{C}}_{{f}_{j}}({X}_{j}^{+},{S}^{\prime})+{\mathcal{C}}_{f}(v,A)>{\mathcal{C}}_{f}({X}_{i}^{+},S)={\mathcal{C}}_{{f}_{j}}({X}_{j}^{+},S)+{\mathcal{C}}_{f}(v,A)$. Therefore, it follows that ${\mathcal{C}}_{{f}_{j}}({X}_{j}^{+},{S}^{\prime})>{\mathcal{C}}_{{f}_{j}}({X}_{j}^{+},S)$. In other words, we have concluded that the value for the row $(b,{\mathtt{P}}_{j},{f}_{j})$ in ${T}_{j}$ is not the optimum value. This contradicts our premise at node j, which is of height at most $h-1$ for which, by induction hypothesis, the table maintains the optimal values. Therefore, our assumption that ${T}_{i}[b,\mathtt{P},f]$ is not optimum is wrong.

**Forget node.**We know that ${X}_{i}^{+}={X}_{j}^{+}$, and v is in ${X}_{j}$ but not in ${X}_{i}$, it follows that $N(v)\cap {X}_{i}=\varnothing $ and $N(v)\subseteq {X}_{j}^{+}$. Define ${f}_{j}$ to be the SN-function at ${X}_{j}^{+}$ such that ${f}_{j}(u)=f(u)$ for each $u\in {X}_{i}$ and ${f}_{j}(v)=N(v)$. We have assumed ${T}_{i}[b,\mathtt{P},f].\mathtt{Value}<{\mathcal{C}}_{f}({X}_{i}^{+},{S}^{\prime})$. Further, since ${X}_{i}^{+}={X}_{j}^{+}$ and due to the definition of ${f}_{j}$, ${\mathcal{C}}_{f}({X}_{i}^{+},{S}^{\prime})={\mathcal{C}}_{{f}_{j}}({X}_{j}^{+},{S}^{\prime})$. Since ${T}_{i}[b,\mathtt{P},f]$ is computed identically from some row in ${T}_{j}$, let us say $(b,{\mathtt{P}}_{j},{f}_{j})$, it follows that ${T}_{j}[b,{\mathtt{P}}_{j},{f}_{j}].\mathtt{Value}<{\mathcal{C}}_{{f}_{j}}({X}_{j}^{+},{S}^{\prime})$. This contradicts the premise that the table ${T}_{i}$ is at the lowest height in the tree decomposition at which an entry is sub-optimal. Therefore, our premise is wrong.

**Join node.**We assume that ${S}^{\prime}$ is indeed a better solution than S for the table entry $(b,\mathtt{P},f)$ of ${T}_{i}$. Let ${S}_{j}^{\prime}=S\cap ({X}_{j}^{+}\backslash {X}_{j})$ and ${S}_{g}^{\prime}=S\cap ({X}_{g}^{+}\backslash {X}_{g})$. Let ${b}_{j}^{\prime}=\left|{S}_{j}^{\prime}\right|$. Let ${\mathtt{P}}_{j}^{\prime}=(A,C\cup R,{L}_{j}^{\prime},{R}_{j}^{\prime},{B}_{j}^{\prime})$ and ${\mathtt{P}}_{g}^{\prime}=(A,C\cup L,{L}_{g}^{\prime},{R}_{g}^{\prime},{B}_{g}^{\prime})$ be the partitions of ${X}_{j}$ and ${X}_{g}$ defined using ${S}_{j}^{\prime}$ and ${S}_{g}^{\prime}$, respectively. Note that, ${L}_{j}^{\prime}\cup {R}_{j}^{\prime}\cup {B}_{j}^{\prime}=L\cup B$ and ${L}_{g}^{\prime}\cup {R}_{g}^{\prime}\cup {B}_{g}^{\prime}=R\cup B$. Let ${f}_{j}^{\prime}$ be the SN function on ${X}_{j}^{+}$ such that ${f}_{j}^{\prime}(u)=f(u)$ for $u\in A\cup C\cup L\cup B$ and ${f}_{j}^{\prime}(u)=\varnothing $ for $u\in R$. Let ${f}_{g}^{\prime}$ be the SN function on ${X}_{g}^{+}$ such that ${f}_{g}^{\prime}(u)=f(u)$ for $u\in A\cup C\cup R$, ${f}_{g}^{\prime}(u)=\varnothing $ for $u\in L$ and ${f}_{g}^{\prime}(u)=f(u)\backslash {S}_{j}^{\prime}$. The coverage ${\mathcal{C}}_{f}({X}_{i}^{+},{S}^{\prime})$ can be written as ${\mathcal{C}}_{f}({X}_{i}^{+},{S}^{\prime})={\mathcal{C}}_{{f}_{j}^{\prime}}({X}_{j}^{+},{S}_{j}^{\prime})+{\mathcal{C}}_{{f}_{g}^{\prime}}({X}_{g}^{+},{S}_{g}^{\prime})-w(A)-{\mathcal{C}}_{f}(C,A)$, where the coverage of ${X}_{j}^{+}$ by ${S}_{j}^{\prime}$ and ${X}_{g}^{+}$ by ${S}_{g}^{\prime}$ are restricted to the partitions ${\mathtt{P}}_{j}^{\prime}$ and ${\mathtt{P}}_{g}^{\prime}$.

**Running Time.**There are $\mathcal{O}(n\phantom{\rule{4pt}{0ex}}tw)$-many nodes in the nice tree decomposition H. Each node $i\in H$ has a maximum of $(k+1){(5\xb7{2}^{\Delta})}^{tw}$ entries. The ${2}^{\Delta \xb7tw}$ comes from the fact that at each vertex in a bag, we enumerate all subsets of neighbors to come up with the SN-functions. It is clear from the description that at the leaf nodes, introduce nodes, and forget nodes, the update time is $\mathcal{O}(tw)$. At a join node, the time taken to compute an entry depends on three basic operations. The optimal partitions ${\mathtt{P}}_{j}$ and ${\mathtt{P}}_{g}$ are computed in ${3}^{\left|L\right|+\left|R\right|+2\left|B\right|}$ time and budget distribution can be done in $\mathcal{O}(k)$ time. The costliest operation is to enumerate the different SN-functions ${f}_{j}$ and ${f}_{g}$ for the given SN-function f. Since we need to consider all the ${2}^{\Delta}$-possible ways of distributing the $f(v)$ for each vertex v, the distribution takes $\mathcal{O}({({2}^{\Delta})}^{\left|B\right|})$ time. Therefore, the running time for an entry $(b,\mathtt{P},f)$ in a join node takes $\mathcal{O}(k\xb7{3}^{\left|L\right|+\left|R\right|+2\left|B\right|}\xb7{({2}^{\Delta})}^{\left|B\right|})$ time, and this is ${2}^{\mathcal{O}(\Delta \xb7tw)}$. This analysis of the running time and Lemma 5 complete the proof of the following theorem which is our main result.

**Theorem**

**1.**

## 5. Parameterized Complexity of Probabilistic-Core Problem

#### 5.1. An Exact Algorithm for the Probabilistic-Core-LRO Problem

Algorithm 1:Probabilistic-Core-LRO |

**Lemma**

**6.**

**Proof.**

**Observation**

**3.**

**Proof.**

#### 5.2. Parameterized Complexity of the Probabilistic-Core-RF Problem

**Theorem**

**2.**

**Proof.**

#### 5.3. The Probabilistic-Core-RF Problem is FPT by Treewidth

- for each $v\in {X}_{i}\cap K$, $|N(v)\cap K|=\alpha (v)+\beta (v)$ and $|N(v)\cap K\cap {X}_{i}|=\alpha (v)$, and
- for each $v\in K\backslash {X}_{i}$, $|N(v)\cap K|=d$.

#### 5.3.1. Dynamic Programming

**Leaf node.**Let i be a leaf node with bag ${X}_{i}=\varnothing $. We have one row in the table ${T}_{i}$. Let $\alpha ,\beta :\varnothing \to \left\{0\right\}$ be the pair of functions corresponding to the row and the value of the table entry is given as:

**Insert node.**Let i be an insert node with a child j, and let ${X}_{i}={X}_{j}\cup \left\{v\right\}$ for some $v\notin {X}_{j}$. The row ${T}_{i}[\alpha ,\beta ]$ is computed based on the value of $\alpha (v)$ and $\beta (v)$. If $\beta (v)>0$, then the row ${T}_{i}[\alpha ,\beta ]$ becomes infeasible since $N(v)\cap {X}_{i}^{+}\subseteq {X}_{i}$. That is,

**Forget node.**Let i be a forget node with a child j, and let ${X}_{i}={X}_{j}\backslash \left\{v\right\}$ for some $v\in {X}_{j}$. From the definition of the tree decomposition, it follows that $N(v)\subseteq {X}_{j}^{+}$. Since $N(v)\subseteq {X}_{j}^{+}$, either v is part of solution with constraint $\alpha (v)+\beta (v)=d$ or v is not part of solution. Let $U=(N(v)\cap {X}_{j})\backslash {\alpha}^{-1}(0)$. For each $0\le a\le min(d,|U|)$ and $Y\subseteq U$ of size a, we define ${\alpha}_{a,Y},{\beta}_{a,Y}:{X}_{j}\to \{0,1,\dots ,d\}$ such that

**Join node.**Let i be a join node with children j and g such that ${X}_{i}={X}_{j}={X}_{g}$. For a function $\alpha $ defined on ${X}_{i}$, we consider the function $\alpha $ for the child node ${X}_{j}$ and for the other child node ${X}_{g}$, consider the function ${\alpha}^{\prime}:{X}_{i}\to \{0,1,\dots ,d\}$ such that $\alpha (u)=0$ for all $u\in {X}_{i}$. Since $\beta $ considers the neighbors from outside ${X}_{i}$, each vertex $v\in {X}_{i}$ with $\beta (v)>0$ will get the d-core neighbors from the set ${X}_{i}^{+}\backslash {X}_{i}$. Since ${X}_{i}^{+}\backslash {X}_{i}=({X}_{j}^{+}\backslash {X}_{j})\cup ({X}_{g}^{+}\backslash {X}_{g})$ and both the sets are disjoint, we divide $\beta (v)$ into two parts. For each vertex $v\in {X}_{i}$, we try all possible ways of dividing $\beta (v)$ into two parts. For $x:{X}_{i}\to \{0,1,\dots ,d\}$ such that for each $v\in {X}_{i}$, $0\le x(v)\le \beta (v)$, we define ${\beta}_{x}:{X}_{j}\to \{0,1,\dots ,d\}$ and ${\beta}_{x}^{\prime}:{X}_{g}\to \{0,1,\dots ,d\}$ to be ${\beta}_{x}(v)=x(v)$ and ${\beta}_{x}^{\prime}(v)=\beta (v)-x(v)$. Let

#### 5.3.2. Correctness and Running Time

**Lemma**

**7.**

**Proof.**

**When i is an introduce node.**Let j be the child of i, and ${X}_{i}={X}_{j}\cup \left\{v\right\}$ for some $v\notin {X}_{j}$. If $\beta (v)>0$, then no feasible solution exists since $N(v)\cap ({X}_{i}^{+}\backslash {X}_{i})=\varnothing $. This is captured in our dynamic programming. In the further cases, we consider $\beta (v)=0$. Let $L=N(v)\cap {X}_{i}$. Consider the case when $\alpha (v)=0$. That is, the vertex v is not part of the solution. The recursive definition of the dynamic programming gives

**When i is a forget node.**Let j be the child of i, and ${X}_{i}={X}_{j}\backslash \left\{v\right\}$ for some $v\in {X}_{j}$. We consider two cases depending on whether v belongs to S. We first consider the case when $v\notin S$. Consider the functions ${\alpha}_{0,\varnothing}$ and ${\beta}_{0,\varnothing}$ as defined in the recursive computation of forget node. Since $v\notin S$, v will get zero degree constraint in both functions $\alpha $ and $\beta $, and other vertices will have same constraints as $\alpha $ and $\beta $ values. Then, the probability ${\rho}_{d}(\alpha ,\beta ,{G}_{i}^{\alpha ,\beta},S)$ can be written as follows:

**When i is a join node.**Let j and g be the children of i, and ${X}_{i}={X}_{j}={X}_{g}$. The set $S\backslash {X}_{i}$ can be partitioned into sets ${S}_{j}$ and ${S}_{g}$ where ${S}_{j}=(S\backslash {X}_{i})\cap {X}_{j}$ and ${S}_{g}=(S\backslash {X}_{i})\cap {X}_{g}$. Let $U=\{u\in {X}_{i}\mid \alpha (v)>0\}$ and ${S}_{i}={X}_{i}\backslash {\alpha}^{-1}(0)$. For each vertex $u\in U$, the degree constraint $\alpha (u)$ should be satisfied by the edges from U to u. Since ${X}_{i}={X}_{j}={X}_{g}$, the degree constraint $\alpha (u)$ is either satisfied in the node j or node g and not in both. Without loss of generality we assume that the degree constraint $\alpha $ is satisfied in the node j and no zero degree constraint $\alpha $ in the node g. Then we define ${\alpha}^{\prime}:{X}_{g}\to \{0,1,\dots ,d\}$ to be for every vertex $v\in {X}_{i}$, ${\alpha}^{\prime}(v)=0$. For each vertex $v\in {X}_{i}$ with $\beta (v)>0$, the degree constraint $\beta (v)$ can be satisfied by the sets ${S}_{j}$ and ${S}_{g}$ together. There exists an integer $x(v)$ such that $x(v)$ neighbors in the core are obtained from the set ${S}_{j}$ and $\beta (v)-x(v)$ neighbors in the core are obtained from the set ${S}_{g}$. Then, there exists a function $x:{X}_{i}\to \{0,1,\dots ,d\}$ such that for each vertex $v\in V$, $0\le x(v)\le \beta (v)$. Using the function x, the probability ${\rho}_{d}(\alpha ,\beta ,{\mathcal{G}}_{i}^{\alpha ,\beta},S)$ can be given as follows:

**Lemma**

**8**

**Lemma**

**9.**

**Proof.**

**Theorem**

**3.**

## 6. Discussion

- Are there efficient reductions between distribution models so that we can classify problems based on the efficiency of algorithms under different distribution models? This question is also of practical significance because the distribution models are specified as sampling algorithms. Consequently, the complexity of expectation computation on uncertain graphs under different distribution models is an interesting new parameterization. Further, one concrete question is whether the LRO model is easier than the RF model for other optimization problems on uncertain graphs. For the two case studies considered in this paper, that is the case.
- While our results do support the natural intuition that a tree decomposition is helpful in expectation computation, it is unclear to us how traditional techniques in parameterized algortihms can be carried over to this setting. In particular, it is unclear to us as to whether for any distribution model, a kernelization based algorithm can give an FPT algorithm on uncertain graphs.
- We have considered the coverage and the core problems on uncertain graphs under the LRO and RF models. However, we have not been able to get an FPT algorithm with the parameter treewidth for Max-Exp-Cover-1-RF. Actually, any approach to avoid the exponential dependence on $(\Delta \phantom{\rule{3.33333pt}{0ex}}.\phantom{\rule{3.33333pt}{0ex}}tw\phantom{\rule{3.33333pt}{0ex}})$ would be very interesting and would give a significant insight on other approaches to evaluate the expected coverage.
- Even though the Individual-Core-RF problem and Probabilistic-Core-RF problem are similar, we have not been able to get an FPT algorithm for the Individual-Core-RF problem with treewidth as the parameter. Even for other structural parameters such as vertex-cover number and feedback vertex set number, FPT results will give a significant insight on the Individual-Core problem.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Añez, J.; Barra, T.D.L.; Pérez, B. Dual graph representation of transport networks. Trans. Res. Part B Methodol.
**1996**, 30, 209–216. [Google Scholar] [CrossRef] - Hua, M.; Pei, J. Probabilistic Path Queries in Road Networks: Traffic Uncertainty Aware Path Selection. In Proceedings of the 13th International Conference on Extending Database Technology (EDBT ’10), Lausanne, Switzerland, 22–26 March 2010; pp. 347–358. [Google Scholar] [CrossRef]
- Asthana, S.; King, O.D.; Gibbons, F.D.; Roth, F.P. Predicting protein complex membership using probabilistic network reliability. Genome Res.
**2004**, 14, 1170–1175. [Google Scholar] [CrossRef][Green Version] - Domingos, P.; Richardson, M. Mining the Network Value of Customers. In Proceedings of the Seventh ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD ’01), San Francisco, CA, USA, 26–29 August 2001; pp. 57–66. [Google Scholar] [CrossRef]
- Frank, H. Shortest Paths in Probabilistic Graphs. Oper. Res.
**1969**, 17, 583–599. [Google Scholar] [CrossRef] - Valiant, L.G. The Complexity of Enumeration and Reliability Problems. SIAM J. Comput.
**1979**, 8, 410–421. [Google Scholar] [CrossRef] - Hoffmann, M.; Erlebach, T.; Krizanc, D.; Mihalák, M.; Raman, R. Computing Minimum Spanning Trees with Uncertainty. In Proceedings of the 25th Annual Symposium on Theoretical Aspects of Computer Science, Bordeaux, France, 21–23 February 2008; pp. 277–288. [Google Scholar] [CrossRef]
- Focke, J.; Megow, N.; Meißner, J. Minimum Spanning Tree under Explorable Uncertainty in Theory and Experiments. In Proceedings of the 16th International Symposium on Experimental Algorithms (SEA 2017), London, UK, 21–23 June 2017; pp. 22:1–22:14. [Google Scholar] [CrossRef]
- Frank, H.; Hakimi, S. Probabilistic Flows Through a Communication Network. IEEE Trans. Circuit Theory
**1965**, 12, 413–414. [Google Scholar] [CrossRef] - Evans, J.R. Maximum flow in probabilistic graphs-the discrete case. Networks
**1976**, 6, 161–183. [Google Scholar] [CrossRef] - Hassin, R.; Ravi, R.; Salman, F.S. Tractable Cases of Facility Location on a Network with a Linear Reliability Order of Links. In Algorithms-ESA 2009, Proceedings of the 17th Annual European Symposium, Copenhagen, Denmark, 7–9 September 2009; Springer: Berlin, Germany, 2009; pp. 275–276. [Google Scholar]
- Hassin, R.; Ravi, R.; Salman, F.S. Multiple facility location on a network with linear reliability order of edges. J. Comb. Optim.
**2017**, 34, 1–25. [Google Scholar] [CrossRef] - Narayanaswamy, N.S.; Nasre, M.; Vijayaragunathan, R. Facility Location on Planar Graphs with Unreliable Links. In Proceedings of the Computer Science-Theory and Applications-13th International Computer Science Symposium in Russia, CSR 2018, Moscow, Russia, 6–10 June 2018; pp. 269–281. [Google Scholar] [CrossRef]
- Kempe, D.; Kleinberg, J.M.; Tardos, É. Maximizing the spread of influence through a social network. In Proceedings of the Ninth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, Washington, DC, USA, 24–27 August 2003; pp. 137–146. [Google Scholar] [CrossRef][Green Version]
- Bonchi, F.; Gullo, F.; Kaltenbrunner, A.; Volkovich, Y. Core decomposition of uncertain graphs. In Proceedings of the 20th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD ’14), New York, NY, USA, 24–27 August 2014; pp. 1316–1325. [Google Scholar] [CrossRef]
- Peng, Y.; Zhang, Y.; Zhang, W.; Lin, X.; Qin, L. Efficient Probabilistic K-Core Computation on Uncertain Graphs. In Proceedings of the 34th IEEE International Conference on Data Engineering (ICDE), Paris, France, 16–19 April 2018; pp. 1192–1203. [Google Scholar] [CrossRef]
- Ball, M.O.; Provan, J.S. Calculating bounds on reachability and connectedness in stochastic networks. Networks
**1983**, 13, 253–278. [Google Scholar] [CrossRef] - Zou, Z.; Li, J. Structural-Context Similarities for Uncertain Graphs. In Proceedings of the 2013 IEEE 13th International Conference on Data Mining, Dallas, TX, USA, 7–10 December 2013; pp. 1325–1330. [Google Scholar] [CrossRef]
- Daskin, M.S. A Maximum Expected Covering Location Model: Formulation, Properties and Heuristic Solution. Transp. Sci.
**1983**, 17, 48–70. [Google Scholar] [CrossRef][Green Version] - Ball, M.O. Complexity of network reliability computations. Networks
**1980**, 10, 153–165. [Google Scholar] [CrossRef] - Karp, R.M.; Luby, M. Monte-Carlo algorithms for the planar multiterminal network reliability problem. J. Complex.
**1985**, 1, 45–64. [Google Scholar] [CrossRef][Green Version] - Provan, J.S.; Ball, M.O. The Complexity of Counting Cuts and of Computing the Probability that a Graph is Connected. SIAM J. Comput.
**1983**, 12, 777–788. [Google Scholar] [CrossRef] - Guo, H.; Jerrum, M. A Polynomial-Time Approximation Algorithm for All-Terminal Network Reliability. SIAM J. Comput.
**2019**, 48, 964–978. [Google Scholar] [CrossRef][Green Version] - Ghosh, J.; Ngo, H.Q.; Yoon, S.; Qiao, C. On a Routing Problem Within Probabilistic Graphs and its Application to Intermittently Connected Networks. In Proceedings of the 26th IEEE International Conference on Computer Communications, Joint Conference of the IEEE Computer and Communications Societies, INFOCOM, Anchorage, AK, USA, 6–12 May 2007; pp. 1721–1729. [Google Scholar] [CrossRef]
- Rubino, G. Network Performance Modeling and Simulation; chapter Network Reliability Evaluation; Gordon and Breach Science Publishers, Inc.: Newark, NJ, USA, 1999; pp. 275–302. [Google Scholar]
- Swamynathan, G.; Wilson, C.; Boe, B.; Almeroth, K.C.; Zhao, B.Y. Do social networks improve e-commerce?: a study on social marketplaces. In Proceedings of the first Workshop on Online Social Networks (WOSN 2008), Seattle, WA, USA, 17–22 August 2008; pp. 1–6. [Google Scholar] [CrossRef]
- White, D.R.; Harary, F. The Cohesiveness of Blocks In Social Networks: Node Connectivity and Conditional Density. Soc. Methodol.
**2001**, 31, 305–359. [Google Scholar] [CrossRef][Green Version] - Papadimitriou, C.H.; Yannakakis, M. Shortest paths without a map. Theor. Comput. Sci.
**1991**, 84, 127–150. [Google Scholar] [CrossRef][Green Version] - Khuller, S.; Moss, A.; Naor, J. The Budgeted Maximum Coverage Problem. Inf. Process. Lett.
**1999**, 70, 39–45. [Google Scholar] [CrossRef] - Brown, J.J.; Reingen, P.H. Social Ties and Word-of-Mouth Referral Behavior. J. Consum. Res.
**1987**, 14, 350–362. [Google Scholar] [CrossRef] - Richardson, M.; Domingos, P.M. Mining knowledge-sharing sites for viral marketing. In Proceedings of the Eighth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, Edmonton, AB, Canada, 23–26 July 2002; pp. 61–70. [Google Scholar] [CrossRef]
- Bass, F.M. A New Product Growth for Model Consumer Durables. Manag. Sci.
**1969**, 15, 215–227. [Google Scholar] [CrossRef] - Snyder, L.V. Facility location under uncertainty: A review. IIE Trans.
**2006**, 38, 547–564. [Google Scholar] [CrossRef] - Eiselt, H.A.; Gendreau, M.; Laporte, G. Location of facilities on a network subject to a single-edge failure. Networks
**1992**, 22, 231–246. [Google Scholar] [CrossRef] - Colbourn, C.J.; Xue, G. A linear time algorithm for computing the most reliable source on a series-parallel graph with unreliable edges. Theor. Comput. Sci.
**1998**, 209, 331–345. [Google Scholar] [CrossRef][Green Version] - Ding, W. Computing the Most Reliable Source on Stochastic Ring Networks. In Proceedings of the 2009 WRI World Congress on Software Engineering, Xiamen, China, 19–21 May 2009; Volume 1, pp. 345–347. [Google Scholar] [CrossRef]
- Ding, W.; Xue, G. A linear time algorithm for computing a most reliable source on a tree network with faulty nodes. Theor. Comput. Sci.
**2011**, 412, 225–232. [Google Scholar] [CrossRef][Green Version] - Melachrinoudis, E.; Helander, M.E. A single facility location problem on a tree with unreliable edges. Networks
**1996**, 27, 219–237. [Google Scholar] [CrossRef] - Nemhauser, G.L.; Wolsey, L.A.; Fisher, M.L. An analysis of approximations for maximizing submodular set functions—I. Math. Program.
**1978**, 14, 265–294. [Google Scholar] [CrossRef] - Cygan, M.; Fomin, F.V.; Kowalik, L.; Lokshtanov, D.; Marx, D.; Pilipczuk, M.; Pilipczuk, M.; Saurabh, S. Parameterized Algorithms; Springer: Berlin, Germany, 2015. [Google Scholar] [CrossRef]
- Sigal, C.E.; Pritsker, A.A.B.; Solberg, J.J. The Stochastic Shortest Route Problem. Oper. Res.
**1980**, 28, 1122–1129. [Google Scholar] [CrossRef] - Guerin, R.A.; Orda, A. QoS routing in networks with inaccurate information: Theory and algorithms. IEEE/ACM Trans. Netw.
**1999**, 7, 350–364. [Google Scholar] [CrossRef] - Günneç, D.; Salman, F.S. Assessing the reliability and the expected performance of a network under disaster risk. In Proceedings of the International Network Optimization Conference (INOC), Spa, Belgium, 22–25 April 2007. [Google Scholar]
- Diestel, R. Graph Theory, 4th ed.; Graduate Texts in Mathematics; Springer: Berlin, Germany, 2012; Volume 173. [Google Scholar]
- Bodlaender, H.L. A Tourist Guide through Treewidth. Acta Cybern.
**1993**, 11, 1–21. [Google Scholar] - Kloks, T. Treewidth, Computations and Approximations; Lecture Notes in Computer Science; Springer: Berlin, Germany, 1994; Volume 842. [Google Scholar]
- Downey, R.G.; Fellows, M.R. Fixed-parameter intractability. In Proceedings of the Seventh Annual Structure in Complexity Theory Conference, Boston, MA, USA, 22–25 June 1992. [Google Scholar]
- Koster, A.M.C.A.; Wolle, T.; Bodlaender, H.L. Degree-Based Treewidth Lower Bounds. In Proceedings of the 4th International Workshop, WEA 2005 Experimental and Efficient Algorithms, Santorini Island, Greece, 10–13 May 2005; pp. 101–112. [Google Scholar] [CrossRef][Green Version]

**Figure 1.**(

**a**) A probabilistic graph $\mathcal{G}=(V,E,p)$; (

**b**) A possible world ${H}_{1}$ of $\mathcal{G}$ with $P({H}_{1})=0.0072$; (

**c**) Another possible world ${H}_{2}$ of $\mathcal{G}$ with $P({H}_{2})=0.0588$.

**Figure 3.**An example of leaf $(\mathbf{a})$, introduce $(\mathbf{b})$, forget $(\mathbf{c})$ and join $(\mathbf{d})$ nodes. Directed edges denote child to parent link.

Work | Optimization Problem | Uncertainty Model |
---|---|---|

Frank and Hakimi, 1965 [9] | Probabilistic maximum flow | Capacities on the edges are drawn from an independent continuous distribution. |

Frank, 1969 [5] | Probabilistic shortest path | Length of the edges are drawn from a continuous distribution. |

Evans, 1976 [10] | Probabilistic maximum flow | Capacities on the edges are obtained from an arbitrary but unknown discrete probability distribution. |

Valiant, 1979 [6] | Network reliability | The probability $p(e)$ is the same for each edge and failure of every edge is independent. |

Sigal, Pritsker and Solberg, 1980 [41] | Stochastic shortest path | Edge weights are drawn from a known cumulative distribution function. |

Daskin, 1983 [19] | Expected coverage | Failure probability is the same for each vertex |

Papadimitriou Yannakakis, 1991 [28] | Canadian Traveler Problem | Each edge has a survival probability, edge failure is independent and algorithm knows of the failure during execution. |

Guerin and Orda, 1999 [42] | Most reliable path and flows with bandwidth selection | Each edge e has a survival probability ${p}_{e}(x)$ for the availability of bandwidth x. |

Hassin, Salman and Ravi, 2009 (2017) [11,12] | Expected coverage | Each edge e has a survival probability $p(e)$ and edge failure follows LRO model. |

Bonchi, Gullo, Kaltenbrunner and Volkovich, 2014 [15] | Probabilistic-Core | Each edge e has a survival probability $p(e)$ and edge failure follows RF model. |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Narayanaswamy, N.S.; Vijayaragunathan, R.
Parameterized Optimization in Uncertain Graphs—A Survey and Some Results. *Algorithms* **2020**, *13*, 3.
https://doi.org/10.3390/a13010003

**AMA Style**

Narayanaswamy NS, Vijayaragunathan R.
Parameterized Optimization in Uncertain Graphs—A Survey and Some Results. *Algorithms*. 2020; 13(1):3.
https://doi.org/10.3390/a13010003

**Chicago/Turabian Style**

Narayanaswamy, N. S., and R. Vijayaragunathan.
2020. "Parameterized Optimization in Uncertain Graphs—A Survey and Some Results" *Algorithms* 13, no. 1: 3.
https://doi.org/10.3390/a13010003