A Restart Local Search for Solving Diversified Top-k Weight Clique Search Problem

Abstract

Diversified top-k weight clique (DTKWC) search problem is an important generalization of the diversified top-k clique (DTKC) search problem with practical applications. The diversified top-k weight clique search problem aims to search k maximal cliques that can cover the maximum weight in a vertex weighted graph. In this work, we propose a novel local search algorithm called TOPKWCLQ for the DTKWC search problem which mainly includes two strategies. First, a restart strategy is adopted, which repeated the construction and updating processes of the maximal weight clique set. Second, a scoring heuristic is designed by giving different priorities for maximal weight cliques in candidate set. Meanwhile, a constraint model of the DTKWC search problem is constructed such that the research concerns can be evaluated. Experimental results show that the proposed algorithm TOPKWCLQ outperforms than the comparison algorithm on large-scale real-world graphs.

1. Introduction

Given an undirected graph $G=(V,E)$, a clique is a subset of the graph G, where any two vertices are adjacent. The maximal clique (MC) is a clique with the largest cardinality in the graph G. The maximum weight clique (MWC) is a generalization of MC with a positive integer assigned to each vertex as its weight value. The diversified top-k clique (DTKC) search problem aims to find a set with at most k maximal cliques to occupy as many vertices as possible, where k is a parameter that requires to be provided. The diversified top-k weight clique (DTKWC) search problem [1] attempts to search a set with at most k maximal weight cliques in the graph G with the largest total weight of covered vertices in these cliques, which can be readily verified as a NP-hard problem [2].

The MC and related problems have lots of applications, especially in real-world applications such as combinatorial auction [3], community detection [4,5] and video object segmentation [6]. Recently, considerable attentions have also been paid to solve top-k problems on large graphs [2,7,8]. This kind of problem can be very well applied to practical applications, such as the influential community [9], motif discovery in molecular biology [10]. For example, citation networks are usually represented as a type of social network with papers and links between citation relationships. In citation networks, denoted as graph G, papers are considered as vertices, and citation relationships are the edges between papers. The influence on the paper is viewed as a weight in G. The problem aims to search the top-k maximal divisive groups with different domains in G, which can be regarded as finding a DTKWC solution.

To solve MC and WMC problems on large-scale graphs effectively, some related methods have been proposed. These algorithms are usually divided into two categories: (1) the first one is the exact algorithms which can guarantee the optimality of the solutions, such as [11,12,13]. But exact algorithms may fail to solve the graphs within a reasonable time when the scale of them are larger. The second one is the local search algorithm, which is considered to find a suboptimal solution within a reasonable time for medium even larger graphs. And a large amount of effort has been devoted to designing different local search algorithms. For example, there exist lots of local search algorithms for solving WMC, e.g., [14,15]. Although there exist many algorithms to solve MC and WMC problems, currently, there are very few methods for diversified top-k cohesive groups. Such as Yuan et al. [7], Yuan et al. [8] (2015, 2016) proposed the concept of DTKC and then provided an approximate algorithm for it. Wu et al. [2] (2020) provides a local search algorithm to solve the DTKC search problem in large graphs, and it is state-of-the-art algorithm for DTKC search problem. Wu and Yin [16] (2021) introduce a problem of finding cohesive groups, named DTKSP problem, and develop a local search method based on some new heuristic strategies for this problem. Zhou et al. [1] (2021) encode the DTKWC search problem into the weighted partial MaxSAT (WPMS) problem, including direct encoding (DE) and independent set partition based encoding (ISPE), and solving WPMS with state-of-the-art solvers. However, this method is limited to solve real-world large graphs, because it is failed to encode large graphs into WPMS.

In this work, we propose a local search algorithm for the DTKWC search problem in large graphs, which provides a local optimal solution within a reasonable time and avoids the generation and storage of all maximal weight cliques. It aims at addressing the aforementioned problem. This algorithm, named TOPKWCLQ (which stands for top-k weight cliques), is based on two main strategies.

The first strategy is a restart method that can deal with the cycling problem. In the process of searching for the maximal weight clique, TOPKWCLQ will repeat to create a new maximal weight clique after initializing the set of maximal weight cliques and update this set through a scoring function. When the algorithm cannot be updated at fixed steps, it performs the restart process with the current best candidate solution.

The second strategy is a scoring function, which is designed by giving different priorities for maximal weight cliques in the candidate solution. During the searching process, TOPKWCLQ constructs and then maintains a candidate solution which size is at most k by adding or removing the maximal weight cliques according to the score value of each one. The score of each maximal weight clique is calculated by the total weight of the vertices that the clique has exclusively in the candidate solution.

To date, there is no suitable comparable algorithm for the DTKWC search problem on large scale of real-world graphs. Thus, we compare TOPKWCLQ with a commercial solver, CPLEX solver, with the constraint formulas proposed in this paper. Extensive performance experiments are executed to demonstrate that our proposed algorithm can achieve both high effectiveness and efficiency on real-world large-scale graphs.

The remainder of the paper is organized as follows. In Section 2, we propose the necessary background knowledge about diversified top-k weight clique search problem and formalize the DTKC and DTKWC search problem. In Section 3, we describe the TOPKWCLQ algorithm and the techniques it implements. In Section 4, we report extensive experimental results to demonstrate DTKWC’s high performance compared to CPLEX with our model in solving the DTKWC search problem, and finally, the conclusions are given in Section 5.

2. Diversified Top-k Weight Clique Search Problem

In this section, some notations and basic definitions which are applied to the DTKWC search problem are introduced. Then the proof of NP-hardness about DTKWC search problem is given. Next, the constraint formulas which are used in CPLEX solver as the mathematical model for DTKC and DTKWC search problem are proposed, respectively.

2.1. Definition and Notations

A weighted graph $G=(V,E,w)$ is a graph including $\left|V\right|$ vertices and $\left|E\right|$ edges, w is a weight function that assigns to each vertex $v_i$ of V a non-negative integer $w\left(v_i\right)$ representing its weight. $v_i^{w\left(v_i\right)}$ represents that vertex $v_i$ has weight $w\left(v_i\right)$.

Definition 1 (Maximal clique (MC)).Given an unweighted graph $G(V,E)$, a clique c in G is a set of vertices such that for any $u\in G$, $v\in c$$(u\ne v)$, we have $(u,v)\in E$. A clique c in G called a maximal clique if there exists no clique $c^{{}^{\prime }}$ in G such that $c\subset c^{{}^{\prime }}$.

Definition 2 (Maximal weight clique (MWC)).Given a weighted graph $G(V,E,w)$, a weight clique c in G is a set of vertices such that for any $u\in G$, $v\in c$$(u\ne v)$, we have $(u,v)\in E$ and the weight of c is $\omega \left(c\right)={\sum }_{v_i\in c}w\left(v_i\right)$. A weight clique c in G called a maximal weight clique if there exists no clique $c^{{}^{\prime }}$ in G such that $\omega \left(c\right)<\omega \left(c^{{}^{\prime }}\right)$.

Given a set of maximal cliques $C=\{c_1,c_2,\cdots \}$, the coverage of C, denoted by $cov\left(C\right)$, is the set of vertices covered by C, i.e., $cov\left(C\right)={\bigcup }_{c_i\in C}{c}_i$.

Definition 3 (Diversified top-k clique (DTKC)). 

Given an unweighted graph $G(V,E)$ and an integer k, the problem of diversified top-k clique search is to compute a set C, such that each $c\in C$ is a maximal clique, $\left|C\right|\le k$, and $cov\left(C\right)$ is maximized. C is called diversified top-k cliques.

Given a set of maximal (weighted) cliques $C=\{c_1,c_2,\cdots \}$, the private vertices of a maximal (weighted) clique c in C, denoted by $priv(c,C)$, are a subset of vertices of c not contained in any other clique in C, i.e., $priv(c,C)=c\backslash cov(C\backslash c)$. The weight of C is the total weight of the set of vertices in G covered by the cliques in C, denoted by $W\left(C\right)$, as below

$$W\left(C\right)=\sum _{v_i\in \left({\bigcup }_{c_j\in C}{c}_j\right)}w\left(v_i\right)$$

(1)

For the DTCK search problem, $w\left(v_i\right)=1,i\in [1,|V\left]\right]$. The overlapping of C, denoted by $overlap\left(S\right)$, is a set of vertices that are covered by maximal cliques in C more than once.

Definition 4 (Diversified top-k weight clique (DTKWC)). 

Given a weighted graph $G(V,E,w)$ and an integer k, the problem of diversified top-k weight clique search is to compute a set C, such that each $c\in C$ is a maximal weight clique, $\left|C\right|\le k$, and $W\left(C\right)$ is maximized. C is called diversified top-k weight cliques.

2.2. Constraint Formulation for DTKWC Search Problem

The DTKWC search problem is a generalization of the DTKC search problem [2] which aims to find a maximal clique set with at most k size with maximum total weight and a lower overlapping among all possible maximal clique sets from a given graph. Hence, we first give the formulas of the DTKC search problem and then expand them to the formulas of the DTKWC search problem. The DTKC search problem can be formulated as a mixed integer linear program (MILP) as follows:

$$\mathrm{OBJ}1:Maximum{W}_1\left(G\right)=\sum _{i\in [1,|V\left|\right]}{X}_i$$

(2)

$$\mathrm{OBJ}2:Minimum{W}_2\left(G\right)=\frac{{\sum }_{h=1}^k{\sum }_{i\in [1,|V\left|\right]}{x}_{ih}-{\sum }_{i\in [1,|V\left|\right]}{X}_i}{k-1}$$

(3)

Subject to:

$$x_{ih}+x_{jh}\le 1,\forall (i,j)\in \overline{E},1\le h\le k$$

(4)

$$X_i\le \sum _{i\in [1,|V\left|\right]}{x}_{ih},\forall i\in [1,|V\left|\right],1\le h\le k$$

(5)

$$x_{ih}\in \{0,1\},\forall i\in [1,|V\left|\right],1\le h\le k$$

(6)

$$X_i\in \{0,1\},\forall i\in [1,|V\left|\right]$$

(7)

where $x_{ih}$ is the binary variable associated with the vertex i, such that $x_{ih}=1$ if vertex $v_i$ is in the h’th maximal clique, $x_{ih}=0$ otherwise. $X_i$ is also a binary variable associated with vertex i. $X_i=1$ if there exists a vertex i in a maximal clique, $X_i=0$ otherwise. And constraint (4) is guaranteed that there is an edge between every two vertices in a clique. Constraint (5) means there is one clique including vertex $v_i$, then $X_i=1$. Constraints (6) and (7) give the range of the variables.

According to the formulas above, we give the MILP of the DTKWC search problem below:

$$\mathrm{OBJ}1:Maximum{W}_1\left(G\right)=\sum _{i\in [1,|V\left|\right]}\left(X_i{w}_i\right)$$

(8)

$$\mathrm{OBJ}2:Minimum{W}_2\left(G\right)=\frac{{\sum }_{h=1}^k{\sum }_{i\in [1,|V\left|\right]}\left(x_{ih}{w}_i\right)-{\sum }_{i\in [1,|V\left|\right]}\left(X_i{w}_i\right)}{k-1}$$

(9)

Subject to:

$$x_{ih}+x_{jh}\le 1,\forall (i,j)\in \overline{E},1\le h\le k$$

(10)

$$X_i\le \sum _{i\in V}{x}_{ih},\forall i\in [1,|V\left|\right],1\le h\le k$$

(11)

$$x_{ih}\in \{0,1\},\forall i\in [1,|V\left|\right],1\le h\le k$$

(12)

$$X_i\in \{0,1\},\forall i\in [1,|V\left|\right]$$

(13)

Similarly, $x_{ih}$ and $X_i$ represent the binary variables corresponding with the vertex i. $x_{ih}=1$, if vertex i appears in the h’th maximal clique, $x_{ih}=0$ otherwise. $X_i=1$, if vertex $v_i$ belongs to any maximal clique of C, $X_i=0$ otherwise. $w_i$ denotes the weight of the vertex i. Constraints (10)–(13) have the same intentions as the above constraints (4)–(7), respectively.

In the above formulas, both of these two problems aim to minimize the value of objective “OBJ1” on basis of maximizing the value of objective “OBJ2”. Thus, we can obtain the optimal solution of DTKC (DTKWC) search problem.

3. TOPKWCLQ: A Local Search Method for the DTKWC Search Problem

In this section, we will outline the framework of our algorithm. We use a restart strategy that interleaves between the construction and updating processes of the maximal weight clique set to enhance the quality of the candidate solution.

The restart procedure of the local search avoids the previous trajectory but turns to explore more different maximal weight clique sets. We construct these different maximal weight clique sets by combing the maximal weight cliques constructed from different starting vertices at each iteration. Thus, in the DTKWC search problem, using this restart strategy, TOPKWCLQ can improve the quality of the current candidate solution step by step.

At each restart iteration, we need to construct a new maximal weight clique one by one and eliminate the original maximal weight clique with the scoring function from the current candidate solution until no further improvement is found in the limited updating steps or the limit time is out. Thus, it can save the search time of a single iteration and restart the algorithm as soon as possible. After the updating procedure, a current candidate solution, that is, a local optimal solution, can be found and the algorithm will update the solution by comparing this local optimal solution with the maintained candidate solution from the previous iterations. Finally, until the time limit runs out, the algorithm returns to a maximal weight clique set as the best solution.

In the following, we will provide a random restart local search algorithm for the DTKWC search problem, called TOPKWCLQ.

3.1. Maximal Weight Clique Scoring Function

Before describing the algorithm framework, we first give a core issue in the algorithm TOPKWCLQ to evaluate the priorities of each maximal weight clique. During the search process, TOPKWCLQ must maintain a maximal weight clique set of size at most k as a candidate solution of the DTKWC search problem. Therefore, it is important to balance the quality and efficiency of the solutions by determining which maximal weight clique should be included or eliminated from the current candidate solution. For this reason, we will define a scoring function based on the total weight of its private vertices presented in Section 2 for each maximal weight clique during the updating process.

Definition 5 (Score function (Score function (score(c))).Given a weighted graph $G=(V,E,w)$, a maximal weight clique set C and a maximal weight clique $c_i$ of G$(c_i\in C)$. We use $score\left(c_i\right)$ to define the benefit of $c_i$ after adding a maximal weight clique to the set C. The score of c in C is defined as

$$score\left(c_i\right)=\sum _{v_j\in priv(c_i,C)}w\left(v_j\right)$$

(14)

The maximal weight clique selection method used in the UpdateSolution procedure is based on this scoring function. It attempts to determine the eliminated maximal weight clique with the smallest $score$ value by computing the scoring function for each maximal weight clique in the candidate solution C after adding a new maximal weight clique into C.

3.2. TOPKWCLQ Algorithm: The Top-Level Algorithm

The proposed TOPKWCLQ algorithm (see the flowchart in Figure 1) combines an initialization procedure aiming to generate a feasible initial solution and a local search procedure aiming at improving the initial solution. The top level of TOPKWCLQ is outlined in Algorithm 1, as described below.

Algorithm 1 TOPKWCLQ($G,k,cutoff$)

1: Input: an weighted graph $G(V,E,W)$, one integer k, $cutoff$ time 2: Output: a set $C^{*}$ containing at most k maximal weight cliques 3: $m\leftarrow m_0,C^{*}\leftarrow Ø$; /* $m_0$ is a parameter used in BMS strategy */ 4: while ($\mathrm{elapsed}\mathrm{time}<cutoff$) do 5: $RemainingSet\leftarrow V$; 6:   /* $m_{max}$ is another parameter used in BMS strategy */ 7:   if ($m<m_{max}$) then 8:     $m\leftarrow 2\ast m$; 9:   else 10:     $m_0\leftarrow m_0+1$; $m\leftarrow m_0$; 11:   end if 12:   $C\leftarrow InitKCliques(G,m,RemainingSet)$; 13:   if ($cov\left(C\right)=V$) then 14:     return C; 15:   end if 16:   $C\leftarrow LocalSearch(G,C,m,RemainingSet)$; 17:   if ($W\left(C^{*}\right)<W\left(C\right)$) then 18:     $C^{*}\leftarrow C$; 19:   end if 20: end while 21: return$C^{*}$ ;

First, we introduce the basic framework of our algorithm, which is presented in Algorithm 1. A current best global solution $C^{*}$ will be initialized as an empty set. Then the TOPKWCLQ starts a loop until the limited time reaches the maximum which equals $cutoff$ (lines 4–5). Before this loop, the parameters $m_0$ and $m_{max}$ of the best from multiple selection (BMS) strategy which is used in [2] to solve the DTKC search problem were given first (lines 4.2–4.5) and update the value of m in the loop. Then the TOPKWCLQ adopts a function $InitKCliques$ to construct enough maximal weight cliques as an initialization solution (line 4.6). After the initialization procedure, if $cov\left(C\right)$ equals to V, TOPKWCLQ will return C as a candidate solution (lines 4.7–4.8); Otherwise, update the current candidate solution C by using $LocalSearch$ method (line 4.9). If the total weight of the vertices in $C^{*}$ is smaller than the total weight of C, that is $W\left(C^{*}\right)<W\left(C\right)$, then replace $C^{*}$ with C (lines 4.10–4.11). When the elapsed time is bigger than the cutoff time, TOPKWCLQ stops searching and returns $C^{*}$.

In this section, the technical details of the TOPKWCLQ algorithm are introduced. The function to create a maximal weight clique from a random vertex is introduced in Section 3.3. In Section 3.4, the initialization procedure is presented. Section 3.5 presents the local search updating procedure of our algorithm.

3.3. Constructing a Maximal Weight Clique with Diversity

At each stage of our algorithm, we need constantly to find different maximal weight cliques to add into the candidate solution. Therefore, we design a method called $GetClique$ which uses the vertices in $RemainingSet$ to construct the maximal weight cliques according to the properties of the DTKWC search problem. Let $Candset$ denote the vertices which are adjacent to all vertices already in c. We also design a function $b\left[v\right]$ which will be utilized during the initialization procedure to represent the benefit of a vertex v, the expression is as follows,

$$b\left[v\right]=\sum _{u\in \left(N\right(v)\cap Candset)}w\left(u\right).$$

(15)

The Algorithm 2 shows the pseudo-code of $GetCliques$. First, c is initialized as an empty set. Then, $GetCliques$ iteratively and randomly selects a vertex from $RemainingSet$ which includes all vertices in V but excludes the vertices in the current candidate solution. If the set of $RemainningSet$ is empty (line 2), then the algorithm returns c, and c is empty, which means we cannot create one more maximal weight clique. Otherwise, $GetClique$ selects a vertex v from $RemainingSet$ randomly and then adds it to the set c. Then, the algorithm adds all neighbours of v into $CandSet$. If $CandSet$ is not empty, $GetClique$ will find a maximal weight clique by the BMS strategy (which is proposed by [17] ) used to select the better next vertex as the added vertex to the current partial clique (lines 6–7). In this situation, if the cardinality of $CandSet$ is smaller than the parameter m, the algorithm will pick a vertex v from $CandSet$ with the greatest $\widehat{b}$, breaking ties in favour of the older one; Otherwise, $GetClique$ selects the vertex with biggest benefit from m vertices that randomly selects from $Candset$. After that, we can get a better result by just calculating the score of at most m vertices. $CandSet$ is updated for selecting the next vertex of the maximal weight clique.

Algorithm 2 GetClique$(G,m,RemainingSet)$

1: $c\leftarrow Ø$; 2: if ($RemainingSet=Ø$) then 3:   return c; 4: end if 5: $v\leftarrow$ randomly select a vertex from $RemainingSet$; 6: $c\leftarrow \left\{v\right\}$, $CandSet\leftarrow \left\{u\right|u\in N\left(v\right)\}$; 7: while ($CandSet\ne Ø$) do 8:   if ($\left|CandSet\right|<m$) then 9:     pick the vertex v from $CandSet$ with the greatest $\widehat{b}$, breaking ties in favour of the older one; 10:   else 11:     $v\leftarrow$ randomly select a vertex from $CandSet$; 12:     for ($iter:=1$ to $m-1$) do 13:       $v^{\prime }\leftarrow$ randomly select a vertex from $CandSet$; 14:       if ($\widehat{b}\left[v^{\prime }\right]>\widehat{b}\left[v\right]$) then 15:         $v\leftarrow v^{\prime }$, $\widehat{b}\left[v\right]\leftarrow \widehat{b}\left[v^{\prime }\right]$; 16:       end if 17:     end for 18:   end if 19:   $c\leftarrow c\cup \left\{v\right\}$; 20:   remove v from $RemainingSet$; 21:   $CandSet\leftarrow CandSet\cap N\left(v\right)$; 22: end while 23: returnc

3.4. The Initialization Procedure

In this subsection, we will explain the initialization procedure, which is outlined in Algorithm 3. It is the first stage of our algorithm. At the beginning of this procedure, a current candidate solution C is set to empty. Due to the DTKWC search problem needs to find a solution which is a set including at most k maximal weight cliques, this method attempts to create the maximal weight cliques randomly through $GetClique$ that introduced in the above subsection. If we get an empty result from $GetClique$ until there is no more vertex not belongs to C or we have created k maximal weight cliques. In this method, the starting vertices are generated randomly from a set that includes the vertices never used as a starting vertex. Repeat this random process, we can get the diversified maximal weight cliques that do not depend on the corresponding information acquired during the previous process. Moreover, starting from an unvisited vertex to construct the solution of the DTKWC search problem will overcome the cycling problem, i.e., revisiting the same solution within a short time in the local search algorithm.

Algorithm 3 InitKCliques$(G,m,RemainingSet)$

1: $C\leftarrow Ø$; 2: while ($\left|C\right|<k$) do 3:   if ($cov\left(C\right)=V$) then 4:     returnC; 5:   end if 6:   /* create a maximal weight clique with BMS strategies */ 7:   $c\leftarrow GetClique(G,m,RemainingSet)$; 8:   if ($c\ne Ø$) then 9:     $C\leftarrow C\cup \left\{c\right\}$; 10:   end if 11: end while 12: returnC

3.5. Local Search Updating

The candidate solution initialized by the initialization procedure is just a good candidate solution meeting the requirement of the DTKWC search problem, but there is no guarantee that it is a great candidate solution. Therefore, in this subsection, we design a local search method to improve the quality of this solution by exploring as many new maximal weight cliques as possible (line 2.2).

The proposed local search method in Algorithm 4 finds the different maximal weight clique combinations from a candidate solution. It is a good way to iteratively find a better combination $C^{{}^{\prime }}$ which includes the vertices with a greater total weight. For this reason, we add a new maximal weight clique $c^{{}^{\prime }}$ created by $GetClique$ into the current candidate solution C. Then we compute $k+1$$score$ functions explained in Section 3.1 for all of maximal weight cliques in $C^{{}^{\prime }}$ each time (line 2.5). After this, we gain $k+1$ values of $score$, delete the maximal weight clique with the smallest value among these maximal weight cliques in $C^{{}^{\prime }}$. Such that we maintain a k size maximal weight clique solution as the new candidate solution (line 2.8).

Algorithm 4 LocalSearch($G,C,m,RemainingSet$)

1: $step\leftarrow 0$, $C^{\prime }\leftarrow Ø$; 2: while ($\mathrm{elapsed}\mathrm{time}<cutoff$) do 3:   $step\leftarrow step+1$; 4:   $c^{\prime }\leftarrow GetClique(G,m,RemainingSet)$; 5:   if ($c^{\prime }=Ø$) then 6:     break; 7:   end if 8:   $C^{\prime }\leftarrow C\cup \left\{c^{\prime }\right\}$; 9:   Compute $score\left(c\right)$ of each maximal weight clique c in $C^{\prime }$; 10:   $c_{min}\leftarrow ar{g}_{c\in C^{\prime }}min\left\{score\left(c\right)\right\}$; 11:   Remove $c_{min}$ from $C^{\prime }$, breaking ties in favour of the smaller one; 12:   if ($cov\left(C^{\prime }\right)>cov\left(C\right)$) then 13:     $C\leftarrow C^{\prime }$, $step\leftarrow 0$; 14:   end if 15:   /* $fs$ is the third parameter of TOPKWCLQ */ 16:   if ($step\ge fs$) then 17:     break; 18:   end if 19: end while 20: returnC

Although, we obtain the information that sometimes a candidate solution cannot be improved by the normal local search method in a long time. For this, we add a fixed step, denoted by $fs$, into our local search framework that breaks the loop if the current solution cannot be improved in $fs$ steps.

3.6. An Example of DTKWC Search Problem

 Example 1. 

Let us illustrate how to explore a solution for the DTKWC search problem by using a sample weighted graph in Figure 2.

Figure 2a gives an weighted graph $G(V,E,w)$ with ten vertices, where $v_i^{w_i}$ denotes vertex $v_i$ with $w_i=w\left(v_i\right)$. Assume the integer parameter $k=2$. The best clique weight so far is $\omega \left(C_{max}\right)=13$. During the first phase, InitKCliques creates a maximal weight clique set $C_{max}=\{c_1,c_2\}$, and $c_1=\{v_0^1,v_1^2,v_2^1,v_3^2\}$, $c_2=\{v_3^2,v_4^3,v_5^3\}$. The total weight of $C_{max}$ is 12. In the second phase, LocalSearch tries to determine a new maximal weight clique $c_3$ by GetClique. Suppose $c_3=\{v_5^3,v_7^3\}$. We add $c_3$ into $C_{max}$. Now, $C_{max}$ contains 3 maximal weight cliques. We evaluate the quality of these three maximal weight cliques by the scoring function we proposed. $\{v_0^1,v_1^2,v_2^1\}$, $\left\{v_4^3\right\}$, $\left\{v_7^3\right\}$ are the private vertices set of $c_1$, $c_2$, $c_3$, respectively. $score\left(c_1\right)=4$, $score\left(c_2\right)=3$, $score\left(c_3\right)=3$. After this process, we remove the worst maximal weight clique $c_2$ or $c_3$ to keep the size of the maximal weight clique set equal to 2. Observe that the set of $\{c_1,c_2\}$ or $\{c_1,c_3\}$ are both the solution of DTKWC search problem in this graph with the parameter $k=2$, and $\{c_1,c_3\}$ is the best solution with the lowest overlapping.

4. Experimental Evaluation

In this section, we carry out extensive experiments to evaluate the performance of TOPKWCLQ on weighted real-world large graphs. Since there is no suitable heuristic or exact algorithm for the DTKWC search problem on real-world large graphs in literature, as we know that is a good choice to compare the results of the proposed algorithm to the results obtained by CPLEX solver which is a commercial solver for many combinatorial optimization problems with their constraint formulas of mathematical models. Therefore, the results obtained by CPLEX can be used as reference on the solution quality. We first describe the weighted benchmark and then present the experimental preliminaries and introduce the parameter settings.

4.1. The Benchmark

We evaluate the TOPKWCLQ algorithm on the benchmarks of the weighted real-world graph, which will be shown below.

The weighted real-world large graph benchmark in our experiments was originally from the Network Data Repository online [18] (http://www.graphrepository.com/networks.php, accessed on 1 August 2021). There are millions of vertices and tens of millions of edges on many of the real-world graphs which used in our experiments. This benchmark has been transformed from unweighted graphs to the weighted graphs used the weighting function $w\left(v_i\right)=\left(imod200\right)+1$ (including 102 instances) [19]. Moreover, most of these as the experimental instances used in maximum vertex weight clique problem [6,20,21], coloring problem [22], maximum k-plexes problem [23] and DTKC search problem [2]. Considering the relationship between the DTKWC search problem and these problems, these real-world graphs can naturally be used to evaluate the performance of our algorithm for the DTKWC search problem. These real-world graphs were downloaded from the author’s website (http://lcs.ios.ac.cn/~caisw/Resource/weighted-massive-graphs.zip, accessed on 1 August 2021).

The graphs in our experiments are divided into 11 classes, including biological networks, collaboration networks, interaction networks, infrastructure networks, recommendation networks, retweet networks, scientific computing, social networks, facebook networks, technological networks, and web graphs.

4.2. Experimental Preliminaries and Parameter Tuning

The proposed algorithm TOPKWCLQ was implemented in C++ and compiled on CentOS with 2.4 GHz CPU and 32G RAM with “-O3” flag. We run TOPKWCLQ 10 times independently with the random seed setting from 1 to 10 for each instances. Each one is run until the run time of the algorithm arrives which is a given time limit that is assigned as 600 s in this paper. The termination criterion of CPLEX is either the convergence of lower and upper bounds or a time limit which is assigned as 3600 s. We use the solution values of CPLEX to evaluate the quality of the solution solved by TOPKWCLQ.

For each real-world large graph used in our experiments, we set the parameter k to 10, 20, 30, 40, and 50 to obtain five DTKWC search problem instances. Hence, there were $102\times 5=510$ DTKWC search problem instances in our experiments.

TOPKWCLQ uses three parameters for which well-working values must be found: $m_0$ and $m_{max}$ are the minimum and maximum value of BMS strategy respectively, and $fs$ is the maximum allowed updating steps of the solution per iteration. Parameters $m_0$ and $m_{max}$ are used in the BMS strategy inspired by [17]. The value of these parameters are set in

Table 1. Setting of parameters $m_0$, $m_{max}$, $fs$.

Parameters	Descriptions	Range	Values
$m_0$	minimum number of greediness iterations in BMS strategy	2, 4, 8, 16	8
$m_{max}$	maximum number of greediness iterations in BMS strategy	32, 64, 128, 256	64
$fs$	control restart in update procedure	1000, 2000, 4000, 8000	2000

according to a preliminary tuning experiment.

The next subsection is shown to the evaluation of TOPKWCLQ compared with the lower bound (“$LB$”) and the upper bound (“$UB$”) of CPLEX under all 510 DTKWC search problem instances.

4.3. Experimental Results

We present the comprehensive experiment results on the benchmark instances described in Section 4.1 with 5 values of parameter k in

Table 2. Experiment results on real-world large graphs with $k=10$.

Instance	CPLEX		TOPKWCLQ			Instance	CPLEX		TOPKWCLQ
	$\mathit{LB}$	$\mathit{UB}$	${\mathit{w}}_{\mathit{b}}$	${\mathit{w}}_{\mathit{a}}$	$\mathit{Time}$		$\mathit{LB}$	$\mathit{UB}$	${\mathit{w}}_{\mathit{b}}$	${\mathit{w}}_{\mathit{a}}$	$\mathit{Time}$
bio-celegans	6867	6867	6852	6808.4	250.97	socfb-B-anon	N/A	N/A	21,273	20,893.6	198.57
bio-diseasome	8672	8672	8672	8672	0.11	socfb-Berkeley13	N/A	N/A	38,740	38,354.7	103.04
bio-dmela	N/A	N/A	6094	5994.9	137.32	socfb-CMU	N/A	N/A	32,888	32,488.2	143.52
bio-yeast	4991	5020	4991	4991	4.70	socfb-Duke14	N/A	N/A	30,722	30,284.8	199.64
ca-AstroPh	N/A	N/A	47,912	47,445	152.59	socfb-Indiana	N/A	N/A	45,026	44,249.2	155.71
ca-citeseer	N/A	N/A	76,782	76,398	244.19	socfb-MIT	N/A	N/A	31,207	30,894.1	196.22
ca-coauthors-dblp	N/A	N/A	304,932	302,045	177.41	socfb-OR	N/A	N/A	27,601	27,282.7	186.37
ca-CondMat	N/A	N/A	20,070	19,949.8	187.83	socfb-Penn94	N/A	N/A	39,164	38,537	210.69
ca-CSphd	4059	4059	4059	4059	0.30	socfb-Stanford3	N/A	N/A	38,668	38,137.4	202.05
ca-dblp-2010	N/A	N/A	64,732	64,245.7	168.86	socfb-Texas84	N/A	N/A	43,443	42,722.4	153.66
ca-dblp-2012	N/A	N/A	64,179	64,024.8	209.09	socfb-uci-uni	N/A	N/A	7092	6911.8	174.43
ca-Erdos992	N/A	N/A	5800	5664.5	215.51	socfb-UCLA	N/A	N/A	42,919	42,595	133.01
ca-GrQc	N/A	N/A	25,844	25,844	0.16	socfb-UConn	N/A	N/A	38,261	38,075.5	214.20
ca-HepPh	N/A	N/A	79,624	79,496.8	225.91	socfb-UCSB37	N/A	N/A	42,830	42,442.4	166.30
ca-hollywood-2009	N/A	N/A	91,3861	90,8334.9	223.68	socfb-UF	N/A	N/A	52,626	52,133.5	231.33
ca-MathSciNet	N/A	N/A	24,017	23,826	156.59	socfb-UIllinois	N/A	N/A	46,705	46,144.6	219.00
ca-netscience	7588	7588	7588	7588	0.16	socfb-Wisconsin87	N/A	N/A	33,594	33,384.2	183.56
ia-email-EU	N/A	N/A	8440	8236.2	239.39	soc-flickr	N/A	N/A	26,965	26,228.7	173.67
ia-email-univ	9570	9718	9566	9536.2	236.42	soc-flixster	N/A	N/A	25,505	24,409.3	158.30
ia-enron-large	N/A	N/A	16,124	15,984	194.00	soc-FourSquare	N/A	N/A	12,254	11,894.3	180.01
ia-enron-only	4331	4338	4331	4331	13.55	soc-gowalla	N/A	N/A	19,607	19,371.3	174.70
ia-fb-messages	5738	7441	5792	5737.7	211.06	soc-karate	472	472	472	472	0.00
ia-infect-dublin	10,946	11,001	10,946	10,946	9.17	soc-lastfm	N/A	N/A	13,122	12,833.9	198.74
ia-infect-hyper	4446	5292	4442	4416.3	160.16	soc-livejournal	N/A	N/A	138,283	135,583.1	145.34
ia-reality	N/A	N/A	2882	2852.8	212.69	soc-LiveMocha	N/A	N/A	9869	9608.8	148.29
ia-wiki-Talk	N/A	N/A	10,976	10,754	172.35	soc-orkut	N/A	N/A	42,484	41,987.8	145.63
inf-power	N/A	N/A	6613	6613	37.32	soc-pokec	N/A	N/A	21,995	21,094.4	130.38
inf-roadNet-CA	N/A	N/A	6367	6296.1	141.26	soc-slashdot	N/A	N/A	13,801	13,224.5	169.68
inf-roadNet-PA	N/A	N/A	6084	6082.5	148.54	soc-twitter-follows	N/A	N/A	5535	5346.8	174.89
inf-road-usa	N/A	N/A	6035	6002.5	136.92	soc-wiki-Vote	6376	6768	6341	6253.2	133.06
rec-amazon	N/A	N/A	8931	8931	1.73	soc-youtube	N/A	N/A	12,409	12,255	191.54
rt-retweet	1575	1578	1575	1575	0.06	soc-youtube-snap	N/A	N/A	12,146	11,973.7	140.26
rt-retweet-crawl	N/A	N/A	8742	8650.5	189.03	tech-as-caida2007	N/A	N/A	7240	7111	196.61
rt-twitter-copen	4661	4661	4661	4659.5	128.18	tech-as-skitter	N/A	N/A	29,470	28,331.9	131.52
sc-ldoor	N/A	N/A	40,726	40,704.3	179.99	tech-internet-as	N/A	N/A	7457	7067.2	166.47
sc-msdoor	N/A	N/A	40,670	40,625.9	135.87	tech-p2p-gnutella	N/A	N/A	5829	5816.2	190.24
sc-nasasrb	N/A	N/A	43,776	43,714	173.28	tech-RL-caida	N/A	N/A	12,215	11,994.8	164.86
sc-pkustk11	N/A	N/A	47,548	47,268.1	153.85	tech-routers-rf	3914	207,554	10,002	9871	128.18
sc-pkustk13	N/A	N/A	57,282	56,738	185.36	tech-WHOIS	N/A	N/A	31,459	31,125.1	144.75
sc-pwtk	N/A	N/A	45,504	45,432	179.58	web-arabic-2005	N/A	N/A	92,108	92,108	3.98
sc-shipsec1	N/A	N/A	31,790	31,661.2	214.63	web-BerkStan	N/A	N/A	11,200	11,200	71.12
sc-shipsec5	N/A	N/A	43,260	43,157.4	132.02	web-edu	N/A	N/A	11,250	11,250	0.17
soc-BlogCatalog	N/A	N/A	20,014	19,109.1	242.59	web-google	13,126	13,126	13,126	13,126	0.01
soc-brightkite	N/A	N/A	19,548	19,178.3	173.03	web-indochina-2004	N/A	N/A	44,052	44,052	44.42
soc-buzznet	N/A	N/A	17,620	17,041.3	199.35	web-it-2004	N/A	N/A	415,850	415,850	0.63
soc-delicious	N/A	N/A	11,711	11,553.1	136.82	web-polblogs	5926	6428	5918	5828.2	166.71
soc-digg	N/A	N/A	27,726	27,083.9	226.17	web-sk-2005	N/A	N/A	63,930	63,920.4	132.89
soc-dolphins	1226	1226	1226	1226	0.04	web-spam	N/A	N/A	14,569	14,418.8	150.91
soc-douban	N/A	N/A	8983	8860.1	207.39	web-uk-2005	N/A	N/A	441,613	441,613	0.41
soc-epinions	N/A	N/A	12,942	12,661.1	139.36	web-webbase-2001	N/A	N/A	20,648	20,648	68.16
socfb-A-anon	N/A	N/A	22,551	21,896.8	176.24	web-wikipedia2009	N/A	N/A	29,781	29,063.8	162.08

,

Table 3. Experiment results on real-world large graphs with $k=20$.

Instance	CPLEX		TOPKWCLQ			Instance	CPLEX		TOPKWCLQ
	$\mathit{LB}$	$\mathit{UB}$	${\mathit{w}}_{\mathit{b}}$	${\mathit{w}}_{\mathit{a}}$	$\mathit{Time}$		$\mathit{LB}$	$\mathit{UB}$	${\mathit{w}}_{\mathit{b}}$	${\mathit{w}}_{\mathit{a}}$	$\mathit{Time}$
bio-celegans	11,489	11,747	11,275	11,198.2	147.50	socfb-B-anon	N/A	N/A	38,667	38,284.5	221.32
bio-diseasome	14,313	14,345	14,313	14,313	2.06	socfb-Berkeley13	N/A	N/A	67,518	66,896.4	158.57
bio-dmela	N/A	N/A	10,447	10,370.9	209.09	socfb-CMU	N/A	N/A	55,267	54,811	191.40
bio-yeast	8874	9730	9067	9060	125.32	socfb-Duke14	N/A	N/A	54,844	54,091.1	137.00
ca-AstroPh	N/A	N/A	85,366	85,173.6	185.29	socfb-Indiana	N/A	N/A	80,040	79,382.1	162.68
ca-citeseer	N/A	N/A	131,947	131,244.5	148.71	socfb-MIT	N/A	N/A	55,086	54,421.2	156.99
ca-coauthors-dblp	N/A	N/A	531,666	529,145	235.05	socfb-OR	N/A	N/A	48,737	48,162.2	216.69
ca-CondMat	N/A	N/A	35,029	34,734	249.30	socfb-Penn94	N/A	N/A	70,888	70,417.4	178.75
ca-CSphd	-	-	7943	7942	103.99	socfb-Stanford3	N/A	N/A	66,445	65,926.7	213.33
ca-dblp-2010	N/A	N/A	107,838	107,254.7	192.43	socfb-Texas84	N/A	N/A	75,522	74,283.2	156.21
ca-dblp-2012	N/A	N/A	102,895	102,577.8	214.70	socfb-uci-uni	N/A	N/A	13,586	13,267.4	191.29
ca-Erdos992	N/A	N/A	9766	9580.7	146.01	socfb-UCLA	N/A	N/A	72,363	71,465.5	192.25
ca-GrQc	N/A	N/A	37,559	37,559	24.67	socfb-UConn	N/A	N/A	64,266	63,327	202.52
ca-HepPh	N/A	N/A	113,984	113,639.5	189.33	socfb-UCSB37	N/A	N/A	68,722	68,234.3	135.47
ca-hollywood-2009	N/A	N/A	1,305,029	1,293,108	185.10	socfb-UF	N/A	N/A	90,307	89,178.6	233.87
ca-MathSciNet	N/A	N/A	41,461	41,215.4	137.01	socfb-UIllinois	N/A	N/A	79,952	79,253.7	243.79
ca-netscience	13,178	13,196	13,189	13,189	2.16	socfb-Wisconsin87	N/A	N/A	59,743	59,398.8	215.07
ia-email-EU	N/A	N/A	13,440	13,232.2	151.90	soc-flickr	N/A	N/A	44,279	43,612.1	194.81
ia-email-univ	11,024	18,426	15,647	15,525.2	172.17	soc-flixster	N/A	N/A	41,715	40,413	210.02
ia-enron-large	N/A	N/A	28,592	28,252.6	123.99	soc-FourSquare	N/A	N/A	20,866	20,602.4	213.53
ia-enron-only	6922	7237	6923	6917	177.34	soc-gowalla	N/A	N/A	35,017	34,396	185.05
ia-fb-messages	6204	15,510	10,305	10,207.4	189.05	soc-karate	629	629	629	629	0.00
ia-infect-dublin	15,009	18,694	17,588	17,527.2	215.34	soc-lastfm	N/A	N/A	23,753	22,918.3	113.19
ia-infect-hyper	5972	6554	6086	6068.2	116.70	soc-livejournal	N/A	N/A	222,938	220,213.4	147.43
ia-reality	N/A	N/A	5291	5252.3	161.50	soc-LiveMocha	N/A	N/A	16,907	16,391.4	106.99
ia-wiki-Talk	N/A	N/A	18,644	18,315.4	252.87	soc-orkut	N/A	N/A	79,351	78,363.4	150.28
inf-power	N/A	N/A	11,743	11,712	204.17	soc-pokec	N/A	N/A	38,793	38,128.1	239.65
inf-roadNet-CA	N/A	N/A	12,341	12,292.8	179.86	soc-slashdot	N/A	N/A	21,071	20,868.8	217.39
inf-roadNet-PA	N/A	N/A	12,049	12,041.1	157.62	soc-twitter-follows	N/A	N/A	9722	9588.3	208.03
inf-road-usa	N/A	N/A	11,940	11,827.3	191.68	soc-wiki-Vote	10,649	12,613	10,670	10,620.8	225.30
rec-amazon	N/A	N/A	17,414	17,414	10.02	soc-youtube	N/A	N/A	22,173	21,901.9	199.76
rt-retweet	2732	2758	2754	2754	19.86	soc-youtube-snap	N/A	N/A	21,687	21,423.5	162.58
rt-retweet-crawl	N/A	N/A	15,283	14,983.6	163.67	tech-as-caida2007	N/A	N/A	11,632	11,485.1	114.11
rt-twitter-copen	8364	8525	8293	8256.7	142.45	tech-as-skitter	N/A	N/A	47,288	44,902.5	169.13
sc-ldoor	N/A	N/A	81,165	81,074.7	202.88	tech-internet-as	N/A	N/A	12,591	11,964.9	157.32
sc-msdoor	N/A	N/A	80,703	80,651.2	178.45	tech-p2p-gnutella	N/A	N/A	11,115	11,034.6	168.48
sc-nasasrb	N/A	N/A	84,638	84,212.2	162.43	tech-RL-caida	N/A	N/A	21,048	20,652.7	83.62
sc-pkustk11	N/A	N/A	90,376	90,205.5	204.08	tech-routers-rf	N/A	N/A	15,456	15,301.5	173.66
sc-pkustk13	N/A	N/A	108,096	107,515.5	141.38	tech-WHOIS	N/A	N/A	44,189	43,782.9	188.40
sc-pwtk	N/A	N/A	89,496	89,366	204.28	web-arabic-2005	N/A	N/A	178,434	178,434	3.22
sc-shipsec1	N/A	N/A	60,831	60,323.5	156.03	web-BerkStan	N/A	N/A	18,756	18,724	143.16
sc-shipsec5	N/A	N/A	82,422	81,832.3	133.56	web-edu	N/A	N/A	16,531	16,498	170.07
soc-BlogCatalog	N/A	N/A	30,479	29,811	201.55	web-google	14,174	22,943	21,479	21,479	0.59
soc-brightkite	N/A	N/A	31,062	30,674.7	207.00	web-indochina-2004	N/A	N/A	74,320	74,236.2	218.99
soc-buzznet	N/A	N/A	28,642	28,133.5	193.13	web-it-2004	N/A	N/A	797,123	797,123	1.03
soc-delicious	N/A	N/A	20,402	20,228.4	92.03	web-polblogs	9808	11,906	9949	9909.4	135.08
soc-digg	N/A	N/A	42,903	42,169.9	170.04	web-sk-2005	N/A	N/A	97,196	97,053	192.90
soc-dolphins	1861	1871	1861	1861	6.05	web-spam	N/A	N/A	23,982	23,625.7	175.29
soc-douban	N/A	N/A	15,043	14,670.5	198.98	web-uk-2005	N/A	N/A	789,896	789,896	0.43
soc-epinions	N/A	N/A	22,001	21,671.5	247.93	web-webbase-2001	N/A	N/A	33,438	33,290.2	154.66
socfb-A-anon	N/A	N/A	39,458	39,201.1	210.51	web-wikipedia2009	N/A	N/A	48,222	47,165.3	230.13

,

Table 4. Experiment results on real-world large graphs with $k=30$.

Instance	CPLEX		TOPKWCLQ			Instance	CPLEX		TOPKWCLQ
	$\mathit{LB}$	$\mathit{UB}$	${\mathit{w}}_{\mathit{b}}$	${\mathit{w}}_{\mathit{a}}$	$\mathit{Time}$		$\mathit{LB}$	$\mathit{UB}$	${\mathit{w}}_{\mathit{b}}$	${\mathit{w}}_{\mathit{a}}$	$\mathit{Time}$
bio-celegans	14,528	16,193	14,805	14,740	203.67	socfb-B-anon	N/A	N/A	55,318	54,993.8	155.81
bio-diseasome	12,610	20,103	18,431	18,425.5	91.62	socfb-Berkeley13	N/A	N/A	91,991	90,393.9	162.54
bio-dmela	N/A	N/A	14,497	14,384	180.80	socfb-CMU	N/A	N/A	72,885	72,032.9	162.68
bio-yeast	-	-	12,774	12,750.3	198.29	socfb-Duke14	N/A	N/A	75,543	74,516.6	149.19
ca-AstroPh	N/A	N/A	117,328	116,221.3	241.64	socfb-Indiana	N/A	N/A	110,454	109,407.1	198.30
ca-citeseer	N/A	N/A	178,715	177,726.9	209.84	socfb-MIT	N/A	N/A	74,158	73,595.9	182.28
ca-coauthors-dblp	N/A	N/A	740,505	733,645.3	135.01	socfb-OR	N/A	N/A	67,317	66,836.6	179.21
ca-CondMat	N/A	N/A	48,476	48,219.6	138.60	socfb-Penn94	N/A	N/A	98,255	97,120.6	165.00
ca-CSphd	N/A	N/A	11,739	11,731	150.83	socfb-Stanford3	N/A	N/A	90,442	89,784.3	151.78
ca-dblp-2010	N/A	N/A	146,421	145,317.6	171.76	socfb-Texas84	N/A	N/A	103,092	101,991	212.90
ca-dblp-2012	N/A	N/A	133,948	133,150.3	106.77	socfb-uci-uni	N/A	N/A	19,436	19,111	173.62
ca-Erdos992	N/A	N/A	13,381	13,305.1	173.98	socfb-UCLA	N/A	N/A	97,506	96,350.8	206.58
ca-GrQc	N/A	N/A	46,262	46,161.4	151.01	socfb-UConn	N/A	N/A	84,391	83,777.7	183.23
ca-HepPh	N/A	N/A	140,485	140,068.4	169.58	socfb-UCSB37	N/A	N/A	90,198	89,200.8	181.14
ca-hollywood-2009	N/A	N/A	1,591,287	1,578,513	203.39	socfb-UF	N/A	N/A	119,700	117,876.9	124.65
ca-MathSciNet	N/A	N/A	56,826	56,490.5	202.42	socfb-UIllinois	N/A	N/A	108,948	108,470.3	135.60
ca-netscience	17,341	17,997	17,781	17,780.1	121.27	socfb-Wisconsin87	N/A	N/A	83,837	82,596.5	211.36
ia-email-EU	N/A	N/A	17,861	17,546.7	182.28	soc-flickr	N/A	N/A	59,976	59,342.1	125.18
ia-email-univ	8733	26,220	20,749	20,538.9	201.94	soc-flixster	N/A	N/A	55,676	54,557.7	181.08
ia-enron-large	N/A	N/A	40,233	39,553.5	189.25	soc-FourSquare	N/A	N/A	28,775	28,598.1	233.85
ia-enron-only	8030	9226	8570	8542.2	160.40	soc-gowalla	N/A	N/A	48,365	47,895.4	148.82
ia-fb-messages	7135	23,605	14,396	14,223	152.57	soc-karate	629	629	629	629	0.00
ia-infect-dublin	12,608	26,004	22,783	22,627.5	193.70	soc-lastfm	N/A	N/A	32,614	32,036.3	157.81
ia-infect-hyper	6554	6554	6554	6554	1.68	soc-livejournal	N/A	N/A	297,475	289,923.2	167.77
ia-reality	N/A	N/A	7473	7404.8	205.85	soc-LiveMocha	N/A	N/A	23,056	22,702.8	187.07
ia-wiki-Talk	N/A	N/A	25,589	25,136.2	172.33	soc-orkut	N/A	N/A	114,442	112,839.3	187.46
inf-power	N/A	N/A	16,414	16,356.8	172.16	soc-pokec	N/A	N/A	55,396	54,045.9	170.75
inf-roadNet-CA	N/A	N/A	18,351	18,245.3	232.32	soc-slashdot	N/A	N/A	28,032	27,717.9	261.61
inf-roadNet-PA	N/A	N/A	18,005	17,990.4	186.22	soc-twitter-follows	N/A	N/A	13,690	13,527.6	203.29
inf-road-usa	N/A	N/A	17,728	17,649.3	156.50	soc-wiki-Vote	9944	18,426	14,496	14,390.8	248.59
rec-amazon	N/A	N/A	25,468	25,468	112.68	soc-youtube	N/A	N/A	30,905	30,491.7	185.84
rt-retweet	3609	3609	3609	3609	21.53	soc-youtube-snap	N/A	N/A	30,746	30,309.2	203.55
rt-retweet-crawl	N/A	N/A	21,151	20,857.2	230.71	tech-as-caida2007	N/A	N/A	16,120	15,863.9	138.62
rt-twitter-copen	11,777	12,474	11,590	11,548.8	161.99	tech-as-skitter	N/A	N/A	60,308	58,536.9	151.54
sc-ldoor	N/A	N/A	121,303	121,165.8	214.11	tech-internet-as	N/A	N/A	16,818	16,538	131.74
sc-msdoor	N/A	N/A	120,426	120,330.8	250.58	tech-p2p-gnutella	N/A	N/A	16,023	15,920.1	239.06
sc-nasasrb	N/A	N/A	123,112	122,866	197.63	tech-RL-caida	N/A	N/A	28,826	28,485.9	174.90
sc-pkustk11	N/A	N/A	132,640	132,153	244.31	tech-routers-rf	N/A	N/A	20,062	19,967.9	145.42
sc-pkustk13	N/A	N/A	157,978	156,744.2	221.26	tech-WHOIS	N/A	N/A	53,750	53,301.1	159.00
sc-pwtk	N/A	N/A	132,744	132,648.4	245.68	web-arabic-2005	N/A	N/A	263,602	263,602	7.09
sc-shipsec1	N/A	N/A	87,853	87,048.3	150.32	web-BerkStan	N/A	N/A	25,848	25,786.1	122.40
sc-shipsec5	N/A	N/A	118,524	117,849.5	174.37	web-edu	N/A	N/A	20,805	20,751	198.34
soc-BlogCatalog	N/A	N/A	40,724	39,594.4	184.04	web-google	10,767	28,599	26,866	26,858.8	131.36
soc-brightkite	N/A	N/A	41,189	40,361.4	212.23	web-indochina-2004	N/A	N/A	98,809	98,657.3	176.93
soc-buzznet	N/A	N/A	38,628	37,701	174.26	web-it-2004	N/A	N/A	1,115,823	1,115,823	1.32
soc-delicious	N/A	N/A	28,263	27,792.6	135.52	web-polblogs	8579	17,312	13,549	13,498.1	171.71
soc-digg	N/A	N/A	55,879	54,964.3	164.61	web-sk-2005	N/A	N/A	126,823	126,524.1	144.92
soc-dolphins	2015	2015	2015	2015	0.00	web-spam	N/A	N/A	31,803	31,327.7	134.99
soc-douban	N/A	N/A	20,017	19,862	201.77	web-uk-2005	N/A	N/A	1,050,477	105,0477	0.44
soc-epinions	N/A	N/A	29,866	29,612.2	209.61	web-webbase-2001	N/A	N/A	44,256	43,708.3	156.66
socfb-A-anon	N/A	N/A	57,408	55,907.1	157.96	web-wikipedia2009	N/A	N/A	65,286	63,121.4	198.87

,

Table 5. Experiment results on real-world large graphs with $k=40$.

Instance	CPLEX		TOPKWCLQ			Instance	CPLEX		TOPKWCLQ
	$\mathit{LB}$	$\mathit{UB}$	${\mathit{w}}_{\mathit{b}}$	${\mathit{w}}_{\mathit{a}}$	$\mathit{Time}$		$\mathit{LB}$	$\mathit{UB}$	${\mathit{w}}_{\mathit{b}}$	${\mathit{w}}_{\mathit{a}}$	$\mathit{Time}$
bio-celegans	10,037	22,077	18,057	17,956.7	218.25	socfb-B-anon	N/A	N/A	71,895	70,879.8	170.92
bio-diseasome	10,450	24,138	21,945	21,933.6	188.25	socfb-Berkeley13	N/A	N/A	113,862	111,784.4	173.56
bio-dmela	N/A	N/A	18,403	18,266.6	197.28	socfb-CMU	N/A	N/A	87,972	86,992.2	180.33
bio-yeast	-	-	16,302	16,247.8	182.60	socfb-Duke14	N/A	N/A	93,294	92,967.7	224.83
ca-AstroPh	N/A	N/A	144,660	143,098.9	186.93	socfb-Indiana	N/A	N/A	135,869	135,265	178.45
ca-citeseer	N/A	N/A	218,277	217,906.1	194.61	socfb-MIT	N/A	N/A	90,760	89,700.5	182.88
ca-coauthors-dblp	N/A	N/A	924,023	921,156.3	170.24	socfb-OR	N/A	N/A	85,045	84,407.4	133.30
ca-CondMat	N/A	N/A	61,647	60,964.7	221.77	socfb-Penn94	N/A	N/A	121,319	120,455.7	179.44
ca-CSphd	N/A	N/A	15,398	15,370.2	206.08	socfb-Stanford3	N/A	N/A	112,011	111,138.3	140.08
ca-dblp-2010	N/A	N/A	180,532	179,364.5	182.20	socfb-Texas84	N/A	N/A	128,204	126,768.1	219.27
ca-dblp-2012	N/A	N/A	160,660	159,929.8	169.46	socfb-uci-uni	N/A	N/A	25,650	25,101.6	226.36
ca-Erdos992	N/A	N/A	16,877	16,752.9	125.53	socfb-UCLA	N/A	N/A	119,747	118,738.3	187.82
ca-GrQc	N/A	N/A	53,645	53,532.1	170.29	socfb-UConn	N/A	N/A	102,092	101,497.7	207.71
ca-HepPh	N/A	N/A	161,167	160,718.8	207.33	socfb-UCSB37	N/A	N/A	108,210	107,448.6	210.05
ca-hollywood-2009	N/A	N/A	1,815,958	1,803,757	213.10	socfb-UF	N/A	N/A	143,528	142,751.4	126.65
ca-MathSciNet	N/A	N/A	70,636	70,271.1	143.10	socfb-UIllinois	N/A	N/A	135,222	134,182.4	201.21
ca-netscience	10,519	22,011	21,076	21,072.8	136.11	socfb-Wisconsin87	N/A	N/A	104,664	103,275.1	203.96
ia-email-EU	N/A	N/A	21,652	21,347.9	217.56	soc-flickr	N/A	N/A	74,830	74,224.6	160.25
ia-email-univ	10,977	32,641	25,202	25,079.5	174.42	soc-flixster	N/A	N/A	68,593	67,034.3	182.67
ia-enron-large	N/A	N/A	50,536	50,100.5	197.78	soc-FourSquare	N/A	N/A	36,905	36,496.1	180.98
ia-enron-only	8159	10,121	9610	9590.5	123.20	soc-gowalla	N/A	N/A	61,166	60,477.1	106.86
ia-fb-messages	-	-	18,078	17,939.4	155.67	soc-karate	629	629	629	629	0.00
ia-infect-dublin	17,535	30,618	26,721	26,652.9	207.08	soc-lastfm	N/A	N/A	40,830	40,624.8	191.26
ia-infect-hyper	6554	6554	6554	6554	0.00	soc-livejournal	N/A	N/A	353,252	345,997.4	159.91
ia-reality	N/A	N/A	9474	9427.1	159.85	soc-LiveMocha	N/A	N/A	29,376	28,832	190.46
ia-wiki-Talk	N/A	N/A	31,492	31,299.6	216.93	soc-orkut	N/A	N/A	146,600	145,529.7	135.93
inf-power	N/A	N/A	20,842	20,757.7	129.00	soc-pokec	N/A	N/A	70,454	69,295.1	140.46
inf-roadNet-CA	N/A	N/A	24,261	24,204.7	150.68	soc-slashdot	N/A	N/A	34,450	33,981	205.31
inf-roadNet-PA	N/A	N/A	23,940	23,929.1	134.36	soc-twitter-follows	N/A	N/A	17,550	17,336.8	220.63
inf-road-usa	N/A	N/A	23,594	23,458.5	146.11	soc-wiki-Vote	8107	23,864	18,068	17,883.6	175.77
rec-amazon	N/A	N/A	33,363	33,356.2	161.28	soc-youtube	N/A	N/A	39,556	38,949	91.83
rt-retweet	4161	4169	4161	4161	10.27	soc-youtube-snap	N/A	N/A	38,929	38,679.1	195.84
rt-retweet-crawl	N/A	N/A	27,011	26,594	138.17	tech-as-caida2007	N/A	N/A	19,973	19,779.8	135.87
rt-twitter-copen	14,772	16,130	14,657	14,635.9	175.68	tech-as-skitter	N/A	N/A	72,571	71,620	209.40
sc-ldoor	N/A	N/A	161,126	160,998.6	178.39	tech-internet-as	N/A	N/A	21,029	20,875.3	175.95
sc-msdoor	N/A	N/A	159,967	159,737.1	224.65	tech-p2p-gnutella	N/A	N/A	20,773	20,588	110.00
sc-nasasrb	N/A	N/A	160,746	160,572.2	148.46	tech-RL-caida	N/A	N/A	36,579	36,137.1	133.97
sc-pkustk11	N/A	N/A	174,249	173,122.1	163.16	tech-routers-rf	N/A	N/A	24,391	24,223	156.74
sc-pkustk13	N/A	N/A	204,876	204,369.1	137.80	tech-WHOIS	N/A	N/A	61,598	60,786.6	181.99
sc-pwtk	N/A	N/A	175,800	175,532	183.57	web-arabic-2005	N/A	N/A	348,027	348,027	5.92
sc-shipsec1	N/A	N/A	113,485	112,618.6	214.02	web-BerkStan	N/A	N/A	32,566	32,486.7	199.25
sc-shipsec5	N/A	N/A	153,006	151,986	162.57	web-edu	N/A	N/A	24,780	24,730.8	219.82
soc-BlogCatalog	N/A	N/A	49,422	48,620.8	193.41	web-google	-	-	31,201	31,141.2	148.58
soc-brightkite	N/A	N/A	49,753	49,285.4	204.41	web-indochina-2004	N/A	N/A	119,525	119,336.2	220.28
soc-buzznet	N/A	N/A	47,092	46,465.3	230.39	web-it-2004	N/A	N/A	1,314,144	1,314,144	9.23
soc-delicious	N/A	N/A	35,377	34,841.4	161.33	web-polblogs	9606	21,840	16,844	16,756.7	201.77
soc-digg	N/A	N/A	67,806	66,826.6	248.28	web-sk-2005	N/A	N/A	154,004	153,682.4	175.56
soc-dolphins	2015	2015	2015	2015	0.00	web-spam	N/A	N/A	38,112	37,909.5	198.68
soc-douban	N/A	N/A	25,245	24,822.4	118.98	web-uk-2005	N/A	N/A	1,277,887	1,277,887	0.45
soc-epinions	N/A	N/A	37,345	36,766.1	187.05	web-webbase-2001	N/A	N/A	52,395	52,146	192.70
socfb-A-anon	N/A	N/A	71,908	71,376.4	198.96	web-wikipedia2009	N/A	N/A	79,081	77,242.5	215.47

and

Table 6. Experiment results on real-world large graphs with $k=50$.

Instance	CPLEX		TOPKWCLQ			Instance	CPLEX		TOPKWCLQ
	$\mathit{LB}$	$\mathit{UB}$	${\mathit{w}}_{\mathit{b}}$	${\mathit{w}}_{\mathit{a}}$	$\mathit{Time}$		$\mathit{LB}$	$\mathit{UB}$	${\mathit{w}}_{\mathit{b}}$	${\mathit{w}}_{\mathit{a}}$	$\mathit{Time}$
bio-celegans	14,811	25,659	21,032	20,899.3	242.15	socfb-B-anon	N/A	N/A	87,455	85,964.7	168.6
bio-diseasome	13,041	27,569	25,157	25,132.1	219.96	socfb-Berkeley13	N/A	N/A	132,576	131,226.4	245.12
bio-dmela	N/A	N/A	22,060	21,985	156.03	socfb-CMU	N/A	N/A	101,002	100,407.2	224.22
bio-yeast	N/A	N/A	19,700	19,584.3	130.19	socfb-Duke14	N/A	N/A	110,636	109,983.7	198.54
ca-AstroPh	N/A	N/A	167,224	166,691.2	161.95	socfb-Indiana	N/A	N/A	158,910	157,463.4	175.19
ca-citeseer	N/A	N/A	258,664	256,459.1	200.09	socfb-MIT	N/A	N/A	104,412	10,3962	155.76
ca-coauthors-dblp	N/A	N/A	1,103,639	1,100,063	173.91	socfb-OR	N/A	N/A	101,953	100,836	164.1
ca-CondMat	N/A	N/A	73,224	72,874.1	174.02	socfb-Penn94	N/A	N/A	143,535	141,653.6	171.98
ca-CSphd	N/A	N/A	18,913	18,888.6	215.74	socfb-Stanford3	N/A	N/A	130,978	130,307.6	160.01
ca-dblp-2010	N/A	N/A	211,258	210,658.3	202.01	socfb-Texas84	N/A	N/A	150,919	149,626.2	215.77
ca-dblp-2012	N/A	N/A	185,410	184,644.4	138.06	socfb-uci-uni	N/A	N/A	31,019	30,807.1	192.24
ca-Erdos992	N/A	N/A	20,331	20,133.4	134.71	socfb-UCLA	N/A	N/A	141,024	139,929.2	165.78
ca-GrQc	N/A	N/A	60,410	60,254	174.04	socfb-UConn	N/A	N/A	119,269	118,552.1	183.93
ca-HepPh	N/A	N/A	177,770	177,104.2	157.61	socfb-UCSB37	N/A	N/A	125,680	124,864.5	169.75
ca-hollywood-2009	N/A	N/A	2,017,283	2,004,122	220.66	socfb-UF	N/A	N/A	166,765	165,868.4	242.62
ca-MathSciNet	N/A	N/A	84,035	83,379.4	168.59	socfb-UIllinois	N/A	N/A	159,630	157,918	184.39
ca-netscience	14,917	25,103	23,696	23,684.2	146.82	socfb-Wisconsin87	N/A	N/A	122,934	122,265.3	172.9
ia-email-EU	N/A	N/A	25,603	25,175.5	229.9	soc-flickr	N/A	N/A	89,807	87,883	175.89
ia-email-univ	-	-	29,402	29,187.6	231.77	soc-flixster	N/A	N/A	80,451	79,254.8	212.02
ia-enron-large	N/A	N/A	60,880	60,270.8	212.19	soc-FourSquare	N/A	N/A	44,676	44,272	194.57
ia-enron-only	9157	10,428	10,297	10,266	141.51	soc-gowalla	N/A	N/A	72,608	72,149.7	213.64
ia-fb-messages	-	-	21,654	21,446	135.39	soc-karate	629	629	629	629	0
ia-infect-dublin	21,426	34,156	30,074	29,964.5	221.4	soc-lastfm	N/A	N/A	49,431	48,941.3	182.21
ia-infect-hyper	6554	6554	6554	6554	0	soc-livejournal	N/A	N/A	405,317	399,721.7	142.94
ia-reality	N/A	N/A	11,518	11,445.6	146.6	soc-LiveMocha	N/A	N/A	34,890	34,574.5	194.75
ia-wiki-Talk	N/A	N/A	38,239	37,417.3	192.79	soc-orkut	N/A	N/A	18,1096	178,069.6	139.86
inf-power	N/A	N/A	24,968	24,890.9	224.22	soc-pokec	N/A	N/A	85,010	84,118.3	188.24
inf-roadNet-CA	N/A	N/A	30,251	30,155.4	111.57	soc-slashdot	N/A	N/A	40,150	39,867.5	207.98
inf-roadNet-PA	N/A	N/A	29,887	29,867.4	199.69	soc-twitter-follows	N/A	N/A	21,301	21,065.8	187.84
inf-road-usa	N/A	N/A	29,316	29,201.1	142.64	soc-wiki-Vote	10,097	28,039	21,419	21,210.6	187.91
rec-amazon	N/A	N/A	41,146	41,124.2	148.14	soc-youtube	N/A	N/A	47,160	46,973	183.03
rt-retweet	4526	4620	4526	4526	0.14	soc-youtube-snap	N/A	N/A	47,523	46,964.5	234.3
rt-retweet-crawl	N/A	N/A	32,586	32,128.1	175.81	tech-as-caida2007	N/A	N/A	23,881	23,719.3	164.61
rt-twitter-copen	17,314	19,812	17,652	17,571	210.34	tech-as-skitter	N/A	N/A	84,746	83,760	193.48
sc-ldoor	N/A	N/A	200,767	200,601.1	197.73	tech-internet-as	N/A	N/A	25,236	25,078.6	252.39
sc-msdoor	N/A	N/A	199,076	198,801.3	129.64	tech-p2p-gnutella	N/A	N/A	25,238	25,143.9	187.28
sc-nasasrb	N/A	N/A	197,922	197,611.2	186.05	tech-RL-caida	N/A	N/A	43,728	43,471.3	151.62
sc-pkustk11	N/A	N/A	214,700	213,740.4	140.39	tech-routers-rf	N/A	N/A	28,349	28,125.2	137.27
sc-pkustk13	N/A	N/A	252,454	251,525.5	186.96	tech-WHOIS	N/A	N/A	68,102	67,397.8	165.01
sc-pwtk	N/A	N/A	218,360	217,983.9	161.86	web-arabic-2005	N/A	N/A	430,893	430,893	12.32
sc-shipsec1	N/A	N/A	138,034	137,296.4	181.85	web-BerkStan	N/A	N/A	39,160	38,997.3	183.66
sc-shipsec5	N/A	N/A	185,496	184,589.5	171.17	web-edu	N/A	N/A	28,721	28,667.3	205.28
soc-BlogCatalog	N/A	N/A	57,097	56,401.7	150.99	web-google	-	-	34,977	34,930.2	194.03
soc-brightkite	N/A	N/A	58,014	57,698.3	213.89	web-indochina-2004	N/A	N/A	138,107	137,833.2	161.98
soc-buzznet	N/A	N/A	55,420	54,720	209.46	web-it-2004	N/A	N/A	1,502,580	1,502,580	31.06
soc-delicious	N/A	N/A	42,317	41,594	169.03	web-polblogs	9793	25,434	19,890	19,727.5	208.03
soc-digg	N/A	N/A	79,149	77,969.7	168.98	web-sk-2005	N/A	N/A	180,230	179,935.1	183.74
soc-dolphins	2015	2015	2015	2015	0	web-spam	N/A	N/A	45,228	44,432.1	172.25
soc-douban	N/A	N/A	29,993	29,587.2	135.61	web-uk-2005	N/A	N/A	1,497,314	1,497,314	0.46
soc-epinions	N/A	N/A	43,676	43,427.1	179.86	web-webbase-2001	N/A	N/A	59,569	59,108.5	153.63
socfb-A-anon	N/A	N/A	87,217	86,621.5	203.76	web-wikipedia2009	N/A	N/A	90,898	90,000.9	190.59

. Among them, Table 2, Table 3, Table 4, Table 5 and Table 6 for $k=10,20,30,40,50$, respectively.

For each instance, the column “Instance” indicates the basic information for the name. In TOPKWCLQ, we present the maximum weight value of the DTKWC search problem instances ($w_b$) and the average weight DTKWC search problem results ($w_a$) obtained over 10 runs. We also report the average run time over 10 runs ($Time$, in seconds) to reach the maximum weight for all DTKWC search problem instances by TOPKWCLQ. And ”0” in the time column indicates TOPKWCLQ was able to obtain the best solution in less than 0.01 s. To study the effectiveness of TOPKWCLQ for DTKWC search problem, we compare it with the CPLEX solver (version 12.9) with the mathematical model (8)–(13) introduced in Section 2.2. The best lower bound ($LB$) and upper bound ($UB$) found by CPLEX are listed in the CPLEX columns. If CPLEX was unable to find a bound on an instance, the corresponding entry is marked by “-”. If CPLEX was unable to load the model, the entry is marked by “N/A”. For the items in a column, the bold value indicates that the algorithm obtained the same or better objective values compared to the results of the comparison algorithm.

Table 2, Table 3, Table 4, Table 5 and Table 6 show that TOPKWCLQ obtained the same or better objective values compared with the objective values of CPLEX on most instances. On 5 out of 102 graphs, CPLEX can find better objective values than our algorithm TOPKWCLQ. However, the instances become more challenging for CPLEX with a larger k, and TOPKWCLQ is becoming more effective. For 19 out of 510 real-world instances, we can prove the optimal solutions, where the values of the lower bound (“$LB$”) and upper bound (“$UB$”) are equivalent in CPLEX column. In terms of computational time, TOPKWCLQ can obtain the optimal values in less than one second (at most hundreds of seconds) in most cases. For example, on the graph rt-twitter-copen, CPLEX always finds better objective values than TOPKWCLQ except the instance with parameter $k=50$. For another 84 out of 102 larger graphs, TOPKWCLQ can also obtain good objective values where CPLEX failed.

Based on the benchmark introduced in Section 4.1,

Table 7. Summary of comparison between CPLEX and TOPKWCLQ on real-world graphs. #Better denotes the number of graphs where an algorithm finds better objective values. #N/A denotes the number of graphs where an algorithm fails to find an objective value.

Benchmark	k	CPLEX		TOPKWCLQ
		#Better	#N/A	#Better	#N/A
real-world graphs (102)	10	5	84	87	0
	20	2	85	97	0
	30	1	86	97	0
	40	1	86	97	0
	50	0	87	98	0

summarizes the computational results of CPLEX and TOPKWCLQ on 102 real-world graphs. From Table 7, we observe that for almost all instances under the five values of the parameter k, our TOPKWCLQ algorithm can obtain better solutions than the lower bound of CPLEX. It indicates the superiority of the proposed algorithm TOPWCLQ.

5. Conclusions

In this paper, we propose the diversified top-k weight clique search problem and formalize DTKC and DTKWC search problem. The scoring strategy is proposed to find diversified maximal weight cliques for our algorithm. A local search algorithm for the DTKWC search problem based on the scoring strategy and random restart strategy is then proposed, called TOPKWCLQ. This algorithm interleaves maximal weight clique set construction and updating. Experiments on the real-world benchmark show the effectiveness and efficiency of our algorithm. Moreover, further work is to investigate the enhanced configuration checking strategy used in [2] to enhance the performance of the algorithm.