A VNS Metaheuristic with Stochastic Steps for Max 3-Cut and Max 3-Section

Abstract

A heuristic algorithm based on VNS is proposed to solve the Max 3-cut and Max 3-section problems. By establishing a neighborhood structure of the Max 3-cut problem, we propose a local search algorithm and a variable neighborhood global search algorithm with two stochastic search steps to obtain the global solution. We give some numerical results and comparisons with the well-known 0.836-approximate algorithm. Numerical results show that the proposed heuristic algorithm can obtain efficiently the high-quality solutions and has the better numerical performance than the 0.836-approximate algorithm for the NP-Hard Max 3-cut and Max 3-section problems.

1. Introduction

Given a graph 𝐺(𝑉;𝐸), with nodes set 𝑉and edges set 𝐸, the Max 3-cut problem is to find a partition 𝑆₀⊂𝑉, 𝑆₁⊂𝑉 and 𝑆₂⊂𝑉, of the set 𝑉, such that 𝑆₀⋃𝑆₁⋃𝑆₂=𝑉, 𝑆_𝑖⋂𝑆_𝑗=∅ (𝑖≠𝑗) and the sum of the weights on the edges connecting the different parts is maximized. Similar to the Max cut problem, the Max 3-cut problem has long been known to be NP complete [1], even for any un-weighted graphs [2], and has also applications in circuit layout design, statistical physics, and so on [3]. However, due to the complexity of this problem, its research progresses is much lower than that of the Max cut problem. Based on the semidefinite programming relaxation proposed by Goemans and Williamson [4], Frieze and Jerrum [5] obtained a 0.800217-approximation algorithm for the Max 3-Cut problem. Recently, Goemans and Williamson [6] and Zhang and Huang [7] improved Frieze and Jerrum’s 0.800217-approximation ratio to 0.836 using a complex semidefinite programming relaxation of the Max 3-cut problem.

For the purpose of our analysis, we first introduce some notations. We denote the complex conjugate of 𝑦=𝑎+𝑖𝑏 by 𝑦=𝑎−𝑖𝑏, where i =√−1 is the pure image number and the real part and image part of a complex number by Re(⋅) and Im(⋅), respectively. For an 𝑛 dimensional complex vector 𝐲∈ℂ^𝑛 written as bold letter and 𝑛 dimensional complex matrix 𝑌∈ℂ^𝑛×𝑛, we write 𝐲^∗ and 𝑌^∗ to denote their conjugate and transpose. That is, 𝐲^∗=𝐲^𝑇 and 𝑌^∗=𝑌^𝑇. The set of 𝑛 dimensional real symmetric (semidefinite positive) matrices and the set of 𝑛 dimensional complex Hermitian (semidefinite positive) matrices are denoted by 𝒮_𝑛(𝒮^+𝑛) and ℋ_𝑛(ℋ^+𝑛), respectively. We sometimes use 𝐴≽0 to show 𝐴∈𝒮^+𝑛(or 𝐴∈ℋ^+𝑛). For any two complex vector 𝐮,  𝐯∈ℂ^𝑛, we write ⟨𝐮,𝐯⟩=𝐮⋅𝐯=𝐮^∗𝐯 as their inner product. For any two complex matrices 𝐴, 𝐵∈ℋ_𝑛, we write ⟨𝐴,𝐵⟩=𝐴⋅𝐵 as their inner product; that is, ⟨𝐴,𝐵⟩=𝐴⋅𝐵=Tr(𝐵^∗𝐴)=∑_𝑖,𝑗𝑏_𝑖𝑗𝑎_𝑖𝑗, where 𝐴=(𝑎_𝑖𝑗) and 𝐵=(𝑏_𝑖𝑗). ‖⋅‖ means the module of a complex number or the 2-norm of a complex vector or the 𝐹-norm of a complex matrix.

Let the third root of unity be denoted by 𝜔⁰=1, 𝜔=𝜔¹=𝑒^{𝑖(2𝜋/3)}, 𝜔²=𝑒^{𝑖(4𝜋/3)}. Introduce a complex variable 𝑦_𝑖∈{1,𝜔,𝜔²}, 𝑖=1,…,𝑛, then it is not hard to know that 23−13𝑦^𝑖⋅𝑦_𝑗−13𝑦^𝑗⋅𝑦_𝑖=231−Re𝑦^𝑖⋅𝑦_𝑗.(1.1) Denote 𝑆_𝑘={𝑖∶𝑦_𝑖=𝜔^𝑘,𝑘=0,1,2} and 𝐲=(𝑦₁,…,𝑦_𝑛)^𝑇. Then the Max 3-cut problem can be expressed as M3C∶max𝑓(𝐲)=23_𝑖<𝑗s.t.𝐲∈{1,𝜔,𝜔₂𝑤_𝑖𝑗1−Re𝑦_𝑖⋅𝑦_𝑗}^𝑛,(1.2) here 𝐲∈{1,𝜔,𝜔²}^𝑛 means that 𝑦_𝑖∈{1,𝜔,𝜔²}, 𝑖=1,…,𝑛, 𝑊=(𝑤_𝑖𝑗) is the weight-valued matrix of a given graph.

By relaxing the complex variable 𝑦_𝑖 into an 𝑛 dimensional complex vector 𝐲_𝑖, we get a complex semidefinite programming (CSDP) relaxation of (M3C) as follows: CSDP∶max23_𝑖<𝑗s.t.‖‖𝐲_𝑖𝑤_𝑖𝑗1−Re𝐲_𝑖⋅𝐲_𝑗𝐴‖‖=1,𝑖=1,2,…,𝑛,_𝑘𝑖𝑗𝑌≽⋅0𝑌,≥−1,𝑖,𝑗=1,2,…,𝑛,𝑘=0,1,2(1.3) where 𝑌_𝑖𝑗=𝐲_𝑖⋅𝐲_𝑗, 𝐴^𝑘𝑖𝑗=𝜔^𝑘𝐞_𝑖𝐞^𝑇𝑗+𝜔^−𝑘𝐞_𝑗𝐞^𝑇𝑖 and 𝐞_𝑖 denotes the vector with zeros everywhere except for an unit in the 𝑖th component. It is easily to verify that constraints 𝐴^𝑘𝑖𝑗⋅𝑌≥−1 can be expressed as Re𝜔^𝑘𝑌_𝑖𝑗≥−12,𝑘=0,1,2.(1.4) To get an approximate solution of M3C, Goemans and Williamson [6] do not directly solve the CSDP, but solve an equivalent real SDP with following form (Although some softwares, such as SeDuMi [8] and the earlier version of SDPT3-4.0 [9], can deal with SDPs with complex data, this does not reduce the dimensions of problems): RSDP∶max12⎡⎢⎢⎣𝑄𝑂𝑄𝑂⎤⎥⎥⎦⋅𝑋s.t.⎡⎢⎢⎣𝐞_𝑖𝑂𝐞^𝑇𝑖𝐞_𝑖𝑂𝐞^𝑇𝑖⎡⎢⎢⎣Re𝐴_𝑘𝑖𝑗⎤⎥⎥⎦⋅𝑋=2,𝑖=1,2,…,𝑛,Im𝐴_𝑘𝑖𝑗−Im𝐴^𝑘𝑖𝑗⎡⎢⎢⎣𝐴Re𝐴^𝑘𝑖𝑗⎤⎥⎥⎦⋅𝑋≥−2,1≤𝑖<𝑗≤𝑛,𝑘=0,1,2_0𝑖𝑗⎡⎢⎢⎣𝑂𝑂𝐴−𝑂𝐴_{0𝑖𝑗0𝑖𝑗}⎤⎥⎥⎦⋅𝑋=0,1≤𝑖<𝑗≤𝑛,⎡⎢⎢⎣𝐴𝑂^0𝑖𝑗𝑂𝐞_𝑖⎤⎥⎥⎦⋅𝑋=0,1≤𝑖<𝑗≤𝑛,𝐞^𝑇𝑖𝑋𝐞_𝑖∈𝐞^𝑇𝑖𝓢_+2𝑛𝑂⎤⎥⎥⎦⋅𝑋=0,𝑖=1,2,…,𝑛,,(1.5) where 𝑄=(1/3)diag(𝑊𝑒)−𝑊 is the Laplace matrix of given graph, 𝑂 is an 𝑛-dimensional full zeros matrix.

In RSDP, the first, third, and forth classes of equality constraints ensure that 𝑋_𝑖𝑖=1, 𝑖=1,2,…,𝑛 and with the form 𝑋=⎡⎢⎢⎣𝑅𝑆−𝑅𝑆⎤⎥⎥⎦.(1.6) The final two classes of equality constraints ensure that 𝑆_𝑖𝑖=0(𝑖=1,…,𝑛) and 𝑆 is a skew-symmetric matrix.

If 𝑋 is an optimal solution of RSDP, then the complex matrix 𝑌=𝑅+𝑖𝑆 is an optimal solution of CSDP. Then one can randomly generate a complex vector 𝜉, such that 𝜉∼𝑁(0,𝑌), and set 𝑦_𝑖=⎧⎪⎪⎪⎨⎪⎪⎪⎩1,ifArg𝜉^𝑖𝜔,ifArg𝜉_𝑖∈0,23𝜋,𝜔₂∈23𝜋,43𝜋,,ifArg𝜉_𝑖∈4𝜋3,2𝜋,(1.7) where Arg(⋅)∈[0,2𝜋] means the complex angle principal value of a complex number. Goemans and Williamson [6] had verified that, see also Zhang and Huang [7], 𝑓(𝐲̂)≥0.836⋅𝑄⋅𝑌.(1.8)

The algorithm proposed by Goemans and Williamson [6] can obtain a very good approximate ratio, and RSDP can be solved by interior point algorithm, but the 0.836-approximate algorithm will be not practical in numerical study for the Max 3-cut problem. From RSDP, one can see that for a graph with 𝑛 nodes, RSDP has 2𝑛+5𝑛(𝑛−1)/2 constraints and 3𝑛(𝑛−1)/2 slack variables via the inequality constraints. That is to say, RSDP has a 2𝑛 dimensional unknown symmetrical semidefinite positive matrix variable and a 3𝑛(𝑛−1)/2 dimensional unknown vector variable, and 2𝑛+5𝑛(𝑛−1)/2 constraints, and has also many matrices without an explicit block diagonal structure although they are sparse. For instance, when 𝑛=100, RSDP becomes a very-high-dimensional semidefinite programming problem with 14850 slack variables and 24950 constraints. Further, as we known, it is only a class of universal and medium-scale instances for Max 3-cut problems with 50 to 100 nodes. Hence, it will be very time consuming to solve such a RSDP relaxation of M3C using the current existing any SDP softwares. This leads that 0.836-approximate algorithm is not suitable for computational study of the Max 3-cut problem. This limitation for solving M3C based on CSDP (or RSDP) relaxation motivates us to find a new efficient and fast algorithm for the practical purpose for the Max 3-cut problem.

In the current paper, we first establish a definition of 𝐾-neighborhood structure of the Max 3-cut problem and design a local search algorithm to find the local minimizer. And then, we propose a variable neighborhood search (VNS) metaheuristic with stochastic steps which is originally considered by Mladenović and Hansen [10], by which we can find efficiently a high-quality global approximate solution of the Max 3-cut problem. Further, combining a greed algorithm, we extend the proposed algorithm to the Max 3-section problem. To the best of our knowledge, it is first time to consider the computational study of the Max 3-cut problem. In order to test the performance of the proposed algorithm, we compare the numerical results with Goemans and Williamson’s 0.836-approximate algorithm.

This paper is organized as follows. In Section 2, we give some definitions and lemmas. In Section 3, we present the VNS metaheuristic for solving the Max 3-cut problem. The VNS is extended to the Max 3-section problem in Section 4. In Section 5, we give some numerical results and comparisons.

2. Preliminaries

In this section, we will establish some definitions and give some facts for our sequel purpose. For the third roots of unity, 1, 𝜔, 𝜔², we can get the following fact: ‖1−𝜔‖²=‖‖𝜔−𝜔²‖‖²=‖‖1−𝜔²‖‖²=3.(2.1) Denote 𝕊={1,𝜔,𝜔²}^𝑛. Then based on (2.1), for any 𝐲∈𝕊, we may definite a 𝐾-neighborhood of 𝐲 as follows.

Definition 2.1. For any 𝐲∈𝕊 and any positive integer number 𝐾(1≤𝐾≤𝑛), one defines the 𝐾-neighborhood, denoted by 𝑁_𝐾(𝐲), of 𝐲 as the set 𝑁_𝐾(𝐲)=𝐳∈𝕊∶‖𝐳−𝐲‖²=∑^𝑛_𝑖=1‖‖𝑧_𝑖−𝑦_𝑖‖‖²≤3𝐾.(2.2) In particular, if 𝐾=1, we write the 1-neighborhood 𝑁₁(𝐲) of 𝐲 as 𝑁(𝐲).

The boundary of the 𝐾-neighborhood 𝑁_𝐾(𝐲) is defined by 𝜕𝑁_𝐾(𝐲)={𝐳∈𝕊∶‖𝐲−𝐳‖²=3𝐾}. Clearly, 𝑁(𝐲)=𝜕𝑁(𝐲). If 𝐳∈𝜕𝑁_𝐾(𝐲), we call 𝐳 a 𝐾-neighbor of 𝐲. From Definition 2.1, the difference of between points 𝐲 and its 𝐾-neighbor 𝐳 is that they have only 𝐾 different components. By computing straightforwardly, we get the number of elements of 𝜕𝑁_𝐾(𝐲), that is |𝜕𝑁_𝐾(𝐲)|=2^𝐾𝐶^𝐾𝑛. Particularly, |𝜕𝑁(𝐲)|=2𝑛 when 𝐾=1.

Example 2.2. Let 𝐲=(𝜔,𝜔,𝜔²)^𝑇∈{1,𝜔,𝜔²}³. Then (1,𝜔,𝜔²)^𝑇∈𝑁(𝐲), (1,𝜔²,𝜔²)^𝑇∈𝜕𝑁₂(𝐲)⊂𝑁₂(𝐲), and (1,𝜔²,𝜔)^𝑇∈𝜕𝑁₃(𝐲)⊂𝑁₃(𝐲).

Definition 2.3. For any 𝑢∈{0,1,2}, define two maps from {1,𝜔,𝜔²} to itself as follows: 𝜏_𝑖(𝜔^𝑢)=𝜔^𝑢+𝑖∈{1,𝜔,𝜔²},𝑖=1,2.

Clearly, for any 𝑢∈{0,1,2}, 𝜏_𝑖(𝜔^𝑢)≠𝜔^𝑢, 𝑖=1,2 and 𝜏₁(𝜔^𝑢)≠𝜏₂(𝜔^𝑢). Applying Definition 2.3, for any 𝐳∈𝑁(𝐲) there exists an unique component, 𝑧_𝑘 say, of 𝐳, such that 𝑧_𝑘≠𝑦_𝑘 and either 𝑧_𝑘=𝜏₁(𝑦_𝑘) or 𝑧_𝑘=𝜏₂(𝑦_𝑘), and other components of 𝐳 and 𝐲 are the same. For simplicity, for any 𝐳∈𝑁(𝐲) with 𝑧_𝑘≠𝑦_𝑘 and 𝑧_𝑖=𝑦_𝑖(𝑖=1,…,𝑛,𝑖≠𝑘), we denote by 𝐳=𝜏^𝑘1(𝐲) or 𝐳=𝜏^𝑘2(𝐲) corresponding to 𝑧_𝑘=𝜏₁(𝑦_𝑘) or 𝑧_𝑘=𝜏₂(𝑦_𝑘). By Definitions 2.1 and 2.3, for any 𝐲∈𝕊, we can structure its 1-neighborhood points using maps defined by Definition 2.3; that is, we have the following result.

Lemma 2.4. Let 𝜏_𝑖(⋅)(𝑖=1,2) be defined by Definition 2.3. Then, for any 𝐲∈𝕊 and any fixed positive integer number 𝑘(1≤𝑘≤𝑛), one has 𝜏^𝑘𝑖(𝐲)∈𝑁(𝐲),𝑖=1,2,(2.3) that is, 𝜏^𝑘1(𝐲) and 𝜏^𝑘2(𝐲) are two 1-neighborhood points of 𝐲.

Definition 2.5. A point 𝐲̂∈𝕊 is called a 𝐾-local maximizer of the function 𝑓 over 𝕊, if 𝑓(𝐲̂)≥𝑓(𝐲), for all 𝐲∈𝑁_𝐾(𝐲̂). Furthermore, if 𝑓(𝐲̂)≥𝑓(𝐲) for all 𝐲∈𝕊, then 𝐲̂ is called a global maximizer of 𝑓 over 𝕊. A 1-local maximizer of the function 𝑓 is also called a local maximizer of the function 𝑓 over 𝕊.

3. VNS for Max 3-Cut

3.1. Local Search Algorithm

Let 𝐲⁰=(𝑦⁰¹,…,𝑦^0𝑛)^𝑇∈𝕊 be a feasible solution of problem M3C. If 𝐲⁰ is not a local maximizer of 𝑓, then for all 𝐲∈𝑁(𝐲⁰), we may find a 𝐲̃∈𝑁(𝐲⁰), such that 𝑓(𝐲̃)=max{𝑓(𝐲)∶𝐲∈𝑁(𝐲⁰)}. It is clear that 𝑓(𝐲̃)≥𝑓(𝐲⁰). If 𝐲̃ is not still a local maximizer of 𝑓, then replacing 𝐲⁰ with 𝐲̃ and repeating the process until a point 𝐲̂ satisfying 𝑓(𝐲̂)=max{𝑓(𝐲)∶𝐲∈𝑁(𝐲̂)} is found, which indicates that 𝐲̂ is a local maximizer of 𝑓.

For any positive integer number 𝑘(1≤𝑘≤𝑛), let 𝐲^𝑘=(𝑦^𝑘1,…,𝑦^𝑘𝑛)^𝑇=𝜏^𝑘𝑖(𝐲⁰)∈𝑁(𝐲⁰)(𝑖=1,2); that is, 𝑦^𝑘𝑖=𝑦^0𝑖,𝑖=1,2𝑦,…,𝑘−1,𝑘+1,…,𝑛;_𝑘𝑘≠𝑦^0𝑘.(3.1) Denote 𝛿(𝑘)=𝑓𝐲⁰−𝑓𝐲^𝑘.(3.2) Then, we have the following result whose proof is clear.

Lemma 3.1. Consider 𝛿(𝑘)=⎧⎪⎪⎨⎪⎪⎩23^𝑘−1_𝑖∑₌₁𝑤_𝑖𝑘Re𝑦^0𝑖⋅𝑦^𝑘𝑘−𝑦^0𝑘+23^𝑛2∑_𝑗=𝑘+1𝑤_𝑘𝑗Re𝑦^𝑘𝑘−𝑦^0𝑘⋅𝑦^0𝑗,𝑘>1;3^𝑛_𝑗=∑_𝑘+1𝑤_𝑘𝑗Re𝑦^𝑘𝑘−𝑦^0𝑘⋅𝑦^0𝑗,𝑘=1.(3.3)

Based on Lemma 3.1, if we know the value of 𝑓(𝐲⁰), then we can obtain the value function 𝑓(𝐲^𝑘) at next iterative point 𝐲^𝑘 by calculating 𝛿(𝑘) by (3.3), instead of calculating directly the values 𝑓(𝐲^𝑘), which reduces sharply the computational cost. By Definition 2.1, there exist two points satisfying (3.1) for fixed k; that is, when 𝐲^𝑘∈𝑁(𝐲⁰) and (3.1) is satisfied, then either 𝑦^𝑘𝑘=𝜏₁(𝑦^0𝑘) or 𝑦^𝑘𝑘=𝜏₂(𝑦^0𝑘). For our convenience, we denote 𝛿(𝑘) by 𝛿₁(𝑘) when 𝑦^𝑘𝑘=𝜏₁(𝑦^0𝑘) and by 𝛿₂(𝑘) when 𝑦^𝑘𝑘=𝜏₂(𝑦^0𝑘). In what follows, we describe the local search algorithm for the Max 3-cut problem denoted by LSM3C; by this algorithm, we can get a local maximizer of function𝑓(𝐲) over 𝕊.

For LSM3C, one has the following.(1)Input any initial feasible solution 𝐲⁰ of problem (M3C). (2)For 𝑘 from 1 to 𝑛, set 𝐳^𝑘1=𝜏^𝑘1(𝐲⁰), calculate 𝛿₁(𝑘), and set again 𝐳^𝑘2=𝜏^𝑘2(𝐲⁰), calculate 𝛿₂(𝑘). (3)Find 𝛿_𝑖∗(𝑘^∗) by the following way: 𝛿_𝑖∗𝑘^∗=min𝛿₁(1),𝛿₂(1),…,𝛿₁(𝑘),𝛿₂(𝑘),…,𝛿₁(𝑛),𝛿₂(𝑛).(3.4)(4)If 𝛿_𝑖∗(𝑘^∗)≥0, then set 𝐲̂=𝐲⁰, return 𝐲̂, and stop. Otherwise, go to next. (5)Set 𝐲⁰=𝜏^𝑘∗_𝑖∗(𝐲⁰); go to Step 2.

3.2. Variable Neighborhood Stochastic Search

Let 𝐲̂ be a local maximizer obtained by LSM3C and 𝐾_max(1<𝐾_max≤𝑛) a fixed positive integer number. we now describe the variable neighborhood search (VNS) with stochastic steps, by which we can find an approximate global maximizer of problem (M3C). The proposed VNS algorithm actually has three phases: First, for any given positive integer number 𝐾<𝐾_max, a 𝐾-neighborhood point, 𝐲 say, is randomly selected; that is, 𝐲∈𝑁_𝐾(𝐲̂). Next, a solution, 𝐲̂̂ say, is obtained by applying algorithm LSM3C to 𝐲. Finally, the current solution jumps from 𝐲̂ to 𝐲̂̂ if it improves the former one. Otherwise, the order 𝐾 of the neighborhood is increased by one when 𝐾<𝐾_max and the above steps are repeated until some stopping condition is met. The VNS that is also called k-max [11] can be illustrated as follows.

For VNS-k, one has the following.(1)Arbitrary choose a point 𝐲⁰∈𝕊, implement LSM3C starting from 𝐲⁰∈𝕊 and denote the obtained local maximizer by𝐲̂.Set 𝐾=1. (2)Randomly take a point 𝐲∈𝜕𝑁_𝐼(𝐾)(𝐲̂) and implement again LSM3C from 𝐲, and denote the obtained new local maximizer by 𝐲̂̂. (3)If 𝑓(𝐲̂̂)>𝑓(𝐲̂), then set 𝐲̂=𝐲̂̂ and 𝐾=1; go to Step 2. (4)If 𝐾<𝐾_max(≤𝑛), set 𝐾=𝐾+1; go to Step 2. Otherwise, return 𝐲̂ as an approximate global solution of problem M3C and stop.

The subscript 𝐼(𝐾) in Step 2 is a function of 𝐾 and is also a positive integer number not greater than 𝑛. 𝐼(𝐾) reflects the main skill of converting the current neighborhood of local maximizer 𝐲̂ into another neighborhood of 𝐲̂. For a given 𝐾_max, let 𝑚=⌊𝑛/𝐾_max⌋ and 𝐾₀=𝑛−𝑚𝐾_max, where ⌊𝑎⌋ means the integral part of 𝑎. We divide the 𝑛 neighborhoods of 𝐲̂, 𝑁(𝐲̂),𝑁₂(𝐲̂),…,𝑁_𝐾(𝐲̂),…,𝑁_𝑛(𝐲̂) into 𝐾_max neighborhood blocks 𝑁_𝐼(1)(𝐲̂),…,𝑁_{𝐼(𝐾max)}(𝐲̂), such that, for 𝐾=1,2,…,𝐾_max−𝐾₀, 𝑁_{(𝐾−1)𝑚+1}(𝐲̂)⊆𝑁_𝐼(𝐾)(𝐲̂)⊆𝑁_𝐾𝑚+1(𝐲̂),(3.5) and, for 𝐾=𝐾_max−𝐾₀+1,…,𝐾_max−𝐾₀+𝑗,…,𝐾_max, 𝑁_{(𝐾−1)(𝑚+1)+1}(𝐲̂)⊆𝑁_𝐼(𝐾)(𝐲̂)⊆𝑁_𝐾(𝑚+1)(𝐲̂).(3.6) In order to obtain the 𝐾_max neighborhood blocks of 𝐲̂, 𝑁_𝐼(𝐾)(𝐲̂), 𝐾=1,…,𝐾_max, we divide the set {1,2,…,𝑛} into 𝐾_max disjoint subsets, where each subset of the first 𝐾_max−𝐾₀ subsets has 𝑚 integers and each subset of the last 𝐾₀ subsets has 𝑚+1 integers. For any integer 𝐾(≤𝐾_max), let 𝐼(𝐾)=(𝐾−1)⋅𝑚+[𝑐⋅𝑚]+1,𝐾=1,2,…,𝐾_max−𝐾₀,(3.7) or 𝐼(𝐾)==[(𝑚𝐾+_max1)−⋅𝐾𝑐]₀+𝑚1++([𝑚(𝑚++11))(𝐾⋅𝑐−]1+)1−+𝐾(𝑚+_max1)𝐾−𝐾_max−𝐾₀−1𝐾=𝐾_max−𝐾₀,−𝐾₀+1,…,𝐾_max−𝐾₀+𝑗,…,𝐾_max.(3.8) Then we can randomly choose a point 𝐲 in 𝜕𝑁_𝐼(𝐾)(𝐲̂), where 𝑐∈(0,1) is a random number from uniformly distribution 𝒰(0,1), such that 𝑁_𝐼(𝐾)(𝐲̂) satisfies (3.5) or (3.6).

VNS-k stops when the maximum 𝐾 neighborhood is reached. Additionally, we also consider another termination criterion of VNS based on the maximum CPU-time and denoted by VNS-t. VNS-t can obtain a better solution than VNS-k since VNS-t actually runs several times VNS-k in the maximum allowing time 𝑡_max, but it generally has to spend more computational time. The VNS-t can be stated as follows.

For VNS-t, one has the following.(1)Set 𝑡_CPU=0, running VNS-k for an arbitrary initial point 𝐲⁰∈𝕊, and let a local optimal solution 𝐲̂ be obtained. (2)If 𝐾=𝐾_max(≤𝑛), go to Step 3. (3)If 𝑡_CPU<𝑡_max, then set 𝐾=1; go to Step 2 in VNS-k. Otherwise, return 𝐲̂ as an approximate global solution of problem M3C and stop.

We mention that it differs from the classical variable neighborhood search metaheuristic that is originally proposed by Mladenović and Hansen [10]. In order to obtain a global optimal solution or a high-quality approximate solution of problem M3C, we use two stochastic steps in VNS. First, for a fixed 𝐾, a 𝐾-neighbor of 𝐲̂ is chosen randomly. Second, by the definition of 𝐼(𝐾), when we change the neighborhood of 𝐲̂ from 𝑁_{𝐼(𝐾−1)} to 𝑁_𝐼(𝐾), 𝑁_𝐼(𝐾) may take any a neighborhood among 𝑁_{(𝐾−1)𝑚+𝑗}, 𝑗=1,2,…,𝑚 of 𝐲̂, which is decided by random number 𝑐. In VNS, positive integer 𝐾_max decides the maximum search neighborhood block of 𝐲̂, which also decides directly the CPU-time of VNS. Based on the second stochastic step, we may choose a relative small 𝐾_max comparing with 𝑛. This can decrease our computational time.

4. A Greedy Algorithm for Max 3-Section

When the number of nodes 𝑛 is a multiple of three and the condition |𝑆₀|=|𝑆₁|=|𝑆₂|=𝑛/3 is required, the Max 3-cut problem becomes the Max 3-section problem. Notice that 1+𝜔+𝜔²=0, then the Max 3-section problem can be formulated as the following programming problem M3S: M3S∶max23_𝑖<𝑗s.t._𝑛𝑤_𝑖𝑗1−Re𝑦_𝑖⋅𝑦_𝑗_𝑖=1𝐲∈𝕊,𝑦_𝑖=0,(4.1) and its CSDP relaxation is CSDP1∶max23_𝑖<𝑗s.t.𝐞𝐞_𝑇𝑤_𝑖𝑗1−Re𝐲_𝑖⋅𝐲_𝑗‖‖𝐲⋅𝑌=0,_𝑖𝐴‖‖=1,𝑖=1,2,…,𝑛,_𝑘𝑖𝑗𝑌≽⋅0𝑌,≥−1,𝑖,𝑗=1,2,…,𝑛,𝑘=0,1,2(4.2) where 𝐞 is the column vector of all ones. Andersson [12] extended Frieze and Jerrum’s random rounding method to M3S and obtained a (2/3)+𝑂(1/𝑛³)-approximate algorithm, which is the current best approximate ratio for M3S; also see the recent research of Gaur et al. [13]. The author of the current paper considers a special the Max 3-Section problem and obtains a 0.6733-approximate algorithm; see Ling (2009) [14].

Clearly, the feasible region of problem M3S is a subset of 𝕊, and the optimal value of problem M3S is not greater than that of problem M3C. Assume that we have get a global optimal solution or a high-quality approximate solution 𝐲̂ of problem M3C. It is clear that 𝐲̂ may not satisfy the condition ∑^𝑛𝑖=1𝑦_𝑖̂=0. But we may adjust 𝐲̂ to get a new feasible solution 𝐲^𝑠 using a greedy algorithm, such that 𝐲^𝑠 satisfies ∑^𝑛𝑖=1𝑦^𝑠𝑖=0. This is the motivation that we propose the greedy algorithm for the Max 3-section problem.

For the sake of our analysis, without loss of generality, we assume that the local maximizer 𝐲̂ satisfies |𝑆₀|=max{|𝑆₀|,|𝑆₁|,|𝑆₂|}. This means that 𝑆₀={𝑖∶𝑦_𝑖̂=1} is the subset of 𝑉 with maximum cardinal number. If |𝑆_𝑘|=max{|𝑆₀|,|𝑆₁|,|𝑆₂|}(𝑘≠0,𝑘=1,2), then we may set 𝑦^𝑁𝑖=𝑤^𝑘𝑦_𝑖̂, 𝑖=1,…,𝑛. The resulted new solution 𝐲^𝑁=(𝑦^𝑁1,…,𝑦^𝑁𝑖) will not change the objective value since 𝑓(𝐲̂)=𝑓(𝑤^𝑘𝐲̂)(𝑘≠0,𝑘=1,2); moreover, the new partition {𝑆^𝑁0,𝑆^𝑁1,𝑆^𝑁2} based on 𝐲^𝑁 satisfies |𝑆^𝑁0|=max{|𝑆^𝑁0|,|𝑆^𝑁1|,|𝑆^𝑁2|}. By our assumption, the partition 𝑆={𝑆₀,𝑆₁,𝑆₂} still exist four possible cases.

Case 1. |𝑆₀|≥|𝑆₁|≥𝑛/3≥|𝑆₂|.

Case 2. |𝑆₀|≥𝑛/3≥|𝑆₁|≥|𝑆₂|.

Case 3. |𝑆₀|≥|𝑆₂|≥𝑛/3≥|𝑆₁|.

Case 4. |𝑆₀|≥𝑛/3≥|𝑆₂|≥|𝑆₁|.
The sizes adjusting greedy algorithm of Cases 3 and 4 are similar to Cases 1 and 2. Hence, we mainly consider Cases 1 and 2 for adjusting the partition of 𝑉 from 𝑆={𝑆₀,𝑆₁,𝑆₂} to 𝑆={𝑆₀,𝑆₁,𝑆₂} such that |𝑆_𝑘|=𝑛/3, 𝑘=0,1,2. Denote 𝛿₀₁𝛿₀(𝑖)=_{𝑗∈𝑆1∪𝑆2}𝑤_𝑖𝑗,𝑖∈𝑆₀,𝛿₀₂(𝑖)=_𝑗∈𝑆1𝑤_𝑖𝑗,𝑖∈𝑆₀,𝛿₁₀(𝑖)=_𝑗∈𝑆0𝑤_𝑖𝑗,𝑖∈𝑆₁,𝛿₁₂(𝑖)=_𝑗∈𝑆2𝑤_𝑖𝑗,𝑖∈𝑆₀,𝛿₂₀(𝑖)=_𝑗∈𝑆0𝑤_𝑖𝑗,𝑖∈𝑆₂,(𝑖)=_𝑗∈𝑆2𝑤_𝑖𝑗,𝑖∈𝑆₁,𝛿₂₁(𝑖)=_𝑗∈𝑆1𝑤_𝑖𝑗,𝑖∈𝑆₂.(4.3) Then, it follows from simple computation that 𝛿_{0𝑖∈𝑆𝑘}(𝑖)=𝛿₀₁(𝑖)+𝛿₀₂(𝑖),foreach𝑖∈𝑆₀,𝑓(𝐲̂)=𝛿_𝑘𝑙∑(𝑖)=_𝑖∈𝑆0_{𝑖∈𝑆𝑙}𝛿_𝑙𝑘(𝑖),𝑘,𝑙=0,1,2,𝑘≠𝑙,=∑𝛿₀(𝑖)+∑_𝑖∈𝑆1𝛿₁₂(𝑖)_𝑖∈𝑆0=𝑑₀₁𝛿₀₁(𝑖)+∑_𝑖∈𝑆0𝛿₀₂(𝑖)+∑_𝑖∈𝑆1𝛿₁₂(𝑖)+𝑑₀₂+𝑑₁₂,(4.4) where 𝑑₀₁=∑_𝑖∈𝑆0𝛿₀₁(𝑖), 𝑑₀₂=∑_𝑖∈𝑆0𝛿₀₂(𝑖), 𝑑₁₂=∑_𝑖∈𝑆1𝛿₁₂(𝑖).
In what follows, we describe the size adjusting greedy algorithms (SAGAs) for Cases 1 and 2, and denote the greedy algorithms for the two cases by SAGA1 and SAGA2, respectively.
For SAGA1, one has the following.(1)Calculate 𝑚₀₂=∑^𝑖∈𝑆0𝛿₀₂(𝑖)||𝑆₀||,𝑚¹²=∑^𝑖∈𝑆1𝛿₁₂(𝑖)||𝑆₁||.(4.5)(2)If 𝑚₀₂≥𝑚₁₂, let 𝑆₁={𝑗₁,𝑗₂,…,𝑗_|𝑆1|}, where 𝛿₁₂(𝑗_𝑙)≥𝛿₁₂(𝑗_𝑙+1), 𝑙=1,2,…,|𝑆₁|. Set 𝑆₁={𝑗₁,𝑗₂,…,𝑗_𝑛/3}, 𝑆₂=𝑆₂⋃(𝑆₁⧵𝑆₁) and renew to calculate 𝛿^02(𝑖)=_𝑗∈𝑆₂𝑤_𝑖𝑗,(4.6) for each 𝑖∈𝑆₀. Let 𝑆₀={𝑖₁,𝑖₂,…,𝑖_|𝑆0|}, where 𝛿^02(𝑖_𝑘)≥𝛿^02(𝑖_𝑘+1). Set 𝑆₀={𝑖₁,𝑖₂,…,𝑖_𝑛/3} and 𝑆₂=𝑆₂⋃(𝑆₀⧵𝑆₀).(3)If 𝑚₀₂<𝑚₁₂, let 𝑆₀={𝑖₁,𝑖₂,…,𝑖_|𝑆0|}, where 𝛿₀₂(𝑖_𝑘)≥𝛿₀₂(𝑖_𝑘+1), 𝑘=1,2,…,|𝑆₀|, set 𝑆₀={𝑖₁,𝑖₂,…,𝑖_𝑛/3}, 𝑆₂=𝑆₂⋃(𝑆₀⧵𝑆₀), and then renew to calculate 𝛿^12(𝑖)=_𝑗∈𝑆₂𝑤_𝑖𝑗,(4.7) for each 𝑖∈𝑆₁. Set 𝑆₁={𝑗₁,𝑗₂,…,𝑗_𝑛/3} and 𝑆₂=𝑆₂⋃(𝑆₁⧵𝑆₁), where 𝛿^12(𝑗_𝑘)≥𝛿^12(𝑗_𝑘+1) here.(4)Return the current partition 𝑆={𝑆₀,𝑆₁,𝑆₂}; stop.
For SAGA2, one has the following.(1)Calculate 𝑑₀₁=∑_𝑖∈𝑆0𝛿₀₁(𝑖), 𝑑₀₂=∑_𝑖∈𝑆0𝛿₀₂(𝑖), and 𝑚₀₁=𝑑⁰¹||𝑆₀||,𝑚⁰²=𝑑⁰²||𝑆₀||.(4.8)(2)If 𝑚₀₁≤𝑚₀₂, let 𝑆₀=𝑖₁,𝑖₂,…,𝑖_|𝑆0|,(4.9) where 𝛿₀₁(𝑖_𝑘)≥𝛿₀₁(𝑖_𝑘+1), 𝑘=1,2,…,|𝑆₀|. Set 𝑆₀=𝑖₁,𝑖₂,…,𝑖_{|𝑆0|−𝑞1},𝑆₁=𝑆₁𝑆⁰⧵𝑆₀,(4.10) where 𝑞₁=(𝑛/3)−|𝑆₁|. Renew to calculate 𝛿^02(𝑖)=_𝑗∈𝑆2𝑤_𝑖𝑗,𝑖∈𝑆₀.(4.11) and let 𝑆₀=⎧⎪⎨⎪⎩𝑖^1,𝑖^2,…,𝑖^||||𝑆₀^||||⎫⎪⎬⎪⎭,(4.12) where 𝛿^02(𝑖^𝑘)≥𝛿^02(𝑖^𝑘+1), 𝑘=1,2,…,|𝑆₀|. Set 𝑆₀=𝑖^1,𝑖^2,…,𝑖^𝑛/3,𝑆₂=𝑆₂𝑆⁰⧵𝑆₀.(4.13)(3)If 𝑚₀₁>𝑚₀₂, let 𝑆₀=𝑖₁,𝑖₂,…,𝑖_|𝑆0|,(4.14) where 𝛿₀₂(𝑖_𝑘)≥𝛿₀₂(𝑖_𝑘+1), 𝑘=1,2,…,|𝑆₀|. Set 𝑆₀=𝑖₁,𝑖₂,…,𝑖_{|𝑆0|−𝑞2},𝑆₂=𝑆₂𝑆⁰⧵𝑆₀,(4.15) where 𝑞₂=(𝑛/3)−|𝑆₂|. Renew to calculate 𝛿^01(𝑖)=_𝑗∈𝑆1𝑤_𝑖𝑗,𝑖∈𝑆₀.(4.16) and let 𝑆₀=⎧⎪⎨⎪⎩𝑖^1,𝑖^2,…,𝑖^||||𝑆₀^||||⎫⎪⎬⎪⎭,(4.17) where 𝛿^01(𝑖^𝑘)≥𝛿^01(𝑖^𝑘+1), 𝑘=1,2,…,|𝑆₀|. Set 𝑆₀=𝑖^1,𝑖^2,…,𝑖^𝑛/3,𝑆₁=𝑆₁𝑆⁰⧵𝑆₀.(4.18)(4)Return the current partition 𝑆={𝑆_𝟎,𝑆_𝟏,𝑆_𝟐}; stop.

5. Numerical Results

This section describes the obtained experimental results for some instances of Max 3-cut and Max 3-Section problems using the proposed VNS metaheuristic. We also show a quantitative comparison with 0.836-approximate algorithm. The computational experiments are performed in an Intel Pentium 4 processor at 2.0 GHz, with 512 MB of RAM, and all algorithms are coded in Matlab. Because RSDP relaxation of M3C includes many slack variables, many constraints, and matrices variables without a block diagonal structures, in our numerical comparisons, we choose SDPT3-4.0 [9], one of the best and well-known solvers of semidefinite programming, to solve RSDP relaxation of M3C.

All our test problems are generated randomly by the following way. Let 𝑝∈(0,1) be a constant and 𝑟∈(0,1) a random number. If 𝑟≤𝑝, then there is an edge between nodes 𝑖 and 𝑗 with weight 𝑤_𝑖𝑗, that is, a random integer between 1 and 10. Otherwise, 𝑤_𝑖𝑗=0; that is, there is no edge between nodes 𝑖 and 𝑗. Because of the limits of memory of SDPT3, when 𝑛>200, RSDP becomes a huge semidefinite programming problem with not less than 59700 slack variables and 99900 constraints and is out of memory of SDPT3. Hence, in the numerical experiments, we consider 30 instances with 𝑝=0.1,0.3,0.6, and 𝑛 varying from 20 to 200.

Firstly, we check the influence of 𝐾_max on the quality of solution obtained by VNS-k. For a given graph, we take 𝐾_max=3,5,10,15,30; Table 1 presents the results, where Wnp in the first column of this table and the following tables means that a graph is randomly generated with nodes 𝑛 and density 𝑝; for instance, W30.6 presents a graph generated randomly with 𝑛=30 and 𝑝=0.6. We find from Table 1 that the influence of 𝐾_max to objective value denoted by Obj in Table 1 is slight when 𝐾_max>5, but the CPU time increases sharply as 𝐾_max increases. This result is actually not surprising. Indeed, because 𝐼(𝐾)>𝐾, we choose randomly a point 𝐲 in 𝜕𝑁_𝐼(𝐾)(𝐲̂), instead of 𝜕𝑁_𝐾(𝐲̂). This avoids to choose too large 𝐾_max which leads to more CPU-time cost. Hence, in sequel numerical comparisons, we fix 𝐾_max=5 for all test problems.

Table 1 

The objective value obtained by VNS for M3C with different 𝐾_max.

Secondly, we compare VNS (VNS-k, VNS-t) metaheuristic with 0.836-approximate algorithm for all test problems. To avoid the effect of initial points, for each test problem, after RSDP is solved, we run the round procedure of 0.836-approximate algorithm and VNS metaheuristic ten times, respectively.

Table 2 gives the result of numerical comparisons. In the numerical presentations of Table 2, Obj_rsdp is the optimal value of problem RSDP; that is, it is an upper bound of M3C. Obj_GM is the largest value obtained by 0.836-approximate algorithm in the ten tests. Obj_vns stands for the largest value obtained by VNS for M3C in the ten tests, respectively. 𝑚 and s.v. are the number of constraints and slack variables (s.v.), respectively. 𝑡_GM and 𝑡_vns−𝑘 are the average time (second) associated with the two algorithms in the ten tests. For the maximum CPU time of VNS-t, we take 𝑡_max=2𝑡_vns-𝑘, but the real CPU time of VNS-t will be greater than 𝑡_max. Additionally, for measuring the performance of solutions, we take 𝜌=𝙾𝚋𝚓^vns−𝙾𝚋𝚓_rsdp𝙾𝚋𝚓_rsdp=𝙾𝚋𝚓^vns𝙾𝚋𝚓_rsdp−1(5.1) for M3C and 𝜌=𝙾𝚋𝚓^vns+saga−𝙾𝚋𝚓_rsdp𝙾𝚋𝚓_rsdp=𝙾𝚋𝚓^vns+saga𝙾𝚋𝚓_rsdp−1(5.2) for M3S. Clearly, 𝜌 can reflect how close to the solution obtained by VNS from the optimal solution of RSDP. One can see from Table 2 that (1) the VNS metaheuristic not only can obtain a better solution than 0.836-approximate algorithm for all test problems, but also that the elapsed CPU-time of VNS metaheuristic is much less than that of 0.836-approximate algorithm for all test problems, (2) the performance of solution can be improved by VNS-t for most of test problems when the termination criterion of VNS is based on the maximum CPU-time, but VNS-t spends more computational time than VNS-k. The improved performance can be reflected by ∇𝜌=𝜌_𝑡−𝜌_𝑘 in the final column of Table 2. Average speaking, VNS-t improves 0.91 percentage point.

Table 2 

The numerical comparisons of 0.836-approximate algorithm with VNS metaheuristic.

Finally, we consider the solution of M3S by combining VNS-k and greedy sizes-adjusted algorithm SAGA stated in Section 4. Let 𝐲̂ be an approximate solution of M3C obtained by VNS; we can obtain an approximate solution of M3S from SAGA. The numerical results are reported by Table 3 in which Obj_vns+saga stands for the largest value obtained by VNS-k plus SAGA for M3S. Although our sizes-adjusted algorithm may decrease the objective value obtained by VNS, the changes of objective values are very slight from Table 3. Particular, objective values of some problems do not decrease, instead increase, such as W150.3. We do not compare the obtained results with Andersson’s 2/3-approximate algorithm. Because we find that all approximate solutions of M3S obtained by VNS plus SAGA still are better than that of 0.836-approximate algorithm with the exception of only W30.1 and W30.3.

Table 3 

The numerical results of combining VNS-k metaheuristic with SAGA for M3S.

6. Conclusions

A variable neighborhood stochastic metaheuristic was proposed to solve the Max 3-cut and Max 3-section problems in this paper. Our algorithms can solve Max 3-cut and Max 3-section problems with different sizes and densities. Although 0.836-approximate algorithm has the very good theoretic results, in numerical aspects, our comparisons indicate that the proposed VNS metaheuristic is superior to the well-known 0.836-approximate algorithm and can efficiently obtain very high-quality solutions of the Max 3-cut and Max 3-section problems.

We mention that the proposed algorithm in fact can deal with higher dimensional G-set graphs problems created by Pro. Rinaldi using a graph generator, rudy. But, we cannot give numerical comparisons with 0.836-approximate algorithm since RSDP relaxations of these problems are out of memory of the current all SDP software. In additionally, if we increase 𝐾_max or 𝑡_max in numerical implementing, then the quality of solution of M3C will further be improved by VNS.

Funding

This work is supported by the National Natural Science Foundations of China (no. 71001045, 10971162), the China Postdoctoral Science Foundation (no. 20100480491), the Natural Science Foundation of Jiangxi Province of China (no. 20114BAB211008), and the Jiangxi University of Finance and Economics Support Program Funds for Outstanding Youths.

Acknowledgment

The authors would like to thank the editor and an anonymous referee for their numerous suggestions for improving the paper.