Multiple phase tabu search for bipartite boolean quadratic programming with partitioned variables

Highlights

• Propose a multiple phase tabu search algorithm for BBQP-PV.

• Combine the SN-TS phase and the VLSN-TS phase for search intensification.

• Design a hybrid perturbation phase for search diversification.

• Find improved lower bounds for 5 instances.

Abstract

The Bipartite Boolean Quadratic Programming Problem with Partitioned Variables (BBQP-PV) is an NP-hard problem with many practical applications. In this study, we present an effective multiple phase tabu search algorithm for solving BBQP-PV. The algorithm is characterized by a joint use of three key components: two tabu search phases that employ a simple neighborhood and a very large-scale neighborhood to achieve search intensification, and a hybrid perturbation phase that adaptively chooses a greedy perturbation or a recency-based perturbation for search diversification. Experimental assessment on 50 standard benchmarks indicates that the proposed algorithm is able to obtain improved lower bounds for 5 instances and match the previously best solutions for most instances, while achieving this performance within competitive time. Additional analysis confirms the importance of the innovative search components.

Introduction

Let $G=(I,J,E)$ denote a bipartite graph with the two vertex sets $I=\{1,2,\dots ,m\},$ $J=\{1,2,\dots ,n\}$ and the edge set E⊆I × J. The vertex set I is partitioned into p disjoint subsets $S_1,S_2,\dots ,S_p$ and the vertex set J is partitioned into k disjoint subsets $T_1,T_2,\dots ,T_k$. Further, each vertex i ∈ I is associated with a weight c_i, each vertex j ∈ J is associated with a weight d_j, and each edge (i, j) ∈ E is associated with a weight q_ij. A subgraph $G^{\prime }=(I^{\prime },J^{\prime },E^{\prime })$ is said to be a representative subgraph of G, if the vertex set I′ contains only one vertex, say α_r, of each disjoint subset $S_r,r=1,2,\dots ,p,$ the vertex set J′ contains only one vertex, say β_u, of each disjoint subset $T_u,u=1,2,\dots ,k,$ and the edge set E′ contains the edges connecting vertices between I′ and J′. Then, the Binary Quadratic Programming Problem with Partitioned Variables (BBQP-PV) is to find such a representative subgraph G′ that receives the maximum sum of edge weights and vertex weights. Formally, let $s=(s_{\alpha },s_{\beta })$ be a vertex set of G′ where $s_{\alpha }=\{{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_p\}$ and $s_{\beta }=\{{\beta }_1,{\beta }_2,\dots ,{\beta }_k\},$ BBQP-PV is to maximize the following objective:

$$\begin{array}{cc}\hfill \max & f\left(s\right)=\sum _{r=1}^p\sum _{u=1}^k{q}_{{\alpha }_r,{\beta }_u}+\sum _{r=1}^p{c}_{{\alpha }_r}+\sum _{u=1}^k{d}_{{\beta }_u}\hfill \end{array}$$

$$\begin{array}{cc}\hfill s.t.& {\alpha }_r\in S_r,\text{for}r=1,2,\dots ,p\hfill \end{array}$$

$$\begin{array}{cc}& {\beta }_u\in T_u,\text{for}u=1,2,\dots ,k\hfill \end{array}$$

Equivalently, BBQP-PV can be formulated as a constrained 0–1 quadratic program as follows (Punnen and Wang, 2016).

$$\begin{array}{cc}\hfill \max & f(x,y)=\sum _{i=1}^m\sum _{j=1}^n{q}_{ij}{x}_i{y}_j+\sum _{i=1}^m{c}_i{x}_i+\sum _{j=1}^n{d}_j{y}_j+c_0\hfill \end{array}$$

$$\begin{array}{cc}\hfill s.t.& \sum _{i\in S_r}{x}_i=1,forr=1,2,\dots ,p\hfill \end{array}$$

$$\begin{array}{cc}& \sum _{j\in T_u}{y}_j=1,foru=1,2,\dots ,k\hfill \end{array}$$

$$\begin{array}{cc}& x_i,y_j\in \{0,1\},fori\in I,j\in J\hfill \end{array}$$

where x_i (y_j) takes the value of 1 if $i={\alpha }_r$ ($j={\beta }_u$), otherwise the value of x_i (y_j) takes the value of 0. Without loss of generality, the constant c₀ is assumed to be 0.

BBQP-PV is a constrained version of the Bipartite Boolean Quadratic Programming Problem (BBQP) (Duarte, Laguna, Martí, Sánchez-Oro, 2014, Karapetyan, Punnen, Parkes, 2017, Punnen, Sripratak, Karapetyan, 2015). It was recently proposed in Punnen and Wang (2016) and proved to be strongly NP-hard. Moreover, they indicate that when $p=k=n$ and $|S_r|=|T_u|=n$ for any r and u, BBQP-PV turns out to be the Bipartite Quadratic Assignment Problem (BQAP) which in turn is a generalization of well-studied quadratic assignment problem (QAP).

BBQP-PV is a unified model of several classic combinatorial optimization problems, such as the Biclique Problem (Ames, Vavasis, 2011, Gillis, Glineur, 2014, Peeters, 2003), the Max-induced Subgraph Problem (Yang, Evans, Megson, 2005, Zvervich, Zverovich, 1995), the Maximum Cut Problem on a bipartite graph (Festa, Pardalos, Resende, Ribeiro, 2002, Grötschel, Pulleyblank, 1981, Martí, Duarte, Laguna, 2009) and the Matrix Factorization Problem (Miller, 1962, Strintzis, 1972, Zhu, Honeine, Kallas, 2014). Applications of the BBQP-PV model include clustering (Boros, Hammer, 1989, Shen, Ji, Ye, 2009), location problem (Dearing et al., 1992), social network analysis (Kochenberger et al., 2013), bioinformatics (Gupta, Rao, Kumar, 2011, Tanay, Sharan, Shamir, 2002) and many others.

Previous literature has reported many approaches for solving the closely related unconstrained BBQP problem. For example, Duarte et al. (2014) proposed a branch and bound algorithm and several iterated local search algorithms. Glover et al. (2015) proposed multiple hybrid algorithms by combining tabu search and very large-scale neighborhood search strategies. Karapetyan et al. (2017) developed an effective Markov chain search algorithm. Moreover, bilinear programming algorithms are available for solving BBQP-PV (Bloemhof-Ruwaard, Hendrix, 1996, Gallo, Ülkücü, 1977, Jorge, 2005) due to its bilinear objective function. However, without exploiting specific properties and structures of BBQP-PV, these general algorithms can not efficiently solve challenging BBQP-PV problem instances.

To solve BBQP-PV practically, several heuristic and metaheuristic algorithms have been proposed in the literature. The first computational study is proposed in Punnen and Wang (2016), where several tailored local search and hybrid algorithms are developed and computational comparisons among the proposed algorithms are presented. Results show that the hybrid algorithms combining different neighborhoods outperform the algorithms that use these move operators in isolation and tabu search is a critical local search component. Another advanced metaheuristic algorithm recently proposed for solving BBQP-PV is an adaptive tabu search with strategic oscillation (ATS-SO) approach, which combines different move operators to collectively conduct neighborhood exploration and a history information guided strategic oscillation phase to diversify the search when the search gets trapped in local optimum. Computational assessments reveal that the ATS-SO algorithm outperforms the hybrid algorithms proposed in Punnen and Wang (2016).

In this paper, we propose a new multiple phase tabu search (MPTS) algorithm for solving BBQP-PV. The proposed MPTS algorithm consists of a simple neighborhood based tabu search (SN-TS) phase and a very large-scale neighborhood based tabu search (VLSN-TS) phase for search intensification and a hybrid perturbation phase for search diversification (Ahuja, Ergun, Orlin, Punnen, 2002, Ma, Wang, Hao, 2017, Pisinger, Ropke, 2010). The SN-TS phase aims to obtain a high-quality solution within a short period of time, while the VLSN-TS phase is dedicated to further refining the solution returned from the SN-TS phase. To escape local optimality and enable the search to explore new promising regions, the hybrid perturbation phase is employed to build well-diversified solutions. According to the diversification requirement of the current search status, the perturbation phase adaptively chooses to use a greedy perturbation or a recency-based perturbation to improve diversification.

Evaluated on five sets of BBQP-PV benchmarks with a total of 50 instances, the proposed MPTS algorithm is able to find improved solutions for 5 instances and match the best known results for most instances within competitive computation time, performing better than recently proposed state-of-the-art algorithms in the literature. Additional analysis sheds light on the effectiveness of the incorporated components to the performance of the algorithm.

The rest of this paper is organized as follows. In Section 2, move operator definitions and fast evaluation methods are presented. Section 3 describes the main scheme and important components of the proposed algorithm. In Section 4, the computational results of our MPTS and the comparisons with state of the art algorithms in the literature are reported. Section 5 provides an experimental analysis of the key components used in the MPTS algorithm. Section 6 draws conclusions.