A heuristic based on negative chordless cycles for the maximum balanced induced subgraph problem
Introduction
A signed graph is an undirected graph whose edges are labeled either with (positive edges) or with −1 (negative edges) by a signature function . See [40], [41] for a comprehensive survey. The sign of a subset C of edges, e.g., a path or a cycle, is given by the product of the signs of the edges in C. From a theoretical perspective, signed graphs and their incidence matrices are deeply connected with matroid theory [17], [35].
Following the definition by Harary [22], a signed graph is said to be balanced if it has no negative cycles. Harary also proved that a signed graph is balanced if and only if the set of negative edges is a possibly empty cut. Given a signed graph, the Maximum Balanced Induced Subgraph problem (mbis) asks for finding a balanced induced subgraph (BIS) of maximum order. The opposite numerical problem, i.e., finding the smallest number of vertices whose deletion makes the signed graph balanced, was introduced by Harary [23] and called point index for structural balance. Nowadays that number is known as the vertex frustration number; see [42].
Several problems originating in different domains, even far away from each other, can be modeled as mbis. In social networking, for example, relationships between individuals can be represented by a signed graph where vertices are persons and positive (negative) edges express friendships (hostility). Based on the structural balance theory [10], positive cycles are supposed to indicate stable social situations, whereas negative cycles are supposed to be unstable. Hence the maximum balanced subgraph gives a measure of the cohesion of the social group.
Another application lies in the context of portfolio analysis [24]. In that case, the vertices of a signed graph denote stocks, and a positive (negative) edge represents the direct (inverse) correlation between its extremes. It is generally believed that the larger is the maximum balanced induced subgraph, the more predictable is the behavior of the portfolio.
mbis is polynomial on series parallel graphs [4] but in general is NP-hard as it admits the odd cycle transversal (also known as maximum bipartite subgraph or vertex bipartization) and the maximum independent set (mis) problems as special cases, the former with applications, e.g., in VLSI design [2], [12], DNA sequencing [16] and computational biology [43], the latter arising as a subproblem or as a relaxation of many 0–1 integer problems (as a reference, think about the solution of the mis problem on the so-called conflict graph, which is at the basis of many general preprocessing and probing techniques for integer linear programming problems [3], [8]).
In particular, the odd cycle transversal problem on a graph G corresponds to the mbis problem on the signed graph obtained from G by signing all its edges as negative. In fact, odd cycles of G correspond to negative cycles of S, and a minimum odd cycle transversal on G corresponds to a smallest set of vertices to be removed from S to make it balanced. On the other hand, the mis problem on a graph G corresponds to mbis on the signed graph obtained from G by signed expansion [40, Section 7C], i.e., by doubling its edges and assigning opposite signs to each pair of parallel edges.
Our interest in the mbis problem was originally motivated by the study of an equivalent problem known in the domain of integer programming as Maximum Embedded Reflected Network (mern). Nowadays, the most successful methods for solving an integer linear program work by iteratively tightening (cutting planes) or recursively decomposing (branch-and-bound) a polyhedron which represents a continuous formulation of the problem. Such algorithms stop as soon as an integer extreme point is reached and its optimality is proven. When one of such methods is applied, the cases where the polyhedron is integral (or at least has an integer extreme point that is optimal for a given objective function) are of particular interest because the integer linear problem boils down to a continuous linear one. The integrality of the polyhedron depends on the structure of the coefficients of the corresponding integer linear program. A well-known family of such special structures is that of totally unimodular (TU) matrices: one of the famous theorems by Hoffman–Kruskal [27] states that the polyhedron , with , is integral for every integral and if and only if is a TU matrix. Indeed, the recognition of special structures in the coefficient matrix of (integer) linear programs also helps in the solution of large-scale continuous models [6], [7] and, on the other hand, can be exploited in the strategic choice of constraints to be either relaxed in a Lagrangian relaxation scheme [5] or convexified in a Dantzig–Wolfe decomposition.
Total unimodularity can be checked in polynomial time [36]; but most formulations of combinatorial optimization problems do not exhibit a TU matrix. We are therefore interested in finding a maximum embedded TU submatrix. A subclass of TU matrices is that of reflected network matrices; see Section 3. Since, for any given -matrix , a signed graph SA can be defined such that any reflected network submatrix of obtained by row deletion corresponds to a balanced induced subgraph of SA and vice versa [25], [26], the mern problem, i.e., the task of finding a maximum reflected network submatrix by deleting rows, is, in combinatorial terms, the mbis problem. That explains our interest in mbis.
In this paper we propose a greedy heuristic for mbis that generalizes the minimum-degree greedy algorithm for mis. The heuristic is based on a graph transformation (the shortening of negative cycles) that preserves balance of any induced subgraph that is balanced in the original graph, and makes the mbis easier to solve. A broad computational experience shows that the heuristic largely outperforms the best previous algorithm for mbis.
The remainder of the paper is organized as follows. The graph terminology used throughout the paper and the main properties of signed graphs are introduced in Section 2. In Section 3 we describe the link between reflected networks and signed graphs and briefly survey the approaches for solving the mbis problem. In Section 4 we give the details of a heuristic for mbis and Section 5 illustrates our computational experience. Conclusions are sketched in Section 6.
Section snippets
Graphs
Let be a finite undirected graph with vertex set and edge set E that is a set of unordered pairs of distinct vertices. The density of G is given by . The subgraph of G induced by the set of vertices is the graph . The subgraph of G induced by the set of edges is the graph . The set of vertices adjacent to is the neighborhood of u. The cardinality of N(u) is the degree d(u) of u. A path is a
Reflected networks and signed graphs
Throughout the paper we consider only matrices with elements in . Given a -matrix, we say that rows u and v conflict (match) on column j if (). We say that u and v are conflicting (matching) rows if they conflict (match) on each column with two nonzeros.
A reflected network is a -matrix that satisfies the Heller and Tompkins conditions (HT) [25]:
- (i)
each column contains at most two non-zeroes, and
- (ii)
the set V of row indices can be divided into (with V2
A greedy heuristic
The structural weakness of SGA is mostly due to the fact that it does not take into account the set , although any maximum BIS is truly constrained by negative (chordless) cycles. In this section we propose a greedy heuristic for the mbis problem, in fact a generalization of GH, that builds up a balanced induced subgraph of S by choosing, at each iteration, a vertex v by taking into account (an estimation of) the structure of .
Let be a given undirected graph. GH consists of a
Computational results
The algorithm NCCH was coded in C++ and compiled with Microsoft cl compiler (version 12.00.8804) with option /O2. Numerical precision was set to 10−6. Problems were solved on an Intel Core2 Duo E8500 3.17 GHz with 3 Gb RAM. Integer programs were solved by Cplex 12.5 [30] with default setting.
The tests were carried out over three classes (N, P and R) of instances. N-instances are 34 signed graphs used in [15], [20] and obtained from a set of preprocessed and scaled Netlib matrices [32]. N
Concluding remarks
We presented a new greedy heuristic for the mbis problem (NCCH) based on a graph transformation rule that shortens negative cycles. NCCH generalizes the Minimum-Degree Greedy Heuristic for the mis problem.
A broad computational experience shows that, in terms of solution quality, NCCH largely outperforms SGA (as far as we know, the best heuristic to date). In particular, the optimality gap decreases by almost seven times on the reference Netlib instances and by almost five times on random
Acknowledgements
The authors are very grateful to the anonymous referees for their insightful comments and for their time and effort in helping the authors to significantly improve the presentation of this work.
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