The generalized vertex cover problem and some variations

08/11/2017 ∙ by Pooja Pandey, et al. ∙ Simon Fraser University 0

In this paper we study the generalized vertex cover problem (GVC), which is a generalization of various well studied combinatorial optimization problems. GVC is shown to be equivalent to the unconstrained binary quadratic programming problem and also equivalent to some other variations of the general GVC. Some solvable cases are identified and approximation algorithms are suggested for special cases. We also study GVC on bipartite graphs and identify some polynomially solvable cases. We show that GVC on bipartite graphs is equivalent to the bipartite unconstrained 0-1 quadratic programming problem. Integer programming formulations of GVC and related problems are presented and establish half-integrality property on some variables for the corresponding linear programming relaxations. We also discuss special cases of GVC where all feasible solutions are independent sets or vertex covers. These problems are observed to be equivalent to the maximum weight independent set problem or minimum weight vertex cover problem along with some algorithmic results.

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1 Introduction

Let be a graph with . For each edge three real valued weights are associated. Also, for each vertex a weight is prescribed. For any subset , let , , , and

Then the generalized vertex cover problem (GVC) is to find a set such that is minimized. Note that can be viewed as the ‘cost of’ not covering edge , can be viewed as the ‘cost of’ covering edge by selecting exactly one of its end points, and can be viewed as the ‘cost of’ over-covering edge . If the solution is an empty set , then the objective function is defined as .

GVC was introduced by Hassin and Levin [13] and it is a meaningful generalization of the well known minimum weight vertex cover problem (MWVCP) [14, 33] and the maximum weight independent set problem (MWISP) [4, 33]. Note that if for all , a large number and for all then, GVC reduces to MWVCP. Likewise, when , , for all , then GVC is equivalent to MWISP. (It may be noted that MWISP is normally presented as a maximization problem which is equivalent to minimization form indicated above). Hassin and Levin [13] although introduced GVC, they focused primarily on a special case of it where is assumed to be non-negative for all and for all . We denote this special case of GVC by GVC-HL and it may be noted that MWISP is not a special case of GVC-HL. In [13] two 2-approximation algorithms for GVC-HL are proposed , one based on linear programming, and the other based on the local-ratio technique [6]. When for all and , GVC is called uniform cost generalized vertex cover problem (UGVC). In [13] the complexity of UGVC has been studied for all possible values of and . They showed that GVC is polynomial time solvable in the following cases: 1) , 2) , 3) and there exists an integer such that . For the general case, UGVC is NP-hard [13]. Milanovic [23]

proposed a genetic algorithm to solve GVC-HL and reported experimental results comparing their algorithm with CPLEX and the 2-approximation algorithm given in

[13]. Kochenberger et. al. [20] compared an integer linear programming formulation and an integer quadratic programming formulations using the CPLEX solver.

Another special case of GVC where for all was considered by Houchbaum [16] and Bar-Yehuda et al. [7]. This problem is also known as the generalized vertex cover problem in the literature and for definiteness we denote this problem by GVC1. In fact, Houchbaum [16] and Bar-Yehuda et al.  [7] studied primarily a special case of GVC1 where for all , for all . We refer to this version of GVC1 as GVC1-HB. Houchbaum [16] provided an integer linear programming formulation of GVC1-HB and showed that the corresponding linear programming relaxation admits half integrality property. Bar-Yehuda et al. [7] provided an extension of a well known theorem by Nemhauser and Trotter [24] for the vertex cover problem (independent set problem) to GVC1-HB, and presented a -approximation algorithm on graphs with maximum degree of a node is bounded above by . They also presented a polynomial time approximation scheme (PTAS) for GVC1-HB on planner graphs, and a - approximation algorithm for a general graph. In the same paper they showed that GVC1-HB is NP-hard on complete graphs but solvable in polynomial time on bipartite graphs. Note that MWVCP is trivial on complete graphs.

The generalized independent set problem introduced by Hochbaum [16] is yet another special case of GVC. Here are assumed to be zero for all , and we denote this problem by GVC2. Hochbaum and Pathria [15] studied a special case of GVC2 where for all , for all . This version of GVC2 is denoted by GVC2-HP and the model have applications in Forest Harvesting. GVC2-HP in [15] was presented as a maximization problem with and . Clearly, this is equivalent to our definition of GVC2-HP. Kochenberger et al. [21] gave a nonlinear formulation for GVC2-HP and compare the computational effectiveness of this non-linear integer programming formulation with an available linear integer programming formulation.

Another problem closely related to GVC is the unconstrained binary quadratic programming problem (UBQP) studied by various authors [2, 11, 18, 19, 25]. Let be an symmetric matrix and be an

-vector. Then UBQP is to find an

such that

is minimized (or maximized). Without loss of generality is chosen as zero for and we assume UBQP is presented as a minimization problem. Also the assumption that is symmetric is not a restriction, since can be replaced by to obtain an equivalent problem.

Given the matrix , let . The subgraph of the complete graph with the vertex set is called the support graph of . Then UBQP can be reformulated in terms of i.e; UBQP is equivalent to finding such that

is minimized (or maximized), where .

The bipartite unconstrained 0-1 quadratic program problem (BQP01) [10, 30, 31] is closely related to GVC on bipartite graphs. Let be an matrix, , . Then BQP01 is defined as

Consider an instance of BQP01 with cost matrix of dimension . The support bipartite graph of the matrix is the bipartite graph , where , and . The BQP01 can be formulated as a graph theoretic optimization problem on as

Minimized
subject to:

It may be noted that the definition of support bipartite graph is different from that of a support graph. The support graph when the underlying graph is bipartite is different from a support bipartite graph.

In this paper we study the general problems GVC, GVC1, and GVC2. We show that all these problems are equivalent to each other and also equivalent to UBQP. When the underlying graph is bipartite, these are equivalent to BQP01 as well. However, it may be noted that although these equivalences are verified in terms of optimality, domination ratio [12], differential approximation ratio [9], the characteristics of these problems in terms of approximation ratio [33] and various special cases are different and hence it is interesting to explore various special cases of these problems as well. We present several complexity results related to GVC, GVC1, and GVC2 along with some polynomial solvable special cases. Integer programming formulations of GVC, GVC1, and GVC2 are presented and establish half-integrality property on some variables for the corresponding linear programming relaxations. We also present an approximation algorithm for GVC with the approximation ratio when , , and generalizing a result given in [13]. The approximation ratio for the algorithm becomes when for given , and for each edge , all three weights are in . When , the optimal solutions of GVC are vertex covers and this special case leads to the vertex cover problem with node and edge weights (VCPNEW). The problem VCPNEW and two of its special cases are shown to be equivalent to the minimum weight vertex cover problem (MWVCP) and shown that VCPNEW can be solved by an algorithm whenever MWVCP can be solved by an -algorithm for an appropriate if and for all . When , optimal solutions of GVC are independent sets and this special case leads to the independent set problem with node and edge weights (ISPNEW). The problem ISPNEW and two of its special cases are shown to be equivalent to the maximum weight independent set problem (MWISP).

The paper is organized as follows: In Section 2, we study the complexity of GVC. Different NP-hard special cases as well as polynomially solvable special cases are identified. Further we show that GVC, GVC1, GVC2, and UBQP are pairwise equivalent in the sense that an algorithm to compute an optimal solution to one problem can be used to compute an optimal solution to the other. We also show that GVC, GVC1, and GVC2 on bipartite graphs are equivalent to BQP01. Section 3 contains integer programming formulation of GVC and related problems, establish half-integrality property on some variables, and also discuss the approximation algorithm for GVC. Section 4 discusses VCPNEW, ISPNEW and some of their special cases. We also establish that VCPNEW is equivalent to MWVCP, and ISPNEW is equivalent to MWISP.

2 Complexity and Solvable cases

We first show that the problems GVC, GVC1, GVC2, and UBQP are equivalent from an optimality point of view, i.e., any one of these problems can be formulated as another in the sense that from an optimal solution to one, an optimal solution to another can be recovered.

Theorem 2.1.

GVC, GVC1, and GVC2 are equivalent.

Proof.

We first show that GVC can be formulated as GVC1. Given an instance of GVC, define

(1)
(2)

Now, consider the instance of GVC1 where is replaced by and is replaced by . Then, for any , the objective function of this GVC1 can be written as

The objective function of GVC is given by

(3)

It can be verified that

(4)
(5)

From (3), (4) and (5), we have

Thus, for any feasible solution of GVC and the instance of GVC1 constructed above, is a constant. Therefore, an optimal solution to this GVC1 will also be an optimal solution to GVC.

Since GVC1 is a special case of GVC, any GVC1 can be formulated as GVC, establishing equivalence between GVC and GVC1.

To establish the equivalence between GVC and GVC2, define

(6)
(7)

Consider the instance of GVC2 where is replaced by and is replaced by . As in the previous case, we can show that an optimal solution to this GVC2 provides an optimal solution to GVC and the converse follows the fact that GVC2 is a special case of GVC.

Corollary 2.2.

GVC, GVC1, and GVC2 are NP-hard on complete graphs.

Proof.

This follows from Theorem 2.1 and the fact that GVC2-HB is NP-hard on complete graphs [7]. ∎

It may be noted that the special cases MWISP, MWVCP of GVC are trivial on a complete graph.

Theorem 2.3.

GVC, GVC1, and GVC2 are equivalent to UBQP.

Proof.

We first show that GVC2 can be formulated as UBQP. For any feasible solution , let and be the objective function values of GVC2, and UBQP respectively.

Define the matrix as

(8)

and

(9)

Consider the UBQP with and as defined above. For any feasible solution of GVC2 or UBQP, . Thus an optimal solution to UBQP constructed above is also an optimal solution to GVC2.

To show that UBQP can be formulated as GVC2, let be the cost matrix of an instance of UBQP and be the support graph of . Choose , for edge of the support graph . Then, an optimal solution to the resulting GVC2 on solves UBQP. Thus GVC2 and UBQP are equivalent. The equivalence of GVC, GVC1, and UBQP now follow from Theorem 2.1. ∎

Although the equivalence of GVC, GVC1, GVC2 and UBQP are established in Theorem 2.3, the direct reformulation as a UBQP was given only for the GVC2. It is however interesting to present the precise structure of when GVC and GVC1 are reformulated as a UBQP. This is particularly useful in identifying polynomially solvable special cases directly from solvable UBQP instances. For any feasible solution , let and be the objective function values of GVC, GVC1, and UBQP respectively.

Define as

(10)

and

(11)

Consider the UBQP with and are as defined above. For any ,

(12)

It can be verified that

(13)
(14)

From (12), (13) and (14), we have

(15)

Thus an optimal solution to the UBQP constructed above is also an optimal solution to GVC.

Now, define the matrix as

(16)

and

(17)

Consider an instance of UBQP with and as defined above.

Then

Thus an optimal solution to this UBQP is an optimal solution to GVC1.

Let be the matrix defined as in equation (10) and as in equation (11).

Corollary 2.4.
  • GVC is polynomial-time solvable on a series-parallel graph .

  • If the rank of is fixed and is a positive semi-definite, then GVC is polynomial-time solvable.

  • If and are non-negative, then GVC is polynomial-time solvable.

  • GVC is polynomial-time solvable if , where is defined in equation (10).

Proof.

From the construction given in equation (15), the support graph of is a subgraph of on which GVC is defined. Note that UBQP is solvable in polynomial time if the support graph of the cost matrix is series-parallel [5, 22]. Since any subgraph of a series-parallel graph is series-parallel, the result follows.

If the rank of is fixed and is a positive semi-definite, then UBQP is polynomial time solvable [2, 11, 18, 22], and if and are non-negative, UBQP is polynomial time solvable [22]. The result now follows from Theorem 2.3.

When , where is defined in equation (10), Picard et al. [28, 29] showed that UBQP can be reduced to minimum-cut problem on the graph , where , denotes the source and the sink, and with [22]. Therefore, in this case GVC is equivalent to the minimum-cut problem which is polynomial time solvable [28, 29].

Similar results follow for GVC2 if is defined as in (8) and is defined as in equation (9) and for GVC1 when if is defined as in (16) and is defined as in equation (17).

It may be noted that the reductions discussed in Theorem 2.1 and 2.3 above for GVC and GVC1 to UBQP do not preserve -optimality because of the resulting constant terms in the objective function. However, for GVC2, reduction to UBQP preserves -optimality. The equivalence however preserves other performance measures such as differential approximation ratio [3, 9] and domination ratio [12].

For any graph and , is the graph obtained by deleting node and all its incident edges. Then GVC2 on can be solved by solving GVC2 on with two different data sets. Note that in an optimal solution to GVC2 on , either belongs or does not belongs to an optimal solution. If does not belong, then any optimal solution to GVC2 on with original data (restricted to nodes and edges of ) is also an optimal solution to GVC2 on . Otherwise, define

and for all edge in , where .

Let be an optimal solution to GVC2 on with replaced by and is replaced by . Also let be an optimal solution to GVC2 on with original data. Then the best of will be an optimal solution to GVC2 on . This idea (together with Theorem 2.1) can be applied recursively to establish the following.

Theorem 2.5.

GVC, GVC1, and GVC2 can be solved in polynomial time on if GVC2 can be solved in polynomial time on for some where .

It is well known that if is an optimal vertex cover then is an optimal independent set. A similar relationship exist between GVC1 and GVC2.

Lemma 2.6.

is an optimal solution to GVC1 with data and if and only if is an optimal solution to GVC2 with data and

Proof.

Suppose is an optimal solution to the GVC1. Then for any

(18)

Consider the objective function value of the GVC2 given in the statement of the lemma for the solution . We have,

Since any subset of can be represented as , for , it follows that is an optimal solution to GVC2 defined above. The converse can be proved analogously. ∎

2.1 Bipartite graphs

Bipartite graph is a subclass of perfect graphs on which the MWVCP and MWISP can be solved in polynomial time [33]. Houchbaum [16] and Bar-Yehuda et al.  [7] showed that GVC1-HB solvable in polynomial time on bipartite graphs. Hochbaum and Pathria [15] showed that GVC1-HP is solvable in polynomial time on bipartite graphs. Thus it is interesting to investigate the complexity of GVC, GVC1, and GVC2 on bipartite graphs. Let us first define these problems on bipartite graphs using the bi-partitioned structure of the vertex set.

Let be a bipartite graph, where , and . For each vertex a weight is prescribed and for each vertex a weight is prescribed. Let , , , and . Then GVC on this bipartite graph is to find such that

is minimized. Similarly, GVC1 is to minimize

and GVC2 is to minimize

Without loss of generality we assume .

Observation 2.7.

GVC is solvable in polynomial time on bipartite graphs if and or and .

Proof.

If the first set of conditions are satisfied, then from Theorem 2.1, GVC reduces to GVC1-HB, which is solvable in polynomial time on bipartite graphs [7]. If the second set of conditions are satisfied, then GVC reduces to GVC2-HP, which is solvable in polynomial time on bipartite graphs [15]. ∎

Theorem 2.8.

GVC, GVC1, and GVC2 on bipartite graphs are equivalent to BQP01.

Proof.

For any feasible solution , where and , let and be the objective function values of GVC2, and BQP01 respectively.

Define the matrix as

(19)

Set and . Consider the BQP01 with , and are as defined above. For any feasible solution