The max -vertex cover problem is defined as follows: given a graph of order and a constant , determine vertices that cover a maximum number of edges. In the (edge-)weighted version of the max -vertex cover, weights are associated with the edges and the objective is to determine vertices that maximize the total weight of the edges covered by them. The problem is -hard even in bipartite graphs [2, 4].
max -vertex cover is a well-known restriction of max -set cover problem where we are given a family of subsets over a set of elements and an integer and the objective is to determine a subfamily of cardinality , covering a maximum number of elements from . Analogously, in the weighted version of max -set cover, the elements of are provided with positive weights and the objective becomes to determine sets that maximize the total weight of the covered elements.
Both max -set cover and max -vertex cover are well-known problems met in many real-world applications since they model several very natural facility location problems. In particular max -vertex cover is used for modeling problems in areas such as databases, social networks, sensor placement, information retrieval, etc. A non-exhaustive list of references to such applications can be found in .
To the best of our knowledge, the approximation of max -set cover has been studied for the first time in the late seventies by Cornuejols et al. , where an approximation ratio () is proved for the natural greedy algorithm, consisting of iteratively choosing the currently largest-cardinality set, until sets are included in the solution. max -vertex cover being a restriction of max -set cover, the same ratio is achieved for the former problem also and this ratio is tight in (weighted) bipartite graphs . A more systematic study of the greedy approximation of max -vertex cover can be found in . In  it has been proved that the greedy algorithm also achieves ratio . In the same paper, a very simple randomized algorithm is presented, that achieves approximation ratio
. Using a sophisticated linear programming method, the approximation ratio formax -vertex cover, in general graphs was improved to ; this ratio remained the best known one in general graphs until 2018, when  proposed a 0.92-approximation for the problem. Obvously, this ratio remains valid for bipartite graphs. For this class, the best ratio (still based on linear programming) known before  was (). The complexity of this -approximation algorithm is not given in ; a rough evaluation of it, gives a complexity of , where and are the two color-classes of (), which is bounded above by . Finally, max -vertex cover being a generalisation of min vertex cover, its inapproximability by polynomial time approximation schemata (PTAS) in general graphs is immediately derived from the corresponding inapproximability of the latter, ().
Let us note that unweighted max -vertex cover is easy in semi-regular bipartite graphs (where all the vertices of each color class have the same degree). Indeed, any vertices in the color class of maximum degree yield an optimal solution. Obviously, if this color class contains less than vertices, then one can cover all the edges.
In this paper, we propose a PTAS for max -vertex cover in bipartite graphs.
The following notations will be used in what follows:
: an edge-weighted bipartite graph instance of max -vertex cover (
is a weight-vector of dimension);
an optimal solution of max -vertex cover; and the subsets of lying in the color-classes and , respectively;
and the numbers of the optimal vertices in and , respectively; ; we assume that ;
the value of an optimal solution (i.e., the value of the total coverage capacity of ); the total coverage capacity of ; we set , ; denotes the private coverage capacity of , i.e., the edges already covered by are not encountered there; obviously, ;
for a vertex-set cardinality ():
denotes the total weight of the edges covered by the members of covering a maximum weight of edges (the best -elements subset of ; , );
denotes the total weight of the edges covered by the members of covering a minimum weight of edges (the worst -elements subset of );
: the approximation ratio of a known max -vertex cover-algorithm.
The following proposition holds and will be frequently used in what follows.
. For any set of optimal vertices:
Let us note that values of and can be guessed in polynomial time. We simply run the algorithm specified below for any possible pair of integers an such that and take the best result of these runnings. One of the results will be obtained for the pair and the solution returned will dominate the one for this particular pair. It is easy to see that this procedure takes, at worst, time. In what follows we will reason w.r.t. to the solution obtained for pair .
In what follows, we consider that the vertices of and are ordered in decreasing order w.r.t. their initial coverage capacity. Also, we call “best” vertices, a set of vertices that cover the largest total weight of uncovered edges in .
2 A preliminary result
The following proposition shows that we can consider that the values of are polynomially bounded. Denote by max -vertex cover() the instances of max -vertex cover where edge-weights are bounded above by , for a fixed constant .
. There exists an approximation-preserving reduction between max -vertex cover and max -vertex cover().
Proof. Consider an instance of max -vertex cover and produce an instance of max -vertex cover() where , and remain the same and any element of is transformed in in , where is the maximum of the values in . It is easy to see that all the elements in are bounded above by and that the transformation of into can be done in polynomial time.
Assume now that there exists a polynomial -approximation algorithm A for max -vertex cover() computing a solution consisting of edges of total weight and denote by the optimal value for the problem. Denote by the edge-set of an optimal solution of and by the weights of its edges; obviously . We then have:
where inequalities in (2) hold because the maximum number of edges in a bipartite graph of order is bounded above by .
3 Improving an approximation ratio
The following proposition gives a lower bound for , for any , that will be used later (it is easy to see that, symmetrically, the same holds also for ).
where is an upper bound of the approximation ratio of Algorithm 1.
Proof. Denote by A a -approximation algorithm for max -vertex cover, fix an integer and run Algorithm 1.
together with their (optimal) incident edges and, probably, another set of optimal edges incident to a subset ofbelonging to . Step 2 deletes from , additional optimal vertices. After Steps 1 and 2, the optimal value is at least:
Finally, Step 3 produces a solution SOL1 with value:
Since , (4) becomes:
Consider now the following Algorithm 2.
The approximation ratio of Algorithm 2 is:
Proof. Consider Step (2b) of Algorithm 2. If SOL() leaves outside at least optimal vertices, then either , or contains a set of at least optimal vertices with a coverage capacity at least . So, in this case, by adding the best vertices either from , or from and taking into account Proposition 1, Algorithm 2 builds a solution with value at least:
Let now assume that Step 2b of Algorithm 2 leaves outside less than optimal vertices. If there exist vertices (outside of the solution computed) either in , or in , with coverage capacity at least , then the solution computed in Step 2b adds in the solution an additional coverage capacity at least and the discussion just above always holds.
So, the case remaining to be handled, is the one where:
which, after some easy algebra becomes:
Consider now the very simple algorithm consisting of taking the first best vertices of (recall that vertices in are ordered in decreasing order with respect to their coverage capacity). It guarantees at least . So:
Let now consider the other extreme case where Step (2b) of Algorithm 2 leaves outside less than optimal vertices for any from down to . In this case (6) holds for any execution of Step 2 of the algorithm. Then, in Step 3, Algorithm 2 guarantees ratio:
Assume now that from to , Step (2b) of Algorithm 2 leaves outside less than optimal vertices and that for Step (2b) of Algorithm 2 leaves outside more than optimal vertices. This means (following (6)) that until the -th execution of Step 2, and at the -th execution (), becomes smaller than . Denote by the coverage capacity of a vertex and consider the sum , for any . Obviously, there exists at least a such that: , which implies . We can assume that . Indeed, if , then Step 3 of Algorithm 2 computes a set of coverage capacity at least and then discussion above concludes to a ratio:
Assuming we get:
Following the discussion above, for the value of the so-produced solution holds:
Solving the inequality:
The proof of the theorem is now concluded.
4 A PTAS for max -vertex cover
Revisit (10). Some easy algebra allows to conclude that:
and this quantity is decreasing with .
Consider now Algorithm 3 and denote by , the ratio of A (), by , the ratio of Algorithm 3 after the -th execution of the loop for and by the ratio of Algorithm 3 after the last execution. Suppose also that the for loop of Algorithm 3 is executed times. Then, taking 15, into account:
Since the discussion just above holds for any fixed constant , the following theorem holds immediately.
. max -vertex cover in bipartite graphs admits a polynomial time approximation schema.
The very useful discussions with Cécile Murat, Federico Della Croce, Michael Lampis and Aristotelis Giannakos are gratefully acknowledged.
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