Extension of vertex cover and independent set in some classes of graphs and generalizations

Extension of vertex cover and independent set in some classes of graphs and generalizations

Katrin Casel Henning Fernau Universität Trier, Fachbereich 4, Informatikwissenschaften,
                  Trier, Germany
{casel,fernau}@informatik.uni-trier.de
Mehdi Khosravian Ghadikolaei Université Paris-Dauphine, PSL University,
CNRS, LAMSADE, 75016 Paris, France
{mehdi.khosravian-ghadikolaei,jerome.monnot, florian.sikora}@lamsade.dauphine.fr
Jérôme Monnot Université Paris-Dauphine, PSL University,
CNRS, LAMSADE, 75016 Paris, France
{mehdi.khosravian-ghadikolaei,jerome.monnot, florian.sikora}@lamsade.dauphine.fr
Florian Sikora Université Paris-Dauphine, PSL University,
CNRS, LAMSADE, 75016 Paris, France
{mehdi.khosravian-ghadikolaei,jerome.monnot, florian.sikora}@lamsade.dauphine.fr
Abstract

We consider extension variants of the classical graph problems Vertex Cover and Independent Set. Given a graph and a vertex set , it is asked if there exists a minimal vertex cover (resp. maximal independent set) with (resp. ). Possibly contradicting intuition, these problems tend to be -hard, even in graph classes where the classical problem can be solved in polynomial time. Yet, we exhibit some graph classes where the extension variant remains polynomial-time solvable. We also study the parameterized complexity of theses problems, with parameter , as well as the optimality of simple exact algorithms under the Exponential-Time Hypothesis. All these complexity considerations are also carried out in very restricted scenarios, be it degree or topological restrictions (bipartite, planar or chordal graphs). This also motivates presenting some explicit branching algorithms for degree-bounded instances.

We further discuss the price of extension, measuring the distance of to the closest set that can be extended, which results in natural optimization problems related to extension problems for which we discuss polynomial-time approximability.

1 Introduction

We will consider extension problems related to the classical graph problems Vertex Cover and Independent Set. Informally in the extension version of Vertex Cover, the input consists of both a graph  and a subset  of vertices, and the task is to extend  to an inclusion-wise minimal vertex cover of  (if possible). With Independent Set, given a graph and a subset of vertices, we are looking for an inclusion-wise maximal independent set of contained in .

Studying such version is interesting when one wants to develop efficient enumeration algorithms or also for branching algorithms, to name two examples of a list of applications given in [CasFKMS2018].

Related work

In [doi:10.1080/10556789808805708], it is shown that extension of partial solutions is -hard for computing prime implicants of the dual of a Boolean function; a problem which can also be seen as trying to find a minimal hitting set for the prime implicants of the input function. Interpreted in this way, the proof from [doi:10.1080/10556789808805708] yields -hardness for the minimal extension problem for 3-Hitting Set. This result was extended in [BazBCFJKLLMP2018] to prove -hardness for the extension of minimal dominating sets (Ext DS), even restricted to planar cubic graphs.Similarly, it was shown in [BazganBCF16] that extension for minimum vertex cover restricted to planar cubic graphs is -hard. The first systematic study of this type of problems was exhibited in [CasFKMS2018] providing quite a number of different examples of this type of problem.

An independent system is a set system , , that is hereditary under inclusion. The extension problem Ext Ind Sys (also called Flashlight) for independent system was proposed in [LawLenKan80]. In this problem, given as input , one asks for the existence of a maximal independent set including and that does not intersect with . Lawler et al. proved that Ext Ind Sys is -complete, even when  [LawLenKan80]. In order to enumerate all (inclusionwise) minimal dominating sets of a given graph, Kanté et al. studied a restriction of Ext Ind Sys: finding a minimal dominating set containing but excluding . They proved that Ext DS is -complete, even in special graph classes like split graphs, chordal graphs and line graphs [kante2015polynomial, kante2015polynomia]. Moreover, they proposed a linear algorithm for split graphs when is a partition of the clique part [kante2014enumeration].

Organization of the paper

After some definitions and first results in Section 2, we focus on bipartite graphs in Section 3 and give hardness results holding with strong degree or planarity constraints. We also study parameterized complexity at the end of this section and comment on lower bound results based on ETH. In Section 4, we give positive algorithmic results on chordal graphs, with a combinatorial characterization for the subclass of trees. We introduce the novel concept of price of extension in Section 5 and discuss (non-)approximability for the according optimization problems. In Section 8, we generalize our results to -free graphs for some fixed . In Section 6, we prove several algorithmic results for bounded-degree graphs, based on a list of reduction rules and simple branching. Finally, in Section 7, we give some prospects of future research.

2 Definitions and preliminary results

Throughout this paper, we consider simple undirected graphs only, to which we refer as graphs henceforth. A graph can be specified by the set of vertices and the set of edges; every edge has two endpoints, and if is an endpoint of , we also say that and are incident. Let be a graph and ; denotes the neighborhood of in and denotes the closed neighborhood of . For singleton sets , we simply write or , even omitting if clear from the context. The cardinality of is called degree of , denoted . A graph where all vertices have degree is called -regular; 3-regular graphs are called cubic. If three upper-bounds the degree of all vertices we speak of subcubic graphs.

A vertex set induces the graph with vertex set and being an edge in iff both endpoints of are in . A vertex set is called independent if ; is called dominating if ; is a vertex cover if each edge is incident to at least one vertex from . A graph is called bipartite if its vertex set decomposes into two independent sets. A vertex cover  is minimal if any proper subset of is not a vertex cover. Clearly, a vertex cover is minimal iff each vertex in possesses a private edge, i.e., an edge with . An independent set is maximal if any proper superset of is not an independent set. The two main problems discussed in this paper are:

Ext VC Input: A graph , a set of vertices . Question: Does have a minimal vertex cover with ?

Ext IS Input: A graph , a set of vertices . Question: Does have a maximal independent set with ?

For Ext VC, the set is also referred to as the set of required vertices. As complements of maximal independent sets are minimal vertex covers we conclude:

Remark 1.

is a yes-instance of Ext VC iff is a yes-instance of Ext IS, as complements of maximal independent sets are minimal vertex covers.

Since adding or deleting edges between vertices of does not change the minimality of feasible solutions of Ext VC, we can first state the following.

Remark 2.

For Ext VC (and for Ext IS) one can always assume the required vertex set (the set ) is either a clique or an independent set.

The following theorem gives a combinatorial characterization of yes-instances of Ext VC that is quite important in our subsequent discussions.

Theorem 1.

Let be a graph and be a set of vertices. The three following conditions are equivalent:

is a yes-instance of Ext VC.

is a yes-instance of Ext IS.

There exists an independent dominating set of .

Proof.

In the following arguments, let be a graph. Let us first look at conditions and . By our previous discussions, condition is equivalent to: is a yes-instance of Ext VC. Assume there is a minimal vertex cover of with . Hence, in particular we deduce for every by minimality of . Condition therefore entails the existence of an independent set of with and . Hence, condition implies condition . Conversely, let be an independent dominating set of . Clearly, is a vertex cover of . If were not minimal, then there would be a vertex with , as then would not possess a private edge. But then would not be dominated by any vertex from , violating the assumption that is a dominating set of . Hence, conditions and are equivalent.

Now, we will prove the equivalence between items and . Let be a minimal vertex cover of with . Clearly, is a vertex cover of , but notice that it need not be minimal, as private edges of need not lie in the graph induced by . The set is an independent set (as the complement of within ) which dominates all the vertices in . Namely, consider any and assume that . Then, , contradicting minimality of . We turn into a maximal independent set of the induced graph , by adding some vertices from to . Observe that the resulting set is also a maximal independent set in and hence satisfies condition , because each has a private edge (as being part of the minimal vertex cover of , connecting to some . Conversely, assume the existence of an independent dominating set of satisfying . Hence, is an independent set with and . Let be any maximal independents set of , for instance, produced by some greedy procedure. Let . By construction, is an independent set in . If were not maximal, then we would find some with . Clearly, . But as has no neighbors in , it could have been added to by the mentioned greedy procedure. In conclusion, is a maximal independent set. Hence, satisfies the condition . ∎

3 Bipartite graphs

In this section, we focus on bipartite graphs. We prove that Ext VC is -complete, even if restricted to cubic, or planar subcubic graphs. Due to Remark 1, this immediately yields the same type of results for Ext IS. We add some algorithmic notes on planar graphs that are also valid for the non-bipartite case. Also, we discuss results based on ETH. We conclude the section by studying the parameterized complexity of Ext VC in bipartite graphs when parameterized by the size of .

Theorem 2.

Ext VC (and Ext IS) is -complete in cubic bipartite graphs.

Proof.

We reduce from 2-balanced 3-SAT, denoted -SAT, where an instance is given by a set of CNF clauses over a set of Boolean variables such that each clause has exactly literals and each variable appears exactly times, twice negative and twice positive. The bipartite graph associated to is the graph with , and or is literal of . Deciding whether an instance of -SAT is satisfiable is -complete by [ECCC-TR03-049, Theorem 1].

For an instance of -SAT, we build a cubic bipartite graph by duplicating instance (here, vertices and are the duplicate variants of vertices and ) and by connecting gadgets as done in Figure 1. We also add the following edges between the two copies: , and for . The construction is illustrated in Figure 1 and clearly, is a cubic bipartite graph. Finally we set .

Figure 1: Graph for Ext VC built from . Vertices of have a bold border.

We claim that is satisfiable iff admits a minimal vertex cover containing .

Assume is satisfiable and let be a truth assignment which satisfies all clauses. We set . We can easily check that is a minimal vertex cover containing .

Conversely, assume that possesses a minimal vertex cover containing . In order to cover the edges and , for every , either the set of two vertices or belongs to . Actually, for a fixed , we know that ; if or , then is not a minimal vertex cover, because or can be deleted, which is a contradiction. Hence, if (resp., ), then (resp., ), since the edges and (resp., and ) must be covered. In conclusion, by setting if and if we obtain a truth assignment which satisfies all clauses, because . ∎

In the following, we discuss restriction to planar graphs. In order to prove our results, we will present reductions from We use the problem 4-Bounded Planar 3-Connected SAT (4P3C3SAT for short), the restriction of 3-satisfiability to clauses in over variables in , where each variable occurs in at most four clauses (at least one time negative and one time positive) and the associated bipartite variable-gadget graph is planar of maximum degree 4. This restriction is also -complete [Kra94].

Let be an instance of 4P3C3SAT, where and are variable and clause sets of , respectively. By definition, the graph with and is planar. In the following, we always assume that the planar graph comes with an embedding in the plane. Informally, we are building a new graph by putting some gadgets instead of vertices of which satisfy two following conditions: (1) as it can be seen in Fig. 1, the constructions distinguishes between the cases that a variable appears positively and negatively in some clauses (2) the construction preserves planarity.

Suppose that variable appears in the clauses of instance such that in the induced (embedded) subgraph , , , , is an anti-clockwise ordering of edges around . By looking at and considering appears positively and negatively, the construction should satisfy one of the following cases:

  • case 1: and ;

  • case 2: and ;

  • case 3: and .

Note that all other cases are included in these by rotations or replacing with or vice versa.

case 1

case 2

case 3

Figure 2: Construction of Theorem 3. On the left: A variable appearing in four clauses in . On the right, cases 1,2,3: The gadgets in the constructed instance, depending on how appears (negative or positive) in the four clauses. Black vertices denote elements of .
Theorem 3.

Ext IS is -complete on planar bipartite subcubic graphs.

Proof.

The proof is based on a reduction from 4P3C3SAT. We start from graph which is defined already above and build a planar bipartite graph by replacing every node in with one of the three gadgets which are depicted in Fig. 2. Let

F_2={m_i^1, m_i^2, m_i^3, m_i^4:H(x_i) complies with case 2} . The permitted vertex set is , where . This construction is polynomial-time computable and is a planar bipartite subcubic graph. We claim that has a maximal independent set which contains only vertices from iff is satisfiable.If is a truth assignment of which satisfies all clauses, then depending on or , we define the independent set corresponding to three different variable gadgets as follows:

We can see that is a maximal independent set of which contains only vertices from .

Conversely, suppose is a maximal independent set of . By using maximality of , we define an assignment for depending on different types of variable gadgets of as follows:

  • for case 1, one of must be in , hence we set (resp., ) if (resp., ).

  • for case 2, at least one of vertices in each pair must be in . Hence, at most one of and is true. Thus we set (resp., ) if (resp., ).

  • for case 3, one can see, similar to the previous two cases: if one of (resp., ) is in , then none of (resp. ) are in , then we set (resp., ) if (resp., ).

We obtain a valid assignment . This assignment satisfies all clauses of , since for all , (by maximality of ).∎

It is challenging to strengthen the previous result to planar bipartite cubic graphs.

Algorithmic notes for the planar case

By distinguishing between whether a vertex belongs to the cover or not and further, when it belongs to the cover, if it already has a private edge or not, it is not hard to design a dynamic programming algorithm that decides in time if is a yes-instance of Ext VC or not, given a graph together with a tree decomposition of width . With some more care, even can be achieved, but this is not so important here. Rather, below we will make explicit an algorithm on trees that is based on several combinatorial properties and hence differ from the DP approach sketched here for the more general notion of treewidth-bounded graphs.

Moreover, it is well-known that planar graphs of order have treewidth bounded by . In fact, we can obtain a corresponding tree decomposition in polynomial time, given a planar graph . Piecing things together, we obtain:

Theorem 4.

Ext VC can be solved in time on planar graphs of order .

Remarks on the Exponential Time Hypothesis

Assuming ETH, there is no -algorithm for solving -variable, -clause instances of -SAT. As our reduction from -SAT increases the size of the instances only in a linear fashion, we can immediately conclude:

Theorem 5.

There is no -algorithm for -vertex, -edge bipartite subcubic instances of Ext VC or Ext IS, unless ETH fails.

This also motivates us to further study exact exponential-time algorithms. We can also deduce optimality of our algorithms for planar graphs based on the following auxiliary result.

Proposition 1.

There is no algorithm that solves 4P3C3SAT on instances with variables and clauses in time , unless ETH fails.

Corollary 1.

There is no algorithm for solving Ext VC on planar instances of order , unless ETH fails.

Remarks on Parameterized Complexity

We now study our problems in the framework of parameterized complexity where we consider the size of the set of fixed vertices as standard parameter for our extension problems.

Theorem 6.

Ext VC with standard parameter is -complete, even when restricted to bipartite instances.

Proof.

We show hardness by reduction from Multicolored Independent Set, so let be an instance of of this problem, with partition for . W.l.o.g., assume that each induces a clique and . Construct from with built from two copies of , denoted and , and additional vertices , and containing for all and and for all , (see Fig. 3). is bipartite with partition into and . Set and consider as instance of Ext VC. We claim that is a yes-instance for Ext VC iff is a yes-instance for Multicolored Independent Set. Since Multicolored Independent Set is -hard [DBLP:journals/tcs/FellowsHRV09],111The proof is for Multicolored Clique; taking the complement graph is a parameterized reduction showing that Multicolored Independent Set is -hard. this -reduction shows -hardness for Ext VC with standard parameterization.

Suppose is a yes-instance for Ext VC, so there exists a minimal vertex cover for with . Consider . Since is minimal, for all , so especially for each there exists at least one vertex from in and also at least one vertex from . Since has to be an independent set in and for all , (recall that is a clique in ), it follows that if , then is the only vertex independent from in . This means that for all and if , then . The set hence is a multicolored independent set in , since for would imply that which is not possible since is an independent set in .Conversely, it is not hard to see that if there exists a multicolored independent set in , then the set (with ) is a minimal vertex cover for containing .

Membership in is seen as follows. As suggested in [Ces2003], we describe a reduction to Short Nondeterministic Turing Machine. Given a graph and a pre-solution , the constructed Turing machine first guesses vertices , with and then verifies in time if the guessed set is an independent set. As can be greedily extended to an independent dominating set for which, by Theorem 1, is equivalent to being a yes-instance of Ext VC, can be extended to a minimal vertex cover iff one of the guesses is successful. ∎



Figure 3: The graph for Ext VC, Vertices in are drawn bold.

As a remark, it is obvious to see that considering the parameter instead of leads to an -result, as it is sufficient to test if any of the subsets of , together with , form a minimal vertex cover. However, these algorithms are quite trivial and hence not further studied here. The same reasoning shows:

Remark 3.

Ext IS with standard parameter is in .

Theorem 7.

Ext VC with standard parameter is in on planar graphs.

Proof.

Let be an instance of Ext VC such that is planar. By Theorem 1, it suffices to solve Ext VC on , where is the graph induced by . Clearly, is also planar. Moreover, the diameter of each connected component of is upper-bounded by . Therefore, is (at most) -outerplanar and hence according to [Bod96], the treewidth of is at most . Our previous remarks show that Ext VC can be solved in time . ∎

4 Chordal and Circular-arc graphs

An undirected graph is chordal iff each cycle of with a length at least four has a chord (an edge linking two non-consecutive vertices of the cycle) and is circular-arc if it is the intersection graph of a collection of arcs around a circle. We will need the following problem definition.

Minimum Independent Dominating Set (MinISDS for short) Input: A graph . Solution: Subset of vertices which is independent and dominating. Output: Solution that minimizes .

Weighted Minimum Independent Dominating Set (or WMinISDS for short) corresponds to the vertex-weighted variant of MinISDS, where each vertex has a non-negative weight associated to it and the goal consists in minimizing . If with , the weights are called bivaluate, and and corresponds to binary weights.

Remark 4.

MinISDS for chordal graphs has been studied in [Farber82ORL], where it is shown that the restriction to binary weights is solvable in polynomial-time. Bivalued MinISDS with however is already -hard on chordal graphs, see [Chang04]. WMinISDS (without any restriction on the number of distinct weights) is also polynomial-time solvable in circular-arc graphs [Chang98].

Using the mentioned polynomial-time result of binary independent dominating set on chordal graphs [Farber82ORL] and circular-arc graphs [Chang98], we deduce:

Corollary 2.

Ext VC is polynomial-time decidable in chordal and in circular-arc graphs.

Proof.

By Remark 4, we can find, within polynomial-time, an independent dominating set minimizing among the independent dominating sets of a weighted chordal graph or circular-arc graph where and , .

Let be an instance of Ext VC where is a chordal graph (resp., a circular-arc graph). We will apply the result of [Farber82ORL] (resp., [Chang98]) for , where is the subgraph of induced by and if and for . Obviously, is a binary-weighted chordal graph (resp., circular-arc graph). So, an optimal independent dominating set of has a weight iff is a maximal independent set of , otherwise . Using Theorem 1, the result follows. ∎

Farber’s algorithm [Farber82ORL] runs in linear-time and is based on the resolution of a linear programming using primal and dual programs. Yet, it would be nice to find a (direct) combinatorial linear-time algorithm for chordal and circular-arc graphs, as this is quite common in that area. We give a first step in this direction by presenting a characterization of yes-instances of Ext VC on trees.

Consider a tree and a set of vertices . A subtree (ie., a connected induced subgraph) of a tree is called edge full with respect to if , for all . A subtree is induced edge full with respect to if it is edge full with respect to .

For our characterization, we use a coloring of vertices with colors black and white. If is a tree and , we use to denote the colored tree where exactly the vertices from are colored black. Further define the following class of black and white colored trees , inductively as follows. Base case: A tree with a single vertex belongs to if is black. Inductive step: If , the tree resulting from the addition of a (3 new vertices that form a path ) where one endpoint of is black, the two other vertices are white and the white endpoint of is linked to a black vertex of is in .

Theorem 8.

Let be a tree and be an independent set. Then, is a yes-instance of Ext VC if and only if there is no subtree of that is induced edge full with respect to such that .

5 Price of extension

Considering the possibility that some fixed set might not be extendible to any minimal solution, one might ask how wrong is as a fixed choice for an extension problem. One idea to evaluate this, is to ask how much has to be altered when aiming for a minimal solution. Described differently for our extension problems at hand, we want to discuss how many vertices of have to be deleted for Ext VC (added for Ext IS) in order to arrive at a yes-instance of the extension problem. The magnitude of how much has to be altered can be seen as the price that has to be paid to ensure extendibility.

In order to formally discuss this concept, we consider according optimization problems. From an instance of Ext VC or Ext IS, we define two new maximization (resp., minimization) problems, respectively.

Max Ext VC Input: A graph , a set of vertices . Solutions: Minimal vertex cover of . Output: Solution that maximizes .

Min Ext IS Input: A graph , a set of vertices . Solutions: Maximal independent set of . Output: Solution that minimizes .

For Max Ext VC or Min Ext IS, we denote by the value of an optimal solution of Max Ext VC or Min Ext IS, respectively. Since for both of them, iff is a yes-instance of Ext VC or Ext IS, respectively, we deduce that Max Ext VC and Min Ext IS are -hard as soon as Ext VC and Ext IS are -complete.

Notice that alternatively these two optimal quantities can be expressed as

 and
.

Similarly to Remark 1, one observes that the decision variants of Max Ext VC and Min Ext IS areindeed completely equivalent, more precisely:

(1)

We want to discuss polynomial-time approximability of Max Ext VC and Min Ext IS. Considering Max Ext VC on and the particular subset (resp., Min Ext IS with ), we obtain two well known optimization problems called upper vertex cover (UVC for short, also called the maximum minimal vertex cover problem) and the maximum minimal independent set problem (equivalently ISDS for short). In [Manlove99], the computational complexity of these two problems have been studied (among 12 problems), and (in)approximability results are given in [MishraS01, BoriaCP15] for UVC and in [Halldorsson93a] for ISDS where lower bounds of and , respectively, for graphs on vertices are given for every . Analogous bounds can be also derived depending on the maximum degree  of the graph. In particular, we deduce:

Corollary 3.

For any constant and any and , there is no polynomial-time -approximation for Min Ext IS on general graphs of vertices and maximum degree , even when , unless .

Now, we strengthen the above mentioned lower bounds of and for the inapproximability of Max Ext VC.

Theorem 9.

Max Ext VC is as hard as MaxIS to approximate in general graphs even if the set  of required vertices forms an independent set.

Proof.

The proof is based on a simple reduction from MaxIS. Let be an instance of MaxIS. Construct the graph from , where vertex set contains two copies of , denoted by and . The edge set contains together with for all , formally . Consider as instance of Max EXT VC, where the required vertex subset is given by . We claim: has a minimal vertex cover containing vertices from iff has an independent set of size . Let be a maximal independent set of of size ; then is a minimal vertex cover of containing vertices from . Conversely, let be a minimal vertex cover of extending , with . By construction, the set is a vertex cover of and then is an independent set of of size . In particular, we deduce .∎

Using the strong inapproximability results for MaxIS given in [Trevisan01, Zuckerman07], observing and , we deduce the following result.

Corollary 4.

For any constant and any and , there is no polynomial-time -approximation for Max Ext VC on general graphs of vertices and maximum degree , unless .

5.1 Positive results on the price of extension

In contrast to the hardness results on these restricted graph classes from the previous sections, we find that restriction to bipartite graphs or graphs of bounded degree improve approximability of Max Ext VC. For the following results, we assume w.l.o.g. that the input graph is connected, non-trivial and therefore without isolated vertices, as we can solve our problems separately on each connected component and then combine the results.

Theorem 10.

A 2-approximation for Max Ext VC on bipartite graphs can be computed in polynomial time.

Proof.

Let and be an instance of Max Ext VC, where contains only edges connecting and . Since and are both minimal vertex covers ( is without isolated vertices) and also a partition of , then taking one of them containing the largest number of vertices from (assume it is ), we get a 2-approximation, because . ∎

Theorem 11.

A -approximation for Max Ext VC on graphs of maximum degree can be computed in polynomial time.

Proof.

Let be connected of maximum degree , and be an instance of Max Ext VC. Consider the graph induced by the open neighborhood of , i.e., ; subgraph is a graph of maximum degree (at most) and by Brooks’s Theorem, we can color it properly with at most colors in polynomial time. Let be such coloring of , with . Let for . is an independent set which dominates in so it can be extended to satisfy of Theorem 1, so is a yes-instance of Ext VC. Choosing yields a -approximation, since