Parameterized algorithms for Partial vertex covers in bipartite graphs

TL;DR

This paper investigates the fixed-parameter tractability of the weighted partial vertex cover problem in bipartite graphs, revealing conditions under which the problem is tractable or hard, and proposing algorithms for specific cases.

Contribution

It extends existing methods to show FPT results for WPVC in bipartite graphs under certain conditions and establishes hardness results in others.

Findings

WPVCB is FPT with respect to R when c is constant

WPVCB is W[1]-hard with arbitrary c, even if p is constant

WPVC is FPT with respect to L in bipartite graphs

Abstract

In the weighted partial vertex cover problem (WPVC), we are given a graph $G=(V,E)$, cost function $c:V\rightarrow N$, profit function $p:E\rightarrow N$, and positive integers $R$ and $L$. The goal is to check whether there is a subset $V'\subseteq V$ of cost at most $R$, such that the total profit of edges covered by $V'$ is at least $L$. In this paper we study the fixed-parameter tractability of WPVC in bipartite graphs (WPVCB). By extending the methods of Amini et al., we show that WPVCB is FPT with respect to $R$ if $c\equiv 1$. On the negative side, it is $W[1]$-hard for arbitrary $c$, even when $p\equiv 1$. In particular, WPVCB is $W[1]$-hard parameterized by $R$. We complement this negative result by proving that for bounded-degree graphs WPVC is FPT with respect to $R$. The same result holds for the case of WPVCB when we allow to take only one fractional vertex. Additionally, we show that WPVC is FPT with respect to $L$. Finally, we discuss a variant of PVCB in which the edges covered are constrained to include a matching of prescribed size and derive a paramterized algorithm for the same.

Equations

$$S=\left\{v\in V(B):p(\partial (v))\ge \frac{L}{R}\right\}.$$

$$R=\sum _{i=1}^{2k}{2}^i=2^{2k+1}-2$$

$$L=\sum _{i=1}^{2k}(2^i(n+1)+5^i)=(n+1)R+(5/4)(5^{2k}-1).$$

$$\frac{2^i(n+1)+5^i}{2^i}=(n+1)+(5/2{)}^i.$$

$$R_t\cdot (5/2{)}^{t-1}<2^{t+1}(5/2{)}^{t-1}=4\cdot 5^{t-1}<5^t.$$

$$S_B(v)=\{v_i:i=1,2,...,c(v)\}.$$

$$V^{\prime }=\bigcup _{v\in V}{S}_B(v),$$

$$E^{\prime }=\{v_i{u}_j:\text{makebox}{v}_i\in S_B(v),\text{makebox}{u}_j\in S_B(u)\text{makebox}\text{and}\text{makebox}vu\in E\},$$

$$p(v_i{u}_j)=\frac{p(vu)}{c(v)c(u)}.$$

$${\alpha }_v=\frac{|\{v_i:v_i\in S^{\prime }\}|}{c(v)},$$

$$p(E(Z))\ge |E(Z)|\ge \frac{d(z_1)+\dots +d(z_{2L})}{2}\ge \frac{2L}{2}=L.$$

Full text

††journal: Sample Journal

1 Introduction

In this paper, we study several variants of the vertex cover (VC) problem, from the perspectives of parameterized algorithm design and parametric complexity. In particular, we consider the partial vertex cover problem, wherein the goal is to cover a certain threshold of edges (as opposed to all the edges) using the fewest number of vertices. We also look into weighted variants of this

The rest of this paper is organized as follows: The problems studied in this paper are formally described in Section 2. In Section 3, we discuss the motivation for our work and mention related approaches in the literature. Our main results are described in Section 4. A variant of the partial vertex cover problem with applications to computational social choice is detailed in Section 5. We conclude in Section 6, by summarizing our results and outlining avenues for future research.

2 Statement of Problems

We focus on finite, undirected graphs that have no loops or multiple edges. As usual, the degree of a vertex is the number of edges of the graph that are incident to it. The maximum degree of the graph $G$ is just the maximum of all degrees of vertices of $G$. A graph $G=(V,E)$ is bipartite, if its vertex set can be partitioned into two sets $V_{1}$ and $V_{2}$, so that each edge of $G$ joins a vertex from $V_{1}$ to one from $V_{2}$.

Given a graph $G=(V,E)$, and a set ${S\subset V}$ of vertices, an edge $(i,j)\in{E}$ is covered by ${S}$ if $i\in{S}$ or $j\in{S}$. Let $E(S)$ to be the set of edges of $G$ that are covered with at least one vertex of $S$. The classical Vertex Cover problem (VC) is defined as finding the smallest set $S$ of vertices of the input graph $G$, so that $E(S)=E$. The vertex cover problem is a well-known NP-complete problem [16].

In this paper, we study the following variants of VC:

The Partial vertex cover problem (PVC) - In this problem -

Definition 1**.**

Given an undirected graph ${G=(V,E)}$, and a vertex-cardinality parameter $K_{1}$ and an edge-cardinality parameter $K_{2}$, is there a subset $V^{\prime}$ of $V$, such that $|V^{\prime}|\leq K_{1}$ and the number of edges of covered by $V^{\prime}$ is at least $K_{2}$? 2. 2.

The Weighted Partial Vertex Cover problem (WPVC) -

Definition 2**.**

Given an undirected graph ${G=(V,E)}$, weight-functions $c:{V}\rightarrow{N}$ and ${p}:{E}\rightarrow{N}$ and a vertex-weight parameter $R$ and an edge-weight parameter $L$, is there a subset $S$ of $V$, such that $\sum_{e\in{E(S)}}p(e)\geq L$, and $\sum_{v\in{S}}c(v)\leq R$? 3. 3.

The Partial vertex cover problem on bipartite graphs (PVCB) - This is the restriction of the partial vertex cover (PVC) problem to bipartite graphs. 4. 4.

The Weighted Partial vertex cover problem on bipartite graphs (WPVCB) - This is the restriction of the weighted vertex cover (WVC) problem to bipartite graphs. 5. 5.

The Partial vertex cover problem with matching constraint (PVCBM) - This is a variant of the PVCB problem, in which we are given a third parameter $K_{3}$ and the goal is to find a vertex subset of size at most $K_{1}$, covering at least $K_{2}$ edges, such that the edges covered include a matching of size at least $K_{3}$.

The principal contributions of this paper are as follows:

Fixed-parameter tractability for a restricted version of the WPVCB problem. 2. 2.

${\bf W}[1]$ hardness of the WPVCB problem with respect to a certain parameter. 3. 3.

Fixed parameter tractability of the Weighted partial vertex cover problem in bounded degree graphs (not necessarily bipartite). 4. 4.

Fixed parameter tractability of the Weighted partial vertex cover problem with respect to $L$. 5. 5.

A parameterized algorithm for the matching variant of the PVCB problem.

3 Motivation and Related Work

When $c\equiv 1$ and $p\equiv 1$, we get the well-known partial vertex cover problem (PVC). PVC represents a natural theoretical generalization of VC and is motivated by practical applications. Flow-based risk-assessment models in computational systems, for example, can be viewed as instances of PVC [6].

Although VC is polynomial-time solvable in bipartite graphs, the Partial Vertex Cover problem on bipartite graphs is NP-hard. The computational complexity of this problem has been open and recently shown to be NP-hard [3, 14, 9, 10]. Many $2$-approximation algorithms for VC are known [29]. There is an approximation algorithm for the VC problem which has an approximation factor of $2-\theta(\frac{1}{\sqrt{logn}})$ [15]. This is the best known algorithm. The VC problem is also known to be APX-complete [26]. Moreover, it cannot be approximated within a factor of $1.3606$ unless P = NP [12], and not within any constant factor smaller than $2$, unless the unique games conjecture is false [17]. Let us note that in [27], a $(\frac{4}{3}+\epsilon)$-approximation algorithm is designed for WPVC for each $\epsilon>0$ when the input graph is bipartite. This restriction is denoted by WPVCB.

All hardness results for the VC problem directly apply to the PVC problem because the PVC problem is an extension of the VC problem. Since 1990’s the PVC problem and the partial-cover variants of similar graph problems have been extensively studied [8, 7, 23, 19, 28, 20]. In particular, there is an $O(n\cdot\log n+m)$-time $2$-approximation algorithm based on the primal-dual method [20], as well as another combinatorial 2-approximation algorithm [21]. Both of these algorithms are for a more general soft-capacitated version of PVC. There are several older 2-approximations resulting from different approaches [5, 8, 18, 13]. Let us also note that the WPVC problem for trees (WPVCT) is studied in [22], where the authors provide an FPTAS for it, and a polynomial time algorithm for the case when vertices have no weights.

Another problem with close relationship to WPVC is the Budgeted Maximum Coverage problem (BMC). In this problem one tries to find a min-cost subset of vertices, such that the profit of covered edges is maximized. In some sense, this problem can be viewed as a problem “dual” to WPVC, and it can be shown that both problems are equivalent from the perspective of exact solvability. The BMC problem for sets (not necessarily graphs) admits a $(1-\frac{1}{e})$-approximation algorithm [24] but, special cases that beat this bound are rare. The pipage rounding technique gives a $\frac{3}{4}$-approximation algorithm for the BMC problem on graphs [2] which is improved to $\frac{4}{5}$ for bipartite graphs [4]. Finally, let us note that in [9, 10], an $8/9$-approximation algorithm for the problem is presented when the input graph is bipartite and the vertices are unweighted. The result is based on the linear-programming formulation of the problem, and the constant $8/9$ matches the integrality gap of the linear program.

In the present paper, we address these problems from the perspective of fixed-parameter tractability (FPT). Recall that a combinatorial problem $\Pi$ is said to be fixed-parameter tractable with respect to a parameter $k$, if there is an algorithm for solving $\Pi$ exactly, whose running time is bounded by $f(k)\cdot size^{O(1)}$. Here $f$ is some (computable) function of $k$, and $size$ is the length of the input. From the perspective of FPT, the PVC problem is in some sense more difficult than the VC problem. For instance, the PVC problem is W[1]-complete [11], while the VC problem is FPT [25, 11]. Related with this topic, let us note that in [1] the decision version of WPVCB is considered, where in a bipartite graph one needs to check whether there is a subset of cost at most $R$, whose coverage is at least $L$. The authors show that this problem is FPT with respect to $R$, when the vertices and edges of the bipartite graph are unweighted [1].

In this paper, by extending the methods of Amini et al., we show that the decision version of WPVCB is FPT with respect to $R$, if the vertices have cost one, however the edges may have weights. On the negative side, the problem is $W[1]$-hard, even when edges have profit one. We complement this negative result by proving that for bounded-degree graphs WPVC is FPT with respect to $R$. The same result holds for the case of WPVCB when one is allowed to take only one fractional vertex. We finish the paper by showing that WPVC is FPT with respect to $L$. Terms and concepts that we do not define can be found in [11].

4 Main Results

In this section we present our results. Our goal is to investigate the fixed-parameter tractability of the decision version of WPVCB problem. It is formulated as follows:

WPVCBD: Given a bipartite graph $B$, a cost function $c:V(B)\rightarrow{N}$, a profit function $p:E(B)\rightarrow{N}$ and positive integers $R$, $L$, the goal is to check whether there is a set $S\subset V(B)$, such that $c(S)\leq R$ and $p(E(S))\geq L$.

When $c$ and $p$ are identically one, we get the PVCBD problem. When $c$ is identically one, we get EPVCBD. Finally, when $p$ is identically one, we get the VPVCBD problem. We will also use the same scheme of notations when the input graph need not be bipartite. In [1], PVCBD is considered and it is shown that the problem is FPT with respect to $R$. Below we strengthen this result.

Theorem 1**.**

EPVCBD is FPT with respect to $R$.

Proof.

Roughly speaking we obtain the result with the approach of [1] by considering the weighted degree instead of usual degree. Below we present the technical details.

Assume that we have an instance $I$ of EPVCBD. For a vertex $v$ of $B$, let $\partial(v)$ be the set of edges of $B$ incident with $v$. Define the set $S$ of vertices of $B$ as follows:

[TABLE]

We consider two cases based on the size of $S$.

Case 1: $|S|\geq 2R$. Consider the subgraph $H$ of $B$ induced by $S$. Since $B$ is bipartite, $H$ is bipartite, too. Let $(X,Y)$ be the bipartition of $H$, and assume that $|X|\geq|Y|$. Since $|X|+|Y|=|S|\geq 2R$, we have $|X|\geq R$. Take any $R$ vertices of $X$. Observe that $X$ is an independent set in $B$, hence the coverage of the $R$ vertices is at least $L$. This means that $I$ is a “yes” instance.

Case 2: $|S|<2R$. Observe that any feasible solution to $I$ must intersect $S$. Hence, we do recursive guessing, that is, we try each vertex of $S$ one by one as a possible vertex of the feasible solution.

In the Case 1, the algorithm will run in polynomial time, so the most expensive case is Case 2. Since the number of vertices in a feasible solution is at most $R$, we have that the depth of the recursion is at most $R$. Hence the total running time of our algorithm is $O((2R)^{R}\cdot size^{O(1)})$. ∎

Our next result shows that WPVCBD and VPVCBD are W[1]-hard. Our reduction is from the Multi-colored CLIQUE problem [11]. It is formulated as follows:

Multi-colored CLIQUE: Given a graph $G$, a positive integer $k$ and a partition $(V_{1},...,V_{k})$ of vertices of $G$, the goal is to check whether $G$ contains a $k$-clique $Q$, such that $Q$ contains exactly one vertex from each $V_{j}$ for $j=1,...,k$.

Multi-colored CLIQUE is a well-studied problem which is known to be W[1]-hard. Observe that since an edge $e$ connecting two vertices from $V_{i}$ $1\leq i\leq k$ does not lie in a feasible clique, without loss of generality, we can assume that for $i=1,...,k$ $V_{i}$ is an independent set of vertices.

Theorem 2**.**

WPVCBD is W[1]-hard parameterized by $R$.

Proof.

We show an FPT-reduction from Multi-colored Clique. Let $G=(V,E)$ be an instance of this problem with vertices partitioned as $V=V_{1}\cup\ldots\cup V_{k}$. We create a bipartite graph $B=(U^{\prime}\cup V^{\prime}\cup Z,E^{\prime})$ as follows. Let $U^{\prime}$ and $V^{\prime}$ be two copies of $V$, and let $V=V^{\prime}=\{v_{1},\ldots,v_{n}\}$, $U^{\prime}=\{u_{1},\ldots,u_{n}\}$, where for each $i\in[n]$, $u_{i}$ is a copy of $v_{i}$. Here as usual $[n]=\{1,\ldots,n\}.$ For a vertex $v\in V$ let $\chi(v)$ be its color, i.e., $\chi(v)=i$ if $v\in V_{i}$, and extend this to $U^{\prime}\cup V^{\prime}$ so that $\chi(u_{i})=\chi(v_{i})+k$. For a vertex $x\in U^{\prime}\cup V^{\prime}$, let the cost of $x$ be $c(x)=2^{\chi(x)}$. Add an edge $u_{i}v_{j}$ to $B$ if either $\chi(u_{i})=\chi(v_{j})+k$ and $i\neq j$, or $\chi(u_{i})\neq\chi(v_{j})+k$ and $v_{i}v_{j}\notin E(G)$. Give all these edges profit $1$. Observe that a selection of one vertex from every color class of $U^{\prime}\cup V^{\prime}$ forms an independent set in $B$, if and only if it corresponds to two copies of a $k$-clique in $G$.

Add two additional vertices $z_{1}$ and $z_{2}$, let $Z=\{z_{1},z_{2}\}$ and give both the cost $2^{2k+1}$. Finally, for every vertex $x\in U^{\prime}\cup V^{\prime}$ add an edge $xz_{2}$ or $z_{1}x$ (as appropriate, to maintain bipartiteness) and give these edges a profit value so that the total profit of all edges incident with $x$ equals $2^{\chi(x)}(n+1)+5^{\chi(x)}$. (This is clearly possible, since the total profit of all previously created edges incident with $x$ is bounded by $n$.)

Set the budgets of the instance as vertex budget

[TABLE]

and profit threshold

[TABLE]

This finishes the instance description. It is clear that the construction can be performed in polynomial time, and the budget $R$ is a function of $k$. It therefore remains to show that $(B,R,L)$ is a positive instance of WPVCBD if and only if $G$ has a multi-colored clique.

From the multi-colored clique problem to the partial vertex cover problem. Let $X\subseteq V(G)$ be a multi-colored $k$-clique, and let $S=\{u_{i},v_{i}\mid u_{i}\in X\}$. Since $S$ contains one vertex for every color class of $B$ its total cost equals $R$, and since it induces an independent set in $B$ the total profit of the edges covered equals $L$.

From the partial vertex cover problem to the multi-colored clique problem. Now for the more challenging part of the argument. We need to argue that the costs and profits balance out so that the only way to select vertices to a total profit of $L$ is to select one vertex from every color class of $B$. For this, first observe that for a vertex of class $i\in[2k]$, the ratio of the total profit of its incident edges to its cost is

[TABLE]

Let $S$ be a partial vertex cover of profit at least $L$ and cost at most $R$, and for each $i\in[2k]$ let $n_{i}$ be the number of vertices of $S$ with color class $i$. We can consider the contributions to the total profit in two parts.

From the first part of the above expression, it is clear that every selection of cost at most $R$ contributes a profit of at most $(n+1)R$, regardless of the distribution $n_{i}$. Therefore we focus on the contribution of the second part of the formula, with target profit $L-(n+1)R$.

Considering this second part, define $R_{t}=\sum_{i=1}^{t}2^{i}$ and $L_{t}=\sum_{i=1}^{t}5^{i}$. We show by induction that for every $t\in[2k]$, the largest possible contribution of a selection of cost at most $R_{t}$ is $L_{t}$ (which of course is achieved by making one seletion per color class). For $t=1$ this is trivial. Therefore, by induction, let $t>1$ and assume that the claim holds for every value $t^{\prime}<t$. Let $n^{\prime}_{i}$, $i\in[t]$, denote the number of vertices selected in color class $i$, for a selection of total cost at most $R_{t}$. Then if $n_{t}=0$, the maximum possible profit is

[TABLE]

Therefore, the total profit from the selection $n_{i}^{\prime}$ is less than that from a single vertex of class $t$. Therefore $n_{t}^{\prime}\geq 1$. But then the remaining budget is $R_{t}-2^{t}=R_{t-1}$, and by induction the optimal selection has $n_{i}^{\prime}=1$ for every $i\in[t]$, completing the induction step. Therefore we may assume $n_{i}=1$ for every $i\in[2k]$ for our selection $S$.

But then, finally, we observe that a total profit of $L$ is possible only if $S$ is an independent set, since otherwise the profit of some edge will have been double-counted in the above calculations. Therefore, $S$ contains precisely one vertex of each color class of $B$ forming an independent set. In particular, if $v_{i}\in S$ is a selection in color class $j\in[k]$ for some $i\in[n]$, then the selection in color class $j+k$ must be $u_{i}$. Also, for every pair of color classes $i,i^{\prime}\in[k]$ the selections in classes $i$ and $i^{\prime}+k$ are independent in $B$ and therefore the selections in classes $i$ and $i^{\prime}$ are neighbors in $G$. Thus $X=S\cap V^{\prime}$ is a multi-colored clique in $G$, as required. ∎

Corollary 1**.**

The problem is W[1]-hard also in the variant where all edge profits are 1, i.e., the VPVCBD problem is W[1]-hard.

Proof.

The only edges of weight more than 1 in the above reduction are the edges connecting to the special vertices $z_{i}$, and the largest edge weight used is bounded by a function $f(k)(n+1)$. Therefore, instead of using edge weights we can in FPT time simply create the corresponding number of pendant vertices for each vertex. These pendants can be given the same weight as the vertices $z_{i}$. ∎

A class of graphs is said to be bounded-degree, if there is a constant $C$, such that all graphs from the class have maximum degrees at most $C$. It turns out that when the input graphs have bounded-degree and need not be bipartite, we have the following result:

Theorem 3**.**

WPVCD is fixed-parameter tractable with respect to $R$ for bounded-degree graphs.

Proof.

Let $I$ be an instance of the WPVCD problem, where $G=(V,E)$ is a bounded-degree graph, $c:V\rightarrow N$, $p:E\rightarrow N$ are cost and profit functions, and $L$, $R$ are constants. Assume that for every $v\in V$, we have $d(v)\leq d$. For $i=1,2,\ldots,R$, let $M_{i}(G)=\{v:c(v)=i\}$ (we disregard the vertices of cost greater than $R$). Choose a vertex $v_{i}\in M_{i}(G)$ which has the largest coverage, and let $M$ be the set comprised of chosen vertices $v_{i}$. The set of vertices of $G$ which have a neighbor from $M$, we denote by $N$. Consider the set $M\cup N$. It is obvious that $|M\cup N|\leq(d+1)R$, where $d$ is the bound for the degree.

Let us show that if $I$ is a “yes” instance, then we can construct a feasible set, which intersects $M\cup N$. Indeed, let $S$ be a feasible set, which does not have a vertex from $M\cup N$. Then we can replace any vertex $v\in S$ with $v_{c(v)}$ to get a new set $S^{\prime}$. Since there is no vertex of $S$, which is a neighbor of $v_{c(v)}$, we have that the profit of $S^{\prime}$ does not decrease. Moreover, as $c(v)=c(v_{c(v)})$, $S^{\prime}$ is also feasible. Observe that $S^{\prime}$ intersects $M\cup N$.

Now we complete the proof by recursively guessing on $M\cup N$. Since the number of vertices in a feasible set is at most $R$, we have that the depth of the recursion is at most $R$, hence the total running time of the algorithm is $((d+1)R)^{R}size^{O(1)}$. ∎

Corollary 2**.**

WPVCD is fixed-parameter tractable with respect to $R$ for graphs of maximum degree three, in particular, for cubic graphs.

Below we consider a version of the WPVCBD problem denoted by WPVCBFD when one is allowed to take at most one vertex fractionally. When a vertex $v$ is taken to an extent of $\alpha$, ($0\leq\alpha\leq 1$), we assume that it contributes to the cost of the cover by $\alpha\cdot c(v)$. The profits of covered edges are defined as follows: if an edge $e$ is covered solely by $v$ then its profit is $\alpha\cdot p(e)$, otherwise, it is $p(e)$. If a vertex $v$ is taken fractionally, we say that $v$ is a fractional vertex.

Theorem 4**.**

WPVCBFD is fixed-parameter tractable with respect to $R$.

Proof.

Let $I$ be an instance of the WPVCBFD problem, where $B=(V,E)$ is a bipartite graph, $c:V\rightarrow N$, $p:E\rightarrow N$ are cost and profit functions, and $L$, $R$ are constants. For any $v\in V$ let

[TABLE]

$S_{B}(v)$ is called the section corresponding to $v$. We create an instance $I^{\prime}$ of the EPVCBD problem with rational edge profits as follows: the graph is $B^{\prime}=(V^{\prime},E^{\prime})$, where

[TABLE]

[TABLE]

and the profit function is

[TABLE]

The new problem can be turned to an instance of the EPVCBD problem with natural edge profits by multiplying all edge profits with the least common multiplier of all denominators of edge profits. Clearly, this does not change the hardness of the problem. By Theorem 1, we know that EPVCBD is fixed-parameter tractable with respect to $R$. Hence $I^{\prime}$ can be solved in time $f(R)\cdot size^{O(1)}$.

Assume that we have a feasible set $S^{\prime}$ in $I^{\prime}$. We construct a feasible set $S$ of $I$, so that it contains at most one fractional vertex. Let $\alpha_{v}$ be defined as follows:

[TABLE]

and take every vertex $v$ to an extent of $\alpha_{v}$. It is obvious that we have a solution $S$ for $I$, however there can be many vertices which are fractional. Recall that we should allow only one such vertex.

Now, we are going to modify $S$. Let $v$ and $u$ be fractional vertices, and assume that both $v_{i}\in S_{B}(v)$ and $u_{j}\in S_{B}(u)$ do not belong to $S^{\prime}$. Without loss of generality, assume that $v$ has higher coverage. Then, delete one vertex from $S^{\prime}$ that lies in the section of $u$ and add one vertex to $S^{\prime}$ that belongs to the section of $v$. We do this for every such $v$ and $u$. After this only one vertex can be fractional.

It is obvious that we do only polynomially many steps after solving $I^{\prime}$. It means that our problem is fixed-parameter tractable with respect to $R$. ∎

Exercise 5.11 from [11] implies that PVCD is fixed-parameter tractable with respect to $L$. Below, we strengthen the statement of this exercise by showing that the WPVCD problem can be parameterized with respect to $L$.

Theorem 5**.**

WPVCD is fixed-parameter tractable with respect to $L$.

Proof.

Let $I$ be an instance of the WPVCD problem, where $G=(V,E)$ is a graph, $c:V\rightarrow N$, $p:E\rightarrow N$ are cost and profit functions, $L$ and $R$ are constants. We can assume that no vertex of $G$ is isolated. Moreover, without loss of generality, we can assume that for any vertex $v$, we have $c(v)\leq R$.

For every vertex $v$ we denote $p(v)=p(\partial(v))=\sum_{e:v\in e}p(e)$ the total profit of edges incident to $v$. We can assume that $p(v)\leq L-1$ for all vertices, as otherwise we will have a feasible solution comprised of one vertex and, as a result, $I$ is a “yes” instance. This, in particular, means that $d(v)\leq L-1$. For $i=1,\ldots,L-1$ let $v_{i}$ be the vertex which has profit $i$ and for any other vertex $u$, which has profit $i$, we have $c(v_{i})\leq c(u)$. Let $M$ be the set of those vertices $v_{i}$. The set of vertices of $G$ which have a neighbor from $M$ is denoted by $N$. It is obvious that $|M\cup N|\leq(L-1)+(L-1)^{2}<L^{2}$.

Let us show that if $I$ is a “yes” instance and $S$ is a feasible set in $I$, then we can construct a feasible set that intersects $M\cup N$. Indeed, assume that $S$ does not contain any vertex from $M\cup N$. Then we can replace any vertex $v\in S$ by the vertex $v_{p(v)}$. Since there is no vertex in $S$ which is a neighbor of $v_{p(v)}$, it follows that the total vertex cost has not increased, and, as $p(v)=p(v_{p(v)})$, $S^{\prime}$ is also feasible.

Now we complete the proof by recursively guessing on $M\cup N$. We now show that the depth of the recursion is less than $2L$. For the sake of contradiction, assume that during the recursive guessing, the algorithm has considered the vertices $z_{1},\ldots,z_{2L}$. Let $Z=\{z_{1},\ldots,z_{2L}\}$. Since the algorithm has considered these vertices, we have that $c(Z)\leq R$. Then for the profit of edges covered by $Z$, we will have the following bound:

[TABLE]

Thus, $Z$ is a feasible set, hence $I$ is a “yes” instance. This means that there is no need to consider $2L$ or more vertices during the recursive guessing. Hence the depth of the recursion is less than $2L$. Since $|M\cup N|\leq L^{2}$, we have that the running time of our algorithm is bounded by $(L^{2})^{2L}\cdot size^{O(1)}=L^{4L}\cdot size^{O(1)}$. ∎

5 The Matching problem

We now consider a variant of the PVCB problem. In this variant, we are given a bipartite graph $G$ and three integers $k_{1}$, $k_{2}$ and $k_{3}$, and the goal is to check whether there is a subset of $k_{1}$ vertices, that covers at least $k_{2}$ edges, such that the covered edges contain a matching of size at least $k_{3}$. This variant is called PVCBM.

Clearly, this problem is NP-hard, as when $k_{3}=0$ it results into the PVCB. Since PVCB is FPT with respect to $k_{1}$, it would be interesting to parameterize this new version of the problem with respect to $k_{1}$. Observe that we can assume that $k_{3}\leq k_{1}$ otherwise the problem is a trivial NO-instance.

Theorem 6**.**

PVCBM is fixed-parameter tractable with respect to the parameters $k_{1}$, $k_{2}$ and $k_{3}$.

Proof.

Let $PVCB(A,B)$ be the algorithm that in $FPT(A)$-time checks whether there is a subset of $A$ vertices that covers at least $B$ edges of the input bipartite graph. Now, assume that the graph $G$ and the parameters $k_{1}$, $k_{2}$ and $k_{3}$ are given in the matching problem. First, we run $PVCB(k_{1},k_{2})$. If there is no such subgraph, then the answer to the matching problem is also negative. So we can assume that $PVCB(k_{1},k_{2})$ returns such a subgraph. Next, by trying $R=0,1,...,k_{1}$ we can find the smallest $R$ for which $PVCB(R,k_{2})$ is a yes-instance.

Let $H$ be the edge-induced subgraph induced on these $\geq k_{2}$ edges. As usual, let $\nu(G)$ be the size of the largest matching in $G$, and let $\tau(G)$ be the size of the smallest vertex cover in $G$. By the classical König theorem we have $\nu(G)=\tau(G)$ for any bipartite graph $G$.

Observe that we can assume that $R<k_{3}\leq k_{1}$. To see this, observe that $R$ represents the number of vertices required to cover all the edges in $H$. In other words, it is a vertex cover of $H$. Thus, $R=\nu(H)$ and since $H$ is a bipartite subgraph of $G$, $R$ is also the size of a maximum matching in $H$. Thus, if $R\geq k_{3}$, then the edges in $H$, which number at least $k_{2}$ can be covered by $R\leq k_{1}$ vertices and a matching of size $R\geq k_{3}$ is contained in $H$.

Also, observe that if $\tau(G)<k_{3}$, then we have trivial NO-instance, as $G$ contains no matching of size $k_{3}$. Thus, we can assume that $\nu(G)=\tau(G)\geq k_{3}$.

Since $\tau(H)=R<k_{3}\leq\tau(G)$, we have that $E(H)\neq E(G)$. Thus, there is as an edge $e$ lying outside $H$. Add $e$ to $H$. If $\tau(H)$ has increased by adding $e$, define $R:=R+1$, otherwise let $R$ be the same. Repeat this process of adding edges outside $H$. Since $\tau(H)=R<k_{3}\leq\tau(G)$, at some point we will arrive into $H$ such that $R=\tau(H)=k_{3}\leq k_{1}$. Observe that $H$ can be covered with at most $k_{1}$ vertices, it has at least $k_{2}$ edges and it contains a matching of size $k_{3}$. Thus, the problem is a YES-instance.

Finally, let us observe that the running-time of this algorithm is FPT in $k_{1}$. We need at most $k_{1}$ calls of $PVCB(k_{1},k_{2})$. Since the latter is FPT with respect to $k_{1}$, we have ther result. ∎

6 Conclusion

In this paper, we studied the partial vertex cover problem from the perspectives of parameterized tractability and W[1]-hardness. Although our primary focus was on bipartite graphs, we obtained new results for the general case as well. Our main contributions include showing that a restricted version of the WPVCB problem is fixed-parameter tractable and that this problem is W[1]-hard, with respect to the edge-weight parameter. We also showed that the WPVC problem is fixed-parameter tractable in bounded graphs. Finally, we introduced a new variant of the partial vertex cover problem called PVCBM and showed that it is fixed-parameter tractable.

Acknowledgements

We are grateful to Magnus Wahlström for his proof of Theorem $3$.