数学代写|理论计算机代写theoretical computer science代考|On the Complexity of the Upper $r$-Tolerant Edge Cover Problem

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数学代写|理论计算机代写theoretical computer science代考|On the Complexity of the Upper $r$-Tolerant Edge Cover Problem

In this paper we define and study tolerant edge cover problems. An edge cover of a graph $G=(V, E)$ without isolated vertices is a subset of edges $S \subseteq E$ which covers all vertices of $G$, that is, each vertex of $G$ is an endpoint of at least one edge in $S$. The edge cover number of a graph $G=(V, E)$, denoted by ec $(G)$, is the minimum size of an edge cover of $G$ and it can be computed in polynomial time (see Chapter 19 in [29]). An edge cover $S \subseteq E$ is called minimal (with respect to inclusion) if no proper subset of $S$ is an edge cover. Minimal edge cover is also known in the literature as an enclaveless set [30] or as a nonblocker set [14]. While a minimum edge cover can be computed efficiently, finding the largest minimal edge cover is NP-hard [27], where it is shown that the problem is equivalent to finding a dominating set of minimum size. The associated optimization problem is called upper edge cover (and denoted UPPER EC) [1] and the corresponding optimal value will be denoted uec $(G)$ in this paper for the graph $G=(V, E)$.
Here, we are interested in minimal edge cover solutions tolerant to the failures of at most $r-1$ edges. Formally, given an integer $r \geq 1$, an edge subset $S \subseteq E$ of $G=(V, E)$ is a tight $r$-tolenant edge-cover ( $r$-tec for short) if the deletion of

any set of at most $r-1$ edges from $S$ maintains an edge cover ${ }^{1}$ and the deletion of any edge from $S$ yields a set which is not a (tight) $r$-tolerant edge cover. Equivalently, we seek an edge subset $S$ of $G$ such that the subgraph $(V, S)$ has minimum degree $r$ and it is minimal with this property. For the sake of brevity we will omit the word ‘tight’ in the rest of the paper. Note that the case $r=1$ corresponds to the standard notion of minimal edge cover.

As an illustrating example consider the situation in which the mayor of a big city seeks to hire a number of guards, from a security company, who will be constantly patrolling streets between important buildings. An $r$-tolerant edge cover reflects the desire of the mayor to guarantee that the security is not compromised even if $r-1$ guards are attacked. Providing a maximum cover would be the goal of a selfish security company, who would like to propose a patrolling schedule with as many guards as possible, but in which all the proposed guards are necessary in the sense that removing any of them would leave some building not $r$-covered.

Related Work. UPPER EC has been investigated intensively during recent years, mainly using the terminologies of spanning star forests and dominating sets. A dominating set in a graph is a subset $S$ of vertices such that any vertex not in $S$ has at least one neighbor in $S$. The minimum dominating set problem (denoted MinDS) seeks the smallest dominating set of $G$ of value $\gamma(G)$. We have the equality $\operatorname{uec}(G)=n-\gamma(G)[27]$.

Thus, using the complexity results known for MinDS, we deduce that UPPER EDGE COVER is NP-hard in planar graphs of maximum degree 3 [20], chordal graphs [6] (even in undirected path graphs, the class of vertex intersection graphs of a collection of paths in a tree), bipartite graphs, split graphs [5] and $k$-trees with arbitrary $k[12]$, and it is polynomial in $k$-trees with fixed $k$, convex bipartite graphs [13], strongly chordal graphs [16]. Concerning the approximability, an APX-hardness proof with explicit inapproximability bound and a combinatorial 0.6-approximation algorithm is proposed in [28]. Better algorithms with approximation ratio $0.71$ and $0.803$ are given respectively in [9] and [2]. For any $\varepsilon>0$, UPPER EDGE COVER is hard to approximate within a factor of $\frac{259}{260}+\varepsilon$ unless $\mathbf{P}=\mathbf{N P}$ [28]. The weighted version of the problem, denoted as UPPER WEIGHTED EDGE CoVER, have been recently studied in [24], in which it is proved that the problem is not $O\left(\frac{1}{n^{1 / 2-\varepsilon}}\right)$ approximable nor $O\left(\frac{1}{\Delta^{1-\varepsilon}}\right)$ in edge weighted graphs of order $n$ and maximum degree $\Delta$.

数学代写|理论计算机代写theoretical computer science代考|Definitions

Graph Notation and Terminology. Let $G=(V, E)$ be a graph and $S \subseteq V$; $N_{G}(S)={v \in V: \exists u \in S, v u \in E}$ denotes the neighborhood of $S$ in $G$ and $N_{G}[S]=S \cup N_{G}(S)$ denotes the closed neighborhood of $S$. For singleton sets $S={s}$, we simply write $N_{G}(s)$ or $N_{G}[s]$, even omitting $G$ if clear from the context. The maximum degree and minimum degree of a graph are denoted $\Delta(G)$ and $\delta(G)$ respectively. For a subset of edges $S, V(S)$ denotes the vertices that are incident to edges in $S$. A vertex set $U \subseteq V$ induces the graph $G[U]$ with vertex set $U$ and $e \in E$ being an edge in $G[U]$ iff both endpoints of $e$ are in $U$. If $S \subseteq E$ is an edge set, then $\vec{S}=E \backslash S$, edge set $S$ induces the graph $G[V(S)]$, while $G_{S}=(V, S)$ denotes the partial graph induced by $S$. In particular, $G_{\bar{S}}=(V, E \backslash S)$. Let also $\alpha(G)$ and $\gamma(G)$ denote the size of the largest independent and smallest dominating set of $G$, respectively.

An edge set $S$ is called edge cover if the partial graph $G_{S}$ is spanning and it is a matching if $S$ is a set of pairwise non adjacent edges. An edge set $S$ is

minimal (resp., maximal) with respect to a graph property if $S$ satisfies the graph property and any proper subset $S^{\prime} \subset S$ of $S$ (resp., any proper superset $S^{\prime} \supset S$ of $S$ ) does not satisfy the graph property. For instance, an edge set $S \subseteq E$ is a maximal matching (resp., minimal edge cover) if $S$ is a matching and $S+e$ is not a matching for some $e \in \bar{S}$ (resp., $S$ is an edge cover and $S-e$ is not an edge cover for some $e \in \bar{S}$ ).

Problem Definitions. Let $G=(V, E)$ be a graph where the minimum degree is at least $r \geq 1$, i.e., $\delta(G) \geq r$. We assume $r$ is a constant fixed greater than one (but all results given here hold even if $r$ depends on the graph). A $r$-DEGREE EDGE-COVER $^{3}$ is defined as a subset of edges $G^{\prime}=G_{S}=(V, S)$, such that each vertex of $G$ is incident to at least $r \geq 1$ distinct edges $e \in S$. As $r$-tolerant edge-cover (or simply $r$-tec) we will call an edge set $S \subseteq E$ if it is a minimal $r$-degree edge-cover i.e. if for every $e \in S, G^{\prime}-e=(V, S \backslash{e})$ is not an $r$-degree edge-cover. Alternatively, $\delta\left(G^{\prime}\right)=r$, and $\delta\left(G^{\prime}-e\right)=r-1$. If you seek the minimization version, all the problems are polynomial-time solvable. Actually, the case of $r=1$ corresponds to the edge cover in graphs. The optimization version of a generalization of $r$-EC known as the MIN LOWER-UPPER-COVER PROBLEM (MIN LUCP), consists of, given a graph $G$ where $G=(V, E)$ and two non-negative functions $a, b$ from $V$ such that $\forall v \in V, 0 \leq a(v) \leq b(v) \leq d_{G}(v)$, of finding a subset $M \subseteq E$ such that the partial graph $G_{M}=(V, M)$ induced by $M$ satisfies $a(v) \leq d_{G_{M}}(v) \leq b(v)$ (such a solution will be called a lowerupper-cover) and minimizing its total size $|M|$ among all such solutions (if any). Hence, an $r$-EC solution corresponds to a lower-upper-cover with $a(v)=r$ and $b(v)=d_{G}(v)$ for every $v \in V$. Min LUCP is known to be solvable in polynomial time even for edge-weighted graphs (Theorem $35.2$ in Chap. 35 of Volume A in [29]). We are considering two associated problems, formally described as follows.

数学代写|理论计算机代写theoretical computer science代考|Basic Properties of r-Tolerant Solutions

The next property presents a simple characterization of feasible $r$-tec solution generalizing the well known result given for minimal edge covers, i.e., 1-tec, affirming that $S$ is a 1-tec solution of $G$ if and only if $S$ is spanning and the subgraph $(V, S)$ induced by $S$ is $\left(K_{3}, P_{4}\right)$-free.

Property 1. Let $r \geq 1$ and let $G=(V, E)$ be a graph with minimum degree $\delta \geq r$. $S$ is an $r$-tec solution of $G$ if and only if the following conditions meet on $G_{S}=(V, S)$ :
(1) $V=V_{1}(S) \cup V_{2}(S)$ where $V_{1}(S)=\left{v \in V: d_{G_{S}}(v)=r\right}$ and $V_{2}(S)=\left{v \in V: d_{G_{S}}(v)>r\right}$.
(2) $V_{2}(S)$ is an independent set of $G_{S}$.

Proof. Let $r \geq 1$ be a fixed integer and let $G=(V, E)$ be a graph instance of UPPER $r$-EC, i.e., a graph of minimum degree at least $r$. Let us prove the necessary conditions: if $S \subseteq E$ is an $r$-tec solution, then by construction, $V=$ $V_{1}(S) \cup V_{2}(S)$ is a partition of vertices with minimum degree $r$ in $S$. Now, if $u v \in S$ with $u, v \in V_{2}(S)$, then $S-u v$ is also $r$-tec which is a contradiction of minimality.

Now, let us prove the other direction. Consider a subgraph $G^{\prime}=(V, S)$ induced by edge set $S$ satisfying $(1)$ and $(2)$. By (1) it is clear $G S$ has minimum degree at least $r$. If $u v \in S$, then by (2) one vertex, say $u \in V_{1}(S)$ because $V_{2}(S)$ is an independent set. Hence, the deletion of $u v$ leaves $u$ of degree $r-1$ in the subgraph induced by $G_{S \backslash{u v}}$ and then $S$ is an $r$-tec solution.

Property 2. Let $r \geq 1$, for all graphs $G=(V, E)$ of minimum degree at least $r$, the following inequality holds:
$$2 \mathrm{ec}{T}(G) \geq \operatorname{lec}{r}(G)$$
Proof. For a given graph $G=(V, E)$ with $n$ vertices, let $S^{}$ be an optimal solution of UPPER $r$-EC, that is $\left|S^{}\right|=\operatorname{uec}{r}(G)$. Let $\left(V{1}^{}, V_{2}^{}\right)$ be the associated partition related to solution $S^{}$ as indicated in Property 1. Using this characterization, we deduce $\operatorname{uec}{r}(G) \leq r\left|V{1}^{}\right| \leq r n$. On the other side, if $G^{\prime}$ denotes the subgraph induced by a minimum $r$-tec solution of value ec $(G)$, we get $2 \operatorname{ec}{r}(G)=\sum{v \in V} d_{G^{\prime}}(v) \geq r n$. Combining these two inequalities, the results follows.

In particular, inequality (1) of Property 2 shows that any $r$-tec solution is a $\frac{1}{2}$-approximation of UPPER $r$-EC.

The next property is quite natural for induced subgraphs and indicates that the size of an optimal solution of a maximization problem does not decrease with the size of the graph. Nevertheless, this property is false in general when we deal with partial subgraphs; for instance, for the upper domination number, we get $\operatorname{uds}\left(K_{3}\right)=1<2=\operatorname{uds}\left(P_{3}\right)$. It turns out that this inequality is valid for the upper edge cover number.

Property 3. Let $G=(V, E)$ be a graph such that $0<r \leq \delta(G)$. For every partial subgraph $G^{\prime} \subseteq G$ with $\delta\left(G^{\prime}\right) \geq r$, the following inequality holds:
$$\operatorname{uec}{r}(G) \geq \operatorname{uec}{\mathrm{r}}\left(G^{\prime}\right)$$
Proof. Fix an integer $r \geq 1$ and a graph $G=(V, E)$ with $\delta(G) \geq r$. Let $G^{\prime}=$ ( $V^{\prime}, E^{\prime}$ ) with $\delta\left(G^{\prime}\right) \geq r$ be a partial subgraph of $G$, i.e., $V^{\prime} \subseteq V$ and $E^{\prime} \subseteq E$. Consider an upper $r$-tec solution $S^{\prime}$ of $G^{\prime}$ with size $\left|S^{\prime}\right|=$ uec $_{r}\left(G^{\prime}\right)$. We prove inequality (2) by starting from $S=S^{\prime}$ and by iteratively repeating the following procedure:

1. Select a vertex $v \in V$ with $d_{G_{S}}(v)<r$ and $e=u v \in E \backslash S$.
2. If $u$ is covered less or more than $r$ times by $S$, then $S:=S+e$.
3. If vertex $u$ is covered exactly $r$ times by $S$, consider two cases.

数学代写|理论计算机代写theoretical computer science代考|Basic Properties of r-Tolerant Solutions

(1) 在=在1(小号)∪在2(小号)在哪里V_{1}(S)=\left{v \in V: d_{G_{S}}(v)=r\right}V_{1}(S)=\left{v \in V: d_{G_{S}}(v)=r\right}和V_{2}(S)=\left{v \in V: d_{G_{S}}(v)>r\right}V_{2}(S)=\left{v \in V: d_{G_{S}}(v)>r\right}.
(2) 在2(小号)是一个独立的集合G小号.

2和C吨(G)≥莱克⁡r(G)

1. 选择一个顶点在∈在和dG小号(在)<r和和=在在∈和∖小号.
2. 如果在被覆盖少于或多于r次由小号， 然后小号:=小号+和.
3. 如果顶点在被准确覆盖r次由小号，考虑两种情况。

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