On robust clusters of minimum cardinality in networks

Abstract

This paper studies two clique relaxation models, k-blocks and k-robust 2-clubs, used to describe structurally cohesive clusters with good robustness and reachability properties. The minimization version of the two problems are shown to be hard to approximate for $k \ge 3$ and $k \ge 4$, respectively. Integer programming formulations are proposed and a polyhedral study is presented. The results of sample numerical experiments on several graph instances are also reported.

Appendix: Proof of Proposition 1

We use the notion of L-reduction (Papadimitriou and Yannakakis 1991) defined next, that is widely used for establishing APX-hardness results. Given two optimization problems F and G, we say that F L-reduces to G if there are two polynomial-time algorithms f, g and constants $\alpha , \beta >0$ such that for each instance x of F:

• 1. f produces an instance f(x) of G, such that $OPT_G(f(x)) \le \alpha OPT_F(x)$.

• 2. Given any solution of f(x) with cost $c'$, g produces a solution of x with cost c such that $|c-OPT_F(x)| \le \beta |c' - OPT_G(f(x))|$.

If F L-reduces to G and there is a polynomial-time approximation algorithm for G with worst-case error $\epsilon$, then there is a polynomial-time approximation algorithm for F with worst-case error $\alpha \beta \epsilon$. We now give the following L-reduction f from Min Vertex Cover on 3-regular graphs to Min Vertex Cover on k-regular graphs, $k \ge 4$. Given a 3-regular graph $G=(V,E)$ construct a k-regular graph $G'=(V',E')$ as follows. Let $|V(G)|=n$. Consider k identical copies $G_1,G_2,\ldots ,G_k$ of G. Denote the vertex set and edge set of the $r^{th}$ such copy respectively by $V_r$ and $E_r, r = 1,\ldots ,k$, where $V_r = \{1_r,\ldots ,n_r\}$ and $E_r = \{(i_r,j_r): (i,j) \in E(G)\}$. Let $R = \cup _{r=1}^{k}V_r$ and $E_R = \cup _{r=1}^{k}E_r$. For each $v \in V(G)$ consider a set of $(k-3)$ independent vertices $P_v =\{u^v_1,\ldots ,u^v_{k-3}\}$, and let $P = \cup _{v \in V}P_v$. Put $V' = R \cup P$ and $E' = E_R \cup E_P$, where

$$\begin{aligned} E_P=\{(v_r,u^v_j): v \in V, r = 1,\dots ,k, j=1,\ldots ,k-3\} \end{aligned}$$

That is for each $v \in V$, there is an edge between $v_r$ and $u^v_j$ for $j=1,\ldots ,k-3$ and $r=1,\ldots ,k$. This completes the construction of $G'=(V',E')$.

It is easy to see that from every vertex cover $C \subseteq V$ of G we can construct a vertex cover $C' \subseteq V'$ of $G'$ of size exactly $n(k-3)+3|C|$, by choosing $v_r \in V_r, r = 1,\ldots ,k$ if $v \in C$ and by choosing the set $P_v$ if $v \notin C$. Since G is 3-regular we have $n = |V| \le |E| \le \sum _{v \in C}deg(v) = 3|C|$. Then, $|C'| = n(k-3) + 3|C| \le 3(k-3)|C| + 3|C| = 3(k-2)|C|$ and, this satisfies the first property of L-reduction with $\alpha =3(k-2)$.

Conversely, given any vertex cover $C' \subseteq V'$ of $G'$, we can transform it back to a vertex cover $C \subseteq V$ of G as follows. First note that if $v_r \notin C'$ for some $r \in \{1,\ldots ,k\}$, then $P_v\subseteq C'$. Hence given any vertex cover $C'$, if $C_1,C_2,\ldots ,C_k \subseteq C'$ are the subsets of vertices selected respectively from $V_1,\ldots ,V_k$, then $C_i$ must be a vertex cover of $G_i$ for each $i \in \{1,\ldots ,k\}$ and hence, the corresponding vertices in V must be a vertex cover of G. Let h be such that $|C_h|= \min \{|C_1|,\ldots ,|C_k|\}$. Then $C=\{v \in V: v_h \in C_h\}$ is a vertex cover of G and $|C| \le \frac{1}{3}(|C'| -n(k-3))$. Together with the observation $|OPT_{VC}(G')| \le 3|OPT_{VC}(G)| + n(k-3)$ from the previous paragraph, it is easy to see that f is a L-reduction with $\beta = 1$. $\square$