1 Introduction

In the area of robust combinatorial optimization, the central goal is to determine a feasible solution that performs reasonably well under a variety of cost scenarios. We refer the reader to the book by Kouvelis and Yu (1997) for comprehensive information on this area, and to the survey article by Aissi et al. (2009) for a survey centered around complexity and approximation.

In a recent paper Dolgui and Kovalev (2012) investigate the computational complexity of the following robust optimization problem that is motivated by the design process of a production line and that is called the representatives selection problem: The input consists of a finite set \(T\) of tools that is partitioned into \(n\) disjoint sets \(T_1,\ldots ,T_n\), together with \(s\) cost scenarios \(c_1,\ldots ,c_s\). Every such cost scenario \(c_k\) (\(k=1,\ldots ,s\)) specifies non-negative integer costs for the tools in \(T\); hence \(c_k:T\rightarrow \mathbb{N }_0\). A feasible solution \(F\subseteq T\) contains exactly one tool from every set \(T_i\) with \(1\le i\le n\); hence \(|F\cap T_i|=1\) for all \(i\). The cost of a feasible solution \(F\) under the \(k\)th cost scenario is given by \(c_k(F)=\sum _{t\in F}c_k(t)\). The goal in the problem variant DSC (where DSC stands short for Min-Max Discrete Scenario) is to find a feasible solution \(F\) that minimizes the cost \(\max _{1\le k\le s} c_k(F)\) of the worst scenario.

Dolgui and Kovalev (2012) prove that for \(s=2\) cost scenarios, problem DSC is NP-hard in the ordinary sense and solvable in pseudo-polynomial time. Furthermore they show that if the number \(s\) of cost scenarios is part of the input then DSC is NP-hard in the strong sense; the strong NP-hardness proof is built around tool sets of large cardinalities. In their conclusion section the authors of Dolgui and Kovalev (2012) ask about the exact computational complexity of the special case where \(s\) is part of the input and where the cardinality of each tool set \(T_i\) is bounded by a fixed constant.

This technical note settles the question of Dolgui and Kovalev (2012) and establishes strong NP-hardness of the open special case (Sect. 2). Furthermore we derive a \(2-\varepsilon \) in-approximability result for the representatives selection problem, and we also deduce strong NP-hardness of two closely related problem variants (Sect. 3).

2 The NP-hardness proof

The strong NP-hardness proof is done by means of a reduction from the VERTEX COVER problem; see Garey and Johnson (1979). In the decision version of VERTEX COVER, an instance consists of an undirected graph \(G=(V,E)\) together with an integer bound \(k\) with \(k\le |V|\). The question is to decide whether graph \(G\) possesses a vertex cover \(U\) of cardinality at most \(k\), that is, a subset \(U\subseteq V\) of the vertices such that every edge in \(E\) has at least one of its endpoints in \(U\).

We consider an instance of VERTEX COVER and we construct in polynomial time the following instance of DSC from it.

  • For every vertex \(v\in V\), we create two corresponding tools \(t^+(v)\) and \(t^-(v)\) that together form the tool set \(T_v=\{t^+(v),t^-(v)\}\); note that \(T\) is the union of all sets \(T_v\) with \(v\in V\).

  • We create a cost scenario \(c^*\) that assigns cost \(1\) to every tool \(t^+(v)\) and cost \(0\) to every tool \(t^-(v)\) with \(v\in V\).

  • Furthermore for every edge \(e=[u,v]\) in \(E\), we create a corresponding cost scenario \(c_e\). This cost scenario assigns cost \(k\) to the two tools \(t^-(u)\) and \(t^-(v)\), and cost \(0\) to all remaing \(2|V|-2\) tools.

Lemma 1

The constructed instance of DSC possesses a feasible solution \(F\) with worst scenario cost at most \(k\), if and only if the graph \(G\) in the VERTEX COVER instance has a vertex cover of cardinality at most \(k\).

Proof

First assume that \(G\) has a vertex cover \(U\) with \(|U|\le k\). Define a feasible set \(F\) that for \(v\in U\) contains tool \(t^+(v)\) and for \(v\notin U\) contains tool \(t^-(v)\). Note that under cost scenario \(c^*\) the cost of \(F\) equals \(c^*(F)=|U|\le k\). Next consider the cost scenario \(c_e\) for some edge \(e=[u,v]\) in \(E\). Since \(U\) contains at least one of the vertices \(u\) and \(v\), the feasible set \(F\) contains at least one of the tools \(t^+(u)\) and \(t^+(v)\) and hence at most one of the tools \(t^-(u)\) and \(t^-(v)\). Therefore the cost of \(F\) under cost scenario \(c_e\) is also at most \(k\).

Next assume that the DSC instance has a feasible solution \(F\) with worst scenario cost at most \(k\). Define a set \(U\) that contains all vertices \(v\in V\) for which \(t^+(v)\in F\). Then cost scenario \(c^*\) implies \(|U|=c^*(F)\le k\), and cost scenario \(c_e\) implies that edge \(e\) is covered by \(U\). \(\square \)

As every tool set \(T_e\) in the constructed instance has cardinality 2 and as the values of all involved numbers are polynomially bounded in the size of the graph \(G\), we arrive at the following theorem.

Theorem 2

The representatives selection problem DSC is NP-complete in the strong sense, even if the cardinality of each tool set is bounded by 2. \(\square \)

The statement in Theorem 2 is best possible with respect to the size of the tool sets as tool sets of cardinality 1 are trivial to handle.

3 Further consequences of our result

We conclude the note by discussing two consequences of the NP-hardness proof in Sect. 2. The first consequence is the following in-approximability result, which follows by turning the above polynomial time reduction into an approximation preserving reduction. Throughout this section we consider the optimization version of the VERTEX COVER problem, where the goal is to find a vertex cover \(U\) that minimizes the cardinality \(|U|\).

Theorem 3

If the representatives selection problem DSC allows a polynomial time approximation algorithm with worst case guarantee \(2-\varepsilon \), then also the VERTEX COVER problem has a polynomial time approximation algorithm with worst case guarantee \(2-\varepsilon \).

Proof

Assume that DSC has a polynomial time approximation algorithm with worst case guarantee \(2-\varepsilon \). Recall that the optimization version of VERTEX COVER takes an undirected graph \(G=(V,E)\) and asks for a vertex cover of minimum cardinality. For every \(k=1,\ldots ,|V|\) we construct an instance \(I_k\) of DSC as in Sect. 2, and then apply the approximation algorithm to each of these instances to get a feasible solution \(F_k\). Define a vertex set \(U_k\) that contains all vertices \(v\in V\) for which \(t^+(v)\in F_k\).

If graph \(G\) has a vertex cover of cardinality \(k\), then by Lemma 1 the cost of the optimal solution in instance \(I_k\) is at most \(k\) and hence the cost of the approximate solution \(F_k\) is at most \((2-\varepsilon )k\). The corresponding vertex set \(U_k\) has cardinality \(|U_k|=|F_k|\le (2-\varepsilon )k\) (by the corresponding cost scenario \(c^*\)) and covers every edge \(e\) (by the corresponding cost scenario cost scenario \(c_e\)). This suggests the following approximation algorithm with worst case guarantee \(2-\varepsilon \) for VERTEX COVER: we check for every \(k\) whether the vertex set \(U_k\) covers all the edges in \(G\), and we then output the detected vertex cover \(U_k\) with the smallest index \(k\). \(\square \)

The approximation literature contains strong evidence that VERTEX COVER does not allow a polynomial time approximation algorithm with worst case guarantee strictly below \(2\); see for instance Khot and Regev (2008) and the references listed therein. Hence Theorem 3 makes it very unlikely that problem DSC has a polynomial time approximation algorithm with worst case guarantee strictly below \(2\).

The second consequence of our NP-hardness proof concerns two other robust variants of the representatives selection problem, called DSC-R and DSC-RR in Dolgui and Kovalev (2012). Let \(\text{ OPT}(c_k)\) denote the optimal objective value under the cost scenario \(c_k\) (\(k=1,\ldots ,s\)). In problem variant DSC-R the goal is to find a feasible solution \(F\) that minimizes the maximum regret \(\max _{1\le k\le s} c_k(F)-\text{ OPT}(c_k)\). In our reduction in Sect. 2 all cost scenarios \(c_k\) satisfy \(\text{ OPT}(c_k)=0\), so that the maximum regret of \(F\) actually coincides with the worst scenario cost \(\max _{1\le k\le s} c_k(F)\). Hence Theorem 2 also implies strong NP-hardness of DSC-R even if the cardinality of each tool set is bounded by 2, and Theorem 3 makes it unlikely that DSC-R has a polynomial time approximation algorithm with worst case guarantee strictly below \(2\).

In the problem variant DSC-RR the goal is to find a feasible solution \(F\) that minimizes the maximum relative regret \(\max _{1\le k\le s} (c_k(F)-\text{ OPT}(c_k))/\text{ OPT}(c_k)\); this problem variant assumes \(\text{ OPT}(c_k)>0\) for \(k=1,\ldots ,s\). We slighly modify the reduction in Sect. 2 by creating a new dummy tool set containing a single dummy tool that has cost 1 under all considered cost scenarios. Since this dummy tool must be part of every new feasibe solution, this simply boils down to adding 1 to the objective value of every feasible solution. With this, Theorem 2 implies strong NP-hardness of DSC-RR even if the cardinality of each tool set is bounded by 2. Furthermore Theorem 3 implies the in-approximability of DSC-RR. (This can be seen as follows. Consider a MODIFIED VERTEX COVER minimization problem on graphs whose objective value equals the objective value of the standard VERTEX COVER minimization problem plus \(1\). As VERTEX COVER is hard to approximate for large objective values, also MODIFIED VERTEX COVER is hard to approximate for large objective values, and this makes the additive \(1\) negligible.)