A set of vertices of a graph is a resolving set for if for any two distinct , there is such that , where denotes the length of a shortest path between and in the graph . The metric dimension of is the minimum cardinality of a resolving set for . The metric dimension of graphs was introduced independently by Slater  and Harary and Melter . Metric Dimension as a computational problem was first mentioned in the literature by Garey and Johnson  and its decision version is defined as follows.
Garey and Johnson  proved this problem to be NP-complete in general. Their proof was never published, a reduction from 3SAT was provided by Khuller . Diaz  showed that the problem is NP-complete even when restricted to planar graphs of bounded degree but that it is solvable in polynomial time on the class of outer-planar graphs.
Prior to this, not much was known about the computational complexity of this problem except that it is polynomial-time solvable on trees (see [25, 21]), although there are several results proving combinatorial bounds on the metric dimension of various graph classes . Subsequently, Epstein et al.  showed that this problem is NP-complete on split graphs, bipartite and co-bipartite graphs. They also showed that the weighted version of Metric Dimension can be solved in polynomial time on paths, trees, cycles, co-graphs and trees augmented with edges for a fixed . Hoffmann and Wanke  extended the tractability results to a subclass of unit disk graphs, while Foucaud  showed that this problem is NP-complete on interval graphs.
The parameterized complexity of Metric Dimension under the standard parameterization—the metric dimension of the input graph—was open until 2012, when Hartung and Nichterlein  proved that it is -hard. Foucaud  showed the problem becomes fixed-parameter tractable when restricted to interval graphs. The parameterized complexity of Metric Dimension on graphs of bounded treewidth is currently unresolved (the question of whether it is polynomial-time solvable on graphs of treewidth 2 is still open), however, Belmonte  proved that it is when parameterized by the treelength111The length of a tree decomposition is the maximum diameter of the bags in this tree-decomposition and the treelength of a graph is the minimum length over all tree decompositions. Note that this parameter is upper-bounded by treewidth. plus the solution size. In a different line of work, Eppstein  showed that Metric Dimension is when parameterized by the max-leaf number of the input graph alone.
In this paper we initiate the study of the parametric dual of Metric Dimension. To avoid confusion, we will use to denote the (standard) parameter and phrase the parameterized dual as follows:
We call a set of vertices of a co-resolving set if is a resolving set of . Clearly, an instance of Saving Landmarks is positive if and only if there is a co-resolving set of size at least .
This choice of parameterization is informed by previous studies of the parametric dual (see e.g. [1, 4, 15, 24]): problems that are hard with respect to the standard parameter often admit an -algorithms or even polynomial kernels under the dual parameter. A classic example is the Independent Set problem which is -hard while its dual, the Vertex Cover problem is among the earliest problems shown to be in and even admits a linear vertex kernel.
We add yet another entry to the list of hard problems with tractable duals by showing that Saving Landmarks admits a polynomial kernel and a single-exponential algorithm. Concretely, we prove the following two results.
Saving Landmarks admits a kernel with at most vertices.
Saving Landmarks can be solved in time .
We also study the Metric Dimension problem from the kernelization perspective when parameterized by the vertex cover number of the input graph. As Hartung and Nichterlein observed , parameterization of Metric Dimension by the vertex cover number of the input graph (denoted Metric Dimension[VC]) can be easily seen to be in . It is therefore natural to ask whether this structural parameterization allows a polynomial kernel in general graphs, a question we answer in the negative. In fact, we show that not only does the problem not admit a polynomial kernel with the vertex cover as the parameter, even adding the size of the solution (the metric dimension of the graph) to the parameter is unlikely to be helpful in this regard. Specifically, we prove the following result.
Metric Dimension does not admit a polynomial kernel unless the polynomial hierarchy collapses to its third level.
For a graph we denote by the standard distance-metric where is the length of a shortest path between vertices . We denote by and the open and closed neighbourhood of a vertex. We omit the subscript if clear from the context in all these notations. As customary, the number of vertices of a graph under consideration will be denoted by
Two vertices are true twins if (implying that ) and they are false twins if . A twin class is a maximal vertex set in in which all vertices are pairwise true twins or in which all vertices are pairwise false twins.
A vertex set resolves a set if for every pair of distinct vertices there exists at least one vertex such that . We will also say that a pair is resolved by if the above holds and further that sets are distinguished by if every pair , is resolved by . A vertex subset is a resolving set of if resolves . We call the members of such a set landmarks.
Parameterized complexity is a two dimensional framework for studying the computational complexity of a problem. One dimension is the input size and the other is a parameter . A problem is said to be fixed parameter tractable () or in the class , if it can be solved in time for some computable function . We refer to the books of Cygan  and Downey and Fellows  for detailed introductions to parameterized complexity.
Kernelization offers a mathematically rigorous way of analysing and comparing preprocessing algorithms for -hard problems in general and for parameterized problems in particular. A kernel of size for a parameterized problem is a polynomial time algorithm that takes as input an instance of the problem (where is the parameter) and outputs another instance of the same problem such that is a yes- instance of the problem if and only if is a yes-instance of the problem and . The notion of “effective” preprocessing is captured by requiring the function to be polynomially bounded, in which case the kernel is called a polynomial kernel. The reader is referred to Cygan , Downey and Fellows , Fomin  and the surveys [22, 23] for a comprehensive introduction to the topic of kernelization.
Definition (Pruned graph).
For a graph we define the pruned graph as the graph obtained (up to isomorphism) from by iteratively removing vertices from twin-classes of size three or larger. We say that a graph is pruned if .
The following observation simply follows from the fact that among a twin class in , all but one vertex of must be contained in any resolving set.
A graph has a resolving set of size if and only if the pruned graph has a resolving set of size .
Consequently, we call an instance of Metric Dimension or Saving Landmarks reduced if is pruned.
3 Standard parameterization for Saving Landmarks
We present two positive results in this section, namely, that Saving Landmarks admits a polynomial kernel and a single-exponential algorithm.
We begin by describing the kernel. Assume in the following that the input instance is pruned as per Observation 2. This will be the only reduction rule. In the following we will prove that the size of the instance is either bounded polynomially in or it will be a trivial yes-instance. Let us collect some basic observations first.
If contains either an independent set or a clique with vertices, then is a yes-instance.
Let be a set of size such that is either a clique or an independent set. Since is pruned there are at least distinct twin-classes in , which must be distinguished by their neighbourhoods outside of . Hence selecting one vertex from each twin-class in gives a co-resolving set of size at least , and is a yes-instance.
Let us define the function . Note that if , then and are distinguished from each other by any set of landmarks, simply by virtue of having a necessarily different set of landmarks as neighbours. Let us therefore construct an auxiliary graph on where
Observe that if contains an independent set of size then is a yes-instance: the set has size and as such will still resolve all of . This indicates that must be rather dense, however, we can also argue that it cannot have arbitrarily high degree:
Let have degree at least in . Then is a yes-instance.
Let . Note that for every pair it holds that
Now turn our attention to . First consider the case in which every vertex in has degree less than . Then greedily packing closed neighbourhoods gives an independent set in of size at least , and by Lemma 3, is a yes-instance.
Thus consider the alternative that contains a vertex of degree at least . Define and pick any vertex . Note that since it follows that
Consequently, and share at least neighbours in (removing one extra since ). We can repeat this procedure to construct a sequence of distinct vertices and subsets where and is chosen arbitrarily. The sequence terminates with , giving a clique in of size . Since for every , we get since . Thus and , …, induces a clique of size at least in , and again by Lemma 3 we conclude that is a yes-instance.
With these pieces in place, we can prove the first result of this section.
By Lemma 3 we either have that is a yes-instance or that the auxiliary graph has a maximum degree less than . Assuming the latter, if then contains an independent set of size at least and, as observed above, is a yes-instance.
The kernel for Saving Landmarks is therefore the following procedure: for a given instance , compute the reduced instance . If contains more than vertices, return a trivial yes-instance. Otherwise, return .
Let us now move on to the second result, the single-exponential algorithm. To better describe the algorithm, let us introduce a definition. For a set , we say that two vertices and are -equidistant if for every , i.e., if fails to resolve and . Note that this induces an equivalence relation over .
The main ingredient will be fact that a solution to Saving Landmarks is witnessed already by a small resolving set.
Let be a co-resolving set of a graph . Then there exists a set of size at most that resolves .
We construct iteratively as follows. Begin with and pick a pair of -equidistant vertices in . Since resolves , there exists a vertex that distinguishes and . Add to and partition into equivalence classes of -equidistant vertices. Pick a new pair of -equidistant vertices from one of the classes and repeat. Observe that the number of equivalence classes increase with every addition to , hence after at most steps the set resolves every pair in .
We are now ready to complete the proof of Theorem 1.
We may assume that . Let us first show the following claim: there exists a co-resolving set of of size at least if and only if there is a partition of such that contains at least equivalence classes of -equidistant vertices. Suppose that there exists a co-resolving set of of size at least . Then by Lemma 3, there is a set of size at most that resolves . Let and for a partition . Then resolves and hence has at least equivalence classes of -equidistant vertices. Suppose now that there is a partition of such that has at least equivalence classes of -equidistant vertices. Choose a vertex from each equivalence class to form a set . Then is a co-resolving set of .
The above claim leads to the following randomized algorithm. Choose a natural number defined later on. Repeat times the following: uniformly at random partition the vertices of into and , and derive equivalence classes of -equidistant vertices in . If the number of classes is at least , then conclude that is a yes-instance and stop. If after all repetitions we do not conclude that is a yes-instance, then we conclude that is a no-instance.
Let us argue about the success probability of the randomized algorithm and how to choose. The probability that for a random partition the vertices of as , has at least equivalence classes of -equidistant vertices is at least the probability that and , where sets are as in Lemma 3, which is Thus, is enough to achieve a constant success probability .
Observe that every loop in the randomized algorithm can be executed in polynomial time. Thus, the running time of the randomized algorithm is . The randomized algorithm can be derandomized using the standard -universal set technique , which brings an additional to the exponent of the running time.
4 Structural parameterizations for Metric Dimension
As Hartung and Nichterlein observed , Metric Dimension[VC] is trivially by virtue of Observation 2: After reducing the size of each twin class to at most two, any instance with a vertex cover of size will have at most vertices. In sparse graph classes, the twin reduction even results in a polynomial-size kernel: in classes of bounded expansion (planar graphs or graphs excluding a topological minor), the number of twin classes in is bounded linearly in and in nowhere dense classes by (Lemma 4.3 and Corollary 4.4 in ). Furthermore, if the input graphs stem from a -degenerate class, the number of twin-classes and thus the number of vertices in the kernel is bounded by ; a fact that follows easily from the observation that in such a class at most vertices in the independent set can have degree more than .
It is therefore natural to ask whether this structural parameterization allows a polynomial kernel in general graphs, a question we answer in the negative. We will use in the following that Hitting Set parameterized by the size of the universe plus the solution size does not admit a polynomial kernel unless the polynomial hierarchy collapses to the third level 
We provide a polynomial parameter transformation from Hitting Set, i.e. parameterized by the size of the universe and the solution size, to Metric Dimension. Let be a Hitting Set instance with and . We construct a graph as follows (Figure 1):
Begin with the usual bipartite representation of , i.e., create a bipartite graph where for vertices and we have if and only if ;
add vertices to the graph and edges between so that every vertex in has a unique neighbourhood in of size ;
add vertices to the graph and edges between and such that every vertex in has a unique neighbourhood in of size ;
add three vertices where , , and ;
create true twin copies of , and finally
create false twin copies of but remove all edges from to afterwards. For simplicity, we will label the copy of any vertex by .
In summary, the sets connect to only, the sets to and , the edges between encode the hitting set instance and the pairs , , and are apices for the sets , and , respectively. Our construction concludes with as the Metric Dimension instance with the vertex cover and solution size .
Let us first show that if is a yes-instance then so is . Suppose that is a hitting set for of size . We construct a landmark set for by setting ; let us now argue that is indeed a resolving set. First, note that the selected apices , , and make sure that is distinguished from and from . Since and are in , these sets are of course distinguished from their twin counterparts . By construction, every vertex in has a unique neighbourhood in , hence all of is resolved by . The same holds true for all pairs as long as and . The only pairs we have not yet shown to be resolved by are of the form for with its copy . Since is a hitting set for , every set is adjacent to at least one vertex in while has no neighbours at all in . Thus all such pairs are resolved by and we conclude that is a resolving set.
In the other direction, assume that is a resolving set of size for . Since for each pair of twins at least one vertex has to be in any resolving set, we may assume, without loss of generality, that . Let us call this collection of vertices and let us see what it resolves in . As argued above, every pair except those of the form , are certainly resolved. We first need to argue that indeed does not resolve those pairs: this is immediately obvious for landmarks in since share the same neighbours inside this set. For landmarks in , note that all vertices in are at exactly distance two from every vertex in via the apex vertex (or ). Hence cannot resolve any pair and these pairs must then be resolved by the remaining vertices in . All vertices outside of are either selected or twins to selected vertices, hence we may assume that .
First, consider a potential landmark . Since has distance exactly two to every vertex in except itself, such a selection would only distinguish from all other vertices and not resolve any other pair. Thus we can as well choose any vertex in instead and potentially resolve more pairs, thus we may assume that .
Let us split into and . Again, only distinguishes from the rest of . Thus necessarily distinguishes all pairs with and therefore hits all sets for which . We finally construct a hitting set of size as follows: we take all vertices in and for each pair with we select one (arbitrary) neighbour . By the previous observation, is a hitting set for of size and we conclude that is a yes-instance.
This concludes the parameter preserving transformation. Let us conclude by checking that the parameter is polynomial in and :
where we used that .
We note that this reduction also gives evidence against a more general form of kernelization. Where a standard kernel can be understood as a many-one reduction from a problem to itself, with output size bounded by a function of the parameter, a Turing kernel is the corresponding Turing reduction notion. In other words, informally, a Turing kernel is a polynomial-time procedure that solves a parameterized problem, with access to an oracle for the problem but with a bound on the maximum length of the questions it may ask of the oracle. A polynomial Turing kernel is a Turing kernel with a bound on the question size. For a more formal definition, see [18, 5]. It is known that there are parameterized problems that do not allow a polynomial kernel unless the polynomial hierarchy collapses, but which do allow polynomial Turing kernels; cf. [18, 26, 20].
Although we do not have a framework for excluding polynomial Turing kernels that is as powerful as that for excluding standard polynomial kernels, Hermelin  defined a hierarchy of complexity classes, conjectured to represent problems that do not allow polynomial Turing kernels. The most basic and most common of these hardness classes is WK, which is in turn contained in a larger class MK. It is conjectured in  that no WK-hard problem has a polynomial Turing kernel. Since Hitting Set is known to be MK-hard , the above reduction gives the following.
Metric Dimension is MK-hard (hence also WK-hard) under polynomial parameter transformations, and does not allow a polynomial Turing kernel unless CNF-SAT and every other problem in MK does.
We initiated the study of the parameterized complexity of the dual of the classic Metric Dimension problem and obtained a polynomial kernel as well as a single-exponential algorithm. To the best of our knowledge, this is the first non-trivial parameterization for Metric Dimension which leads to a polynomial kernel. Since our focus in this article was on obtaining new classification results, we leave the improvement of the kernel size or a potential proof of a lower bound on the bitsize of our kernel, to future work.
In addition, we note that it remains open whether Metric Dimension is polynomial time solvable even on series-parallel graphs. Since series- parallel graphs are precisely the graphs of treewidth 2, a negative answer would also imply that there is no XP algorithm for Metric Dimension parameterized by the treewidth. Consequently, a natural starting point of enquiry towards addressing this question could be the study of the parameterized complexity of Metric Dimension parameterized by treewidth.
Gutin was partially supported by Royal Society Wolfson Research Merit Award.
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