1.1 Fixed parameter tractability
A parameterized problem instance is created by associating an input instance with an integer parameter . We say that a problem is fixed parameter tractable (FPT) if it admits an algorithm processing an instance in running time , where is a computable function. Such a procedure is called a parameterized algorithm or an FPT algorithm.
To argue that a problem is unlikely to be FPT, we use parameterized reductions analogous to those employed in the classic complexity theory. Here, the concept of W-hardness replaces NP-hardness, and we need not only to construct an equivalent instance in FPT time, but also ensure that the size of the parameter in the new instance depends only on the size of the parameter in the original instance. If there exists a parameterized reduction from a W-hard problem (e.g., -Clique) to another problem , then the problem is W-hard as well. This provides an argument that is unlikely to admit an algorithm with running time .
In recent years new research directions emerged in the intersection of the theory of approximation algorithms and the FPT theory. For example, one may wonder whether one can distinguish graphs with -clique from those with no clique larger than, e.g., , faster than time. In order to prove such a task hard, one would need a reduction from a problem with established hardness which translates YES-instances to graphs with a large clique and NO-instances to graphs with only tiny cliques.
1.2 Recent progress on parameterized approximations
For some problems that are intractable in the exact sense, parameterization still comes in useful when we want to reduce the approximation ratio. Some examples are 1.81-approximation for -Cut  or -approximation for -Median , running in time . The first result beats the factor 2 that is believed to be optimal within polynomial running time and the second one reaches the respective lower bound, what is a long standing open problem for polynomial-time algorithms. For Capacitated -Median, a constant factor FPT approximation has been obtained , whereas the best polynomial approximation gives .
On the other hand several problems have proven resistant to such improvements under the assumption of Gap Exponential Time Hypothesis (Gap-ETH), which states that one requires exponential time to distinguish satisfiable 3-SAT instances from those where only a fraction of clauses can be satisfied at once. Chalermsook et al.  showed that under this assumption there can be no parameterized approximations with ratio for -Clique or -Biclique and none with ratio for -Dominating Set (for any function ). They have also ruled out -approximation for Densest -Subgraph.
Subsequently, Gap-ETH has been replaced with a more established hardness assumption W FPT for -Dominating Set , what involved reduction from an exact problem and required introducing a gap through a distributed PCP theorem.
Marx  proved parameterized inapproximability of Monotone Circuit SAT under even weaker assumption W[P] FPT.
Lokshtanov et al.  introduced Parameterized Inapproximability Hypothesis (PIH), that is weaker than Gap-ETH,
and used it to rule out an FPT approximation scheme for Directed Odd Cycle Transversal
Directed Odd Cycle Transversal. PIH turned out to be a suffcient assumption to prove there can be no FPT algorithm for -Even Set .
1.3 Previous work on Steiner Orientation
One of the first studies on the Steiner Orientation problems were motivated by modeling protein-protein interactions (PPI)  and protein-DNA interaction (PDI) . Whereas PPIs interactions could be represented with undirected graphs, PDIs required introducing mixed graphs. Arkin and Hassin  showed the problem to be NP-hard, but polynomially solvable for . This result was generalized by Cygan et al. , who presented an -time algorithm, which implied that the problem belongs to the class XP. The -Steiner Orientation problem222 We attach the paramater to the problem name when we want to emphasize that we deal with a parameterized problem. has been proved to be -hard by Pilipczuk and Wahlström , via a gadget machinery that was also used to show hardness of -Directed Multicut parameterized by the size of the cutset . The -hardness proof has been later strengthened to work for planar graphs and to give stronger running time lower bounds based on ETH , which are essentially tight with respect to the -time algorithm.
The approximation of Steiner Orientation has been mostly studied when restricted to undirected graphs, where the problem reduces to optimization over trees by contracting 2-connected components. Medvedovsky  presented an -approximation and actually proved that this bound is achievable with respect to the total number of terminals . The approximation factor has been improved to  and later to  by observing that one can compress an undirected instance to a tree of size . A lower bound of (based on P NP) has been obtained via reduction from Max-DiCut .
It is worth mentioning that the decision problem if all the terminal pairs can be satisfied is polynomially solvable on undirected graphs, what makes the maximization version fixed-parameter tractable, by simply enumerating all subsets of terminals. When it comes to FPT-approximability, the reduction by Chitnis et al.  implies that, assuming Gap-ETH, -Steiner Orientation cannot be approximated within factor on mixed graphs, within running time .
1.4 Overview of the results
Our main inapproximability result can be formulated as a -hardness proof for the following problem with a gap. It is a self-reduction that carefully connects copies of an instance via random hash families.
For any function , it is W-hard to distinguish whether for a given instance of -Steiner Orientation:
there exists an orientation satisfying all terminal pairs,
for all orientations the number of satisfied pairs is at most .
The previously known lower bound for parameteried approximation, that is , was obtained via a linear reduction from Clique and was based on Gap-ETH . Our reduction not only raises the inapproximability bar significantly, but also weakens the hardness assumption (Gap-ETH implies ETH, which implies W FPT). In fact, we begin with the exact version of -Steiner Orientation and introduce a gap inside the reduction. What is interesting, we rely on totally different properties of the problem than in the W-hardness proof : that one required gadgets with long undirected paths and we introduce only new directed edges.
The result is also interesting from the perspective of the classic (non-parameterized) approximation theory. The only known approximation lower bound has been , valid also for undirected graphs. Therefore we provide a new inapproximability result for polynomial algorithms, that is based on the assumption W FPT. Such a seemingly strong assumption is necessary because our reduction increases the size of instance by a factor exponential in . A similar phenomenon – a novel polynomial-time hardness based on a parameterized assumption – has appeared in the work on Monotone Circuit SAT , where the reduction involved a perfect hash family. Another example of this kind is a polynomial-time approximation hardness for Densest -Subgraph based on ETH . Moreover, restricting to a purely polynomial running time allows us to rule out approximation depending on (rather than on ): here the lower bound becomes .
So far, the exact -Steiner Orientation problem has only been known to be W-hard  and to belong to XP . We establish its exact location in the W-hierarchy. A crucial new insight is that we can assume the solution to be composed of pieces, for which we only need to check if they fit to each other, and this task reduces to -Clique.
-Steiner Orientation is -complete.
It solves an open problem from . What is more, this shows that the approximate version of -Steiner Orientation (that is, the task of distinguishing whether an instance is totally solvable or only small fraction of terminal pairs can be satisfied) is W-complete. Another problem with this property is Maximum -Subset Intersection, introduced for the purpose of proving W-hardness of -Biclique . We are not aware of any other natural approximate problems being complete in their parameterized complexity class.
As another application of our technique, we present a simple lower bound for the maximization version of Directed Multicut. We show that even if we parameterize the problem with both the number of terminal pairs and the size of the cutset , then we essentially cannot obtain any approximation ratio better than .
For any , it is W-hard to distinguish whether for a given instance of -Directed Multicut:
there is a cut of size that separates all terminal pairs,
all cuts of size separate at most terminal pairs.
When restricted to a polynomial running time, the lower bound of can be improved to , however unlike the case of Steiner Orientation, this time the reduction is polynomial and we can assume only
NP co-RP333 A problem is in co-RP if it admits a polynomial algorithm that is always correct for YES-instances and for NO-instances returns the correct answer with constant probability.
A problem is in co-RP if it admits a polynomial algorithm that is always correct for YES-instances and for NO-instances returns the correct answer with constant probability..
As far as we know, the approximation status of this variant has not been studied yet. If we want to mimimize the number of removed edges to separate all terminal pairs or minimize the ratio of the cutset size to the number of separated terminal pairs (this problem is known as Directed Sparsest Multicut), those cases admit polynomial-time -approximation algorithm  and a lower bound of . Since and , the maximization variant with a hard constraint on the cutset size turns out to be harder.
In the undirected case Multicut is FPT, even when parameterized only by the size of the cutset and allowing arbitrarily many terminals . This is in contrast with the directed case, which gets -hard already for 4 terminals. It is worth mentioning that Steiner Orientation and Directed Multicut were proven to be -hard with a similar gadgeting machinery . The fact that our technique also applies to both of them suggests that the core of hardness is somehow similar inside both problems.
For an instance , of -Steiner Orientation we refer to the vertices as and . We say that an orientation of satisfies the pair with index if there is a directed path from to in . Otherwise we say that the pair is blocked. We define to be the maximal number of pairs that can be satisfied by some orientation in the instance .
We keep the same convention when working with Directed Multicut: we refer to sources and sinks as , , and denote the maximal number of terminal pairs separable by deleting edges with .
We use notation . All logarithms are 2-based.
1.6 Organization of the paper
As our gap amplification technique is arguably the most innovative ingredient of the paper, we begin with informal Section 2, which introduces the main ideas gradually. It is followed by detailed constructions for Steiner Orientation in Section 3 and for Directed Multicut in Section 4. Finally, in Section 5 we prove W-completeness of -Steiner Orientation. This part of the paper is self-contained and can be read independently of the previous sections.
2 The gap amplification technique
We begin with a thought experiment that helps to understand the main idea behind the reduction. Consider copies of the same instance: , that will be treated as the first layer. The output of this layer will form an input for another copy of – let us call it . If is a NO-instance, than for each of the copies, at least one of the terminal pairs is not satisfied. Suppose for now that we know which terminals are those, even before we have finished building our instance.
We add a directed edge from a random sink from to the source number one in . Similarly, we connect a random sink from to (the source number two in ) and so on, as shown in Figure 1. We neglect the unconnected terminals from the first layer and obtain a new instance with terminal pairs. What is the expected number of terminal pairs satisfiable in ?
For each , the probability of choosing an arc reachable from the respective source in is at most . By linearity of expectations, the average number of non-blocked sources connect to is . So far nothing changed. However, with probability we have chosen only non-blocked terminals, but we cannot satisfy all of them within . Therefore the expected optimum is at most : a small gap has been introduced.
Of course we cannot fix the orientation before adding the connecting arcs. Since we can afford an exponential blow-up (with respect to ), we can include in the second layer the whole probabilistic space, that is, choices of connecting the first layer to . This construction suffices to rule out any constant approximation, but we do not have to be so wasteful.
An important observation is that the number of possibilities regarding which terminal pairs form dead ends is relatively small – . If the second layer contained not a single copy but rather such random copies, then by the Hoeffding’s inequality (a generalization of the Chernoff bound) the probability that the fraction of satisfied terminals differs much from the expectation is less than . Surprisingly, this means that there exists a single way of placing the arcs between the layers that guarantees the gap gets amplified no matter how the copies of are oriented.
We need to iterate this trick in order to grow the gap. In further steps we need to add exponential number of copies to the new layer, even when compressing the probabilistic space as above. This is why we need assumption W FPT even for ruling out polynomial-time approximations.
The construction for -Directed Multicut is simpler since it does not involve the layer stacking. Therefore to achieve a polynomial-time hardness it suffices to assume NP co-RP. The phenomenon that both problems admit such strong self-reducing properties can be explained by the fact that when dealing with directed reachability one can compose instances sequentially, what is the first step in both reductions.
3 Inapproximability of Steiner Orientation
We first formulate the properties of the construction and discuss how they are used to prove the claims of the paper. Then, we focus on the construction itself in Section 3.1.
There is a procedure that, for an instance of Steiner Orientation and parameter , constructs a new instance , such that:
if , then (Completeness),
if , then (Soundness).
The construction can be derandomized and takes time .
These properties are proven in the end of Section 3.1. It easily follows from them that the gap can get amplified to any constant . In order to rule out superconstant approximation ratio we additionally need to adjust so that .
Let us fix the function , where is computable. We are going to reduce the exact version of -Steiner Orientation, which is W-hard, to the version with a sufficiently large gap. For a fixed we can treat as a function of : for some constant . For an instance we use Lemma 4 with large enough, so that . Then is a computable function of . We have .
We have obtained a new instance of -Steiner Orientation of size and being a function of . If the original instance is fully satisfiable then the same holds for and otherwise , so the hypothetical approximation algorithm could distinguish these cases. ∎
If we restrict the running time to be purely polynomial, we can slightly strengthen the lower bound (replace with in the approximation ratio) while working with the same hardness assumption. To make this connection, we observe that in order to show that a problem is in FPT, it suffices to solve it in polynomial time for some superconstant bound on the parameter.
Consider a parameterized problem that admits a polynomial-time algorithm for the case , where is some computable function. Then .
Since , it admits an algorithm with running time . Whenever , we execute the polynomial-time algorithm. Otherwise we can solve it in time . ∎
Assuming , for any function , there is no polynomial-time algorithm that given an instance of Steiner Orientation distinguishes between the following cases:
there exists an orientation satisfying all terminal pairs,
for all orientations the number of satisfied pairs is at most .
Suppose there is such an algorithm with coefficient , where . We can assume is non-increasing. Let be the smallest integer , for which . The function is well defined and computable, because is computable. Again, for fixed we have for some constant .
We are going to use the polynomial-time algorithm to solve -Steiner Orientation in time, what would imply , relying on Lemma 5. Since the problem admits an -time algorithm , it suffices to solve instances satisfying . We can thus assume , or equivalently .
Given an instance of -Steiner Orientation, we apply Lemma 4 with .
For large we have . The size of the new instance of polynomially bounded and if the original instance was not fully satisifiable, then the fraction of satisfiable pairs gets less than . The claim follows. ∎
3.1 The gap amplifying step
For a family of functions , a -biased sampler family is a multiset , such that for every it holds
For a given , and , there exists a -biased sampler family of size . This family can be obtained via random sampling.
Let us sample independently elements from (possibly with repetitions): this is the multiset . For a fixed we apply the Hoeffding’s inequality.
For our choice of this bound gets less than
. Consider an event that the estimation did not work for some function. By union bound, the probability of such an event is at most . Since the probability of generating a correct is positive, it clearly exists. ∎
Building the layers
We construct a family of instances with . Let indicate the number of gadgets in the last layer, so and . We construct by taking copies of , denoted and forming a new layer of copies of , where . The derivation of the latter quantity is postponed to Lemma 9.
Let be a family of tuples , . For each copy of in the last layer, we choose a random tuple and add a directed edge from to for each . Next, we add the terminal pair to . The construction is depicted in Figure 1.
Let , . If , then .
First observe that every -path in runs through unique copies of . Therefore -pair is satisfied only if the corresponding terminal pairs in those copies are satisfied.
Recall that we connect each copy of to the terminals from the previous layer given by a random tuple . Let be a family of configurations encoding which the sinks are not reachable from . We have . For a configuration let be a function describing the maximal fraction of satisfied terminal pairs from those with sinks in connected through a tuple .
For a fixed we estimate the expected value of . Let be a random variable indicating that is reachable from for random . Since the maximal number of terminals pairs that can go through is , we have
We can w.l.o.g. assume that , because increasing this value could only increase . By linearity of expectation, . Moreover, , therefore .
Proof of Lemma 4.
We define for .
The completeness is straightforward. If , then we can orient all copies of so that is always reachable from and each requested path in can be formed as a concatenation of paths in copies of . To see the soundness, note that by Lemma 9 we have , what implies .
To estimate , recall that we have and . For , this becomes (we can assume and so ). The size of is at most times the number of layers. We trivially bound to obtain the property (2). ∎
The construction above is randomized and works with probability . It can be derandomized since the sizes of and in applications of Lemma 8 are always bounded by a function of , so we can enumerate all potential sampler families and find the correct one.
4 Inapproximability of Directed Multicut
In the Directed Multicut problem we are given a directed graph with a set of source-sink pairs and a budget . In the maximization variant we want to delete a subset of edges of size at most to disconnect as many source-sink pairs as possible.
There is a procedure that, for an instance of Directed Multicut and parameter , constructs a new instance , such that:
if , then (Completeness),
if , then (Soundness).
The construction is randomized and takes time polynomial in and . It can be derandomized in time .
Consider copies of , denoted . For a random sequence , we add a terminal pair and for each we add directed edges and . We repeat this subroutine times and create that many terminals pairs. We set the budget .
If , then the budget suffices to separate all terminal pairs in all copies of . Otherwise, one needs to remove at least edges from a copy of to separate all 4 pairs, so the budget suffices to cover at most copies. Therefore there are at least copies, where there is at least one terminal pair that is not separated. We define the family of configurations representing information about each copy : whether all terminals pairs are cut and if not – which terminal pair is open. Recall that each terminal pair can be represented by a tuple encoding through which terminal pair it can go in each . For a fixed configuration , function is set to if the pair is separated, or equivalently: if for each the terminal pair is cut. Since there are at least copies with an open path, we have .
The size of is at most . We apply Lemma 8 for and . It follows that random samples from is enough to obtain rounding error of . Therefore with constant probability we have constructed an instance in which for any cutset (and thus for any configuration ) the fraction of separated terminal pairs is at most . ∎
Remark on derandomization
As before, if we allow exponential running time with respect to and , we can find a correct sampler family by enumeration and derandomize the reduction. However, we cannot afford that in a polynomial-time reduction. To circumvent this, we can take advantage of -biased -wise independent hashing to construct binary random variables with few random bits, instead of relying on Lemma 8. This technique simulates -wise independency and provides an analogous bound on additive estimation error for events that depend on at most variables. Such a family can be constructed using random bits .
Since we are interested in having variables, , and -wise independency, the size of the whole probabilistic space becomes . The problem is that we need to optimize the exponent at in order to obtain high lower bounds. Unfortunately we are not aware of any construction of -biased -wise independent hash family, that would optimize this constant. Still, this means that a lower bound of can be obtained for some assuming just PNP.
Let us fix . We are going to reduce the exact version of -Directed Multicut with 4 terminals, which is W-hard, to the version with a sufficiently large gap, parameterized by both and . Let be an integer larger than .
For an instance of -Directed Multicut we apply Lemma 10 with . If the original instance is fully solvable, the new one is as well. Otherwise the maximal fraction of separated terminal pairs is . On the other hand, . The exponent at in the latter formula is , so for large the hypothetical approximation algorithm could detect the weakly separable instance. Both and are functions of , therefore we have obtained a parameterized reduction. ∎
Assuming NP co-RP, for any , there is no polynomial-time algorithm that, given an instance of Directed Multicut, distinguishes between the following cases:
there is a cut of size that separates all terminal pairs,
all cuts of size separate at most terminal pairs.
Suppose there is such an algorithm for some and proceed as in the proof of Theorem 3 with sufficiently large so that and . The reduction is polynomial because is a constant. We have because becomes a dominant term for large and . If the initial instance is fully satisfiable, then always . For a NO-instance, we have . On the other hand, and
We have adjusted to have , so for large the fraction in case (2) is larger than . When the reduction from Lemma 10 is correct (with probability at least ), we are able to detect the NO-instances. This implies that Directed Multicut co-RP. ∎
5 W-completeness of Steiner Orientation
In this section we present a tight upper bound for the parameterized hardness level of -Steiner Orientation, complementing the known -hardness. We construct an FPT reduction to -Clique and thus show that the problem belongs to . The fact that -Steiner Orientation is -complete implies the same for the gap version of the problem studied in the previous chapter, what is uncommon is the theory of parameterized inapproximability.
The main idea in the reduction is to restrict to solutions consisting of subpaths, which can be chosen almost freely between fixed endpoints. This formalizes an intuitive observation that different terminal pairs should obstruct each other only limited number of times, because otherwise one path may exploit the other one as a shortcut and the whole knot of obstructions could be disentangled.
Definition 12 (Canonical path family).
For a graph , a family of paths , defined for all pairs such that is reachable from in , is called canonical if it satisfies the following conditions:
if there exists edge in , then ,
if a vertex lies on a path , then equals concatenated with .
There might be multiple choices for such a family, but in the further arguments we just need to fix an arbitrary one. It is important that we are able to construct it efficiently.
A canonical path family can be constructed in polynomial time.
Let us fix any labeling of vertices with a linearly ordered alphabet. We define to be the shortest path from to , breaking ties lexicographically. In order to construct it, we begin with computing distances between all pairs of vertices. The first edge on the path goes to the lexicographically smallest vertex among those minimizing distance to , and similarly for the further edges. ∎
Definition 14 (Support).
Suppose a path can be represented as a concatenation of canonical paths for some sequence . We will refer to the set as a support of (does not have to be unique). If a path admits a support of size at most , then we say it is -canonical.
Definition 15 (Schedule).
Consider a family of non-empty sets . For each we consider variables and . A schedule of this family is the order relation over those variables.
It is easy to see that the number of possible schedules for sets is at most : for each of the variables we choose its position in the sequence. Note that some variables might have the same value and they can share the same position.
If a -Steiner Orientation instance is satisfiable on an acyclic mixed graph, then it admits a solution in which each path is -canonical.
We say that a set is a support of the solution if it contains supports for all the paths. Due to property , every path admits some, potentially large, support, so we can always find such a set . We are going to show that if an instance is solvable, then there exists a solution with support of size less than . To do so, we assume the contrary: that the minimal size of a support is at least , and then construct a solution with a smaller support. This entails the claim.
Consider a solution with support , . For a vertex , we define to be the set of indices , such that goes though . By counting argument, there must be at least vertices in with the the same set .
There is a natural linear ordering of those vertices, that is coherent with the orientation of paths from the terminal set . For each even index , consider the canonical path from to . If such a path is not in conflict444Two paths are in conflict if there is an undirected edge they use in different directions. with paths from , we could remove from . This would contradict being the minimum size support.
For the canonical path from to , we first consider which other paths are in conflict with : let us call this set . Next, we look at the schedule of sets for with respect to the order given by . Again by counting argument we conclude that there are (at least) paths with the same and the same schedule (factor 2 for choosing only even indices, at most choices of , at most different schedules). Recall that the first vertex of is reachable from the last vertex of .
Having the fixed schedule makes it possible to find a detour from the beginning of to the end of that omits the vertices in between. Let us start by following to the place of the first conflict with . We know that there is a path form the first vertex of to the last vertex of , which is a subpath of ( cannot go in the other direction because we assumed the graph to be acyclic; this is why we needed to be reachable from ). We can follow this path and continue on to the next occurrence of some conflict with , then use the same argument to reach the last vertex of and iterate this procedure. Note that the vertex we arrive at in might be the same where the next detour starts. It is crucial that we have fixed a single schedule so we will never reach another conflict with the same path . We need at most iterations to get behind all conflicts – then the last detour is guaranteed to reach a vertex in (a detour can omit several segments ), from where we can reach the end of . The idea of a detour is depicted in Figure 3.
We can now remove all the endpoints of from (at least vertices) and replace this segment with the detour constructed above for all terminal pairs in . We might need to add vertices to (2 for each iteration) and the rest of the detour is either canonical or follows paths that are already in the solution. Therefore we have constructed a solution with smaller , what finishes the proof.
The problem is known to be -hard , so we just need a parameterized reduction to -Clique. Let be an instance of -Steiner Orientation and let be the sequence from Lemma 16. We can assume to be acyclic by a standard argument that if we can orient some edges to create a cycle, then it can be contracted. We construct the instance of -Clique as follows:
Compute a canonical family of paths for in polynomial time (Lemma 13).
For each pair create an independent set . The vertices in
are given as ordered pairs, such that is reachable from in . If , then we require and if , we require . We allow pairs of form .
For vertices and lying in distinct sets , place an edge between them if the canonical paths and are not in conflict.
If we placed an edge between and , we remove it unless .
The size of is bounded by . If the constructed graph admits a clique of size , then each of its vertices must lie in a different independent set . Due to step (4), we know that canonical paths encoded by the choice of vertices in match and they form a path from to . Step (3) ensures that those paths are not in conflict, therefore the instance is satisfiable.
On the other hand, if admits a solution, we can assume its paths to be -canonical due to Lemma 16. We can thus choose the vertices in
to reflect their supports, padding them with trivial pathsif the support is smaller than . Since none of the paths are in conflict, there is an edge in between all chosen vertices. ∎
6 Final remarks and open problems
I would like to thank Pasin Manurangsi for helpful discussions and, in particular, for suggesting the argument based on Chernoff bounds, which is surprisingly simple and powerful. A question arises whether one can derandomize this argument efficiently and construct a -biased family in some explicit way. The function family has a very special structure in both applications, what also could be exploited. This would allows us to replace the assumption NP co-RP with P NP for Directed Multicut. It is also plausible that this technique may find use in other reductions in parameterized inapproximability.
An obvious question is if any of the studied problems admits an -approximation, or if the lower bounds can be strengthened. Note that for the maximization version of Directed Multicut we do not know anything better than -approximation as we cannot solve the exact problem for . The square root in the lower bound is only an artifact of in the exponent of Chernoff-like bounds. On the other hand, before the exponent was beaten for the minimization version of Directed Multicut, the best approximation ratio was . Can this algorithm be adapted to provide a tight upper bound for the maximization version?
For Steiner Orientation, the reason why the new value of the parameter is so large, is that in each step we can add only exponentially small term to the gap. Getting around this obstacle should lead to higher lower bounds. Also, the approximation status for -Steiner Orientation over planar graphs remains unclear . Here we still cannot rule out a constant approximation and there are no upper bounds known.
Finally, it is an open quest to establish relations between other hard parameterized problems and their gap versions. Is Gap -Clique W-hard or is Gap -Dominating Set W-hard (open questions in )? Or can it be possible that Gap -Dominating Set is in W?
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