In this paper, we consider optimization problems such as:
Maximum -Independent Set, : Given a graph , the objective is to find a largest subset such that distance in between any two vertices in is at least .
Maximum weight induced forest: Given a graph and an assignment of non-negative weights to vertices, the objective is to find a subset such that does not contain a cycle and subject to that, is maximized.
Maximum -Matching, for a fixed connected graph and : Given a graph , the objective is to find a largest subset such that can be partitioned into vertex-disjoint copies of such that distance in between any two vertices belonging to different copies is at least .
To be precise, to fall into the scope of our work, the problem must satisfy the following conditions:
It must be a maximization problem on certain subsets of vertices of an input graph, possibly with non-negative weights. That is, the problem specifies which subsets of vertices of the input graph are admissible, and the goal is to find an admissible subset of largest size or weight.
The problem must be defined in terms of distances between the vertices, up to some fixed bound. That is, there exists a parameter such that for any graphs and , sets and , and a bijection , if holds for all , then is admissible in if and only if is admissible in .
The problem must be monotone (i.e., all subsets of an admissible set must be admissible), or at least near-monotone (as happens for example for Maximum -Matching) in the following sense: There exists a parameter such that for any admissible set in a graph , there exists a system of subsets of such that every vertex belongs to for at most vertices , for each , and for any , the subset is admissible in .
The problem must be tractable in graphs of bounded treewidth, that is, there must exist a function and a polynomial such that given any graph , its tree decomposition of width , an assignment of non-negative weights to the vertices of , and a set , it is possible to find a maximum-weight admissible subset of in time .
Let us call such problems -distance determined -near-monotone -tw-tractable. Note that a convenient way to verify these assumptions is to show that the problem is expressible in solution-restricted Monadic Second-Order Logic () with bounded-distance predicates, i.e., by a formula with one free variable such that the quantification is restricted to subsets and elements of , and using binary predicates , …, , where is interpreted as testing whether the distance between and in the whole graph is at most . This ensures that the problem is -distance determined, and -tw-tractable for some function by Courcelle’s metaalgorithmic result .
Of course, the problems satisfying the assumptions outlined above are typically hard to solve optimally, even in rather restrictive circumstances. For example, Maximum Independent Set is -hard even in planar graphs of maximum degree at most and arbitrarily large (fixed) girth . Moreover, it is hard to approximate it within factor of in graphs of maximum degree at most three . Hence, to obtain polynomial-time approximation schemes (), i.e., polynomial-time algorithms for approximating within any fixed precision, further restrictions on the considered graphs are needed.
A natural restriction that has been considered in this context is the requirement that the graphs have sublinear separators (a set of vertices of a graph is a balanced separator if every component of has at most vertices, and a hereditary class of graphs has sublinear separators if for some , every graph has a balanced separator of size ). This restriction still lets us speak about many interesting graph classes (planar graphs  and more generally proper minor-closed classes , many geometric graph classes , …). Moreover, the problems discussed above admit in all classes with sublinear separators or at least in substantial subclasses of these graphs:
Maximum Independent Set has been shown to admit in graphs with sublinear separators already in the foundational paper of Lipton and Tarjan .
For any positive integer, Maximum -Independent Set and several other problems are known to admit in graphs with sublinear separators by a straightforward local search algorithm .
The problems also admit in graph classes that admit thin systems of overlays , a technical property satisfied by all proper minor-closed classes and by all hereditary classes with sublinear separators and bounded maximum degree.
Bidimensionality arguments  apply to a wide range of problems in proper minor-closed graph classes.
However, each of the outlined approaches has drawbacks. On one side, the local search approach only applies to specific problems and does not work at all in the weighted setting. On the other side of the spectrum, Baker’s approach is quite general as far as the problems go, but there are many hereditary graph classes with sublinear separators to which it does not seem to apply. The approach through thin systems of overlays tries to balance these concerns, but it is rather technical and establishing this property is difficult.
Another option that has been explored is via fractional treewidth-fragility. For a function and a polynomial , a class of graphs is -efficiently fractionally treewidth--fragile if there exists an algorithm that for every and a graph returns in time a collection of subsets such that each vertex of belongs to at most of the subsets, and moreover, for , the algorithm also returns a tree decomposition of of width at most . We say a class is -efficiently fractionally treewidth-fragile if does not depend on its second argument (the number of vertices of ). This property turns out to hold for basically all known natural graph classes with sublinear separators. In particular, a hereditary class of graphs is efficiently fractionally treewidth-fragile if
has sublinear separator and bounded maximum degree ,
In fact, Dvořák conjectured that every hereditary class with sublinear separators is fractionally treewidth-fragile, and gave the following result towards this conjecture.
Theorem 1 (Dvořák ).
There exists a polynomial so that the following claim holds. For every hereditary class of graphs with sublinear separators, there exists a polynomial such that is -efficiently fractionally treewidth--fragile for the function .
Moreover, Dvořák  observed that weighted Maximum Independent Set admits a in any efficiently fractionally treewidth-fragile class of graphs. Indeed, the algorithm is quite simple, based on the observation that for the sets , …, from the definition of fractional treewidth-fragility, at least one of the graphs , …, (of bounded treewidth) contains an independent set whose weight is within the factor of from the optimal solution. A problem with this approach is that it does not generalize to more general problems; even for the Maximum -Independent Set problem, the approach fails, since a -independent set in is not necessarily -independent in . Indeed, this observation served as one of the motivations behind more restrictive (and more technical) concepts employed in [11, 12].
As our main result, we show that this intuition is in fact false: There is a simple way how to extend the approach outlined in the previous paragraph to all bounded distance determined near-monotone tw-tractable problems.
For every class of graphs with bounded expansion, there exists a function such that the following claim holds. Let and be positive integers, and functions and and polynomials. If is -efficiently fractionally treewidth--fragile, then for every -distance determined -near-monotone -tw-tractable problem, there exists an algorithm that given a graph , an assignment of non-negative weights to vertices, and a positive integer , returns in time an admissible subset of whose weight is within the factor of from the optimal one.
Note that the assumption that has bounded expansion is of little consequence—it is true for any hereditary class with sublinear separators  as well as for any fractionally treewidth-fragile class ; see Section 2 for more details. The time complexity of the algorithm from Theorem 2 is polynomial if does not depend on its second argument, and quasipolynomial (exponential in a polylogaritmic function) if is logarithmic in the second argument and is single-exponential (i.e., if ). Hence, we obtain the following corollaries.
Let and be positive integers, a function and a polynomial. Every -distance determined -near-monotone -tw-tractable problem admits a in any efficiently fractionally treewidth-fragile class of graphs.
We say a problem admits a quasipolynomial-time approximation schemes () if there exist quasipolynomial-time algorithms for approximating the problem within any fixed precision. Combining Theorems 1 and 2, we obtain the following result.
Let and be positive integers, a single-exponential function, and a polynomial. Every -distance determined -near-monotone -tw-tractable problem admits a in any hereditary class of graphs with sublinear separators.
The idea of the algorithm from Theorem 2 is quite simple: We consider the sets from the definition of fractional treewidth--fragility, extend them to suitable supersets , …, , and argue that for , any admissible set in disjoint from is also admissible in , and that for some , the weight of the heaviest admissible set in disjoint from is within the factor of from the optimal one. The construction of the sets , …, is based on the existence of orientations with bounded outdegrees that represent all short paths, a result of independent interest that we present in Section 2.
Let us remark one can develop the idea of this paper in further directions. Dvořák proved in (via a substantially more involved argument) that every monotone maximization problem expressible in first-order logic admits a in any efficiently fractionally treewidth-fragile class of graphs. Note that this class of problems is incomparable with the one considered in this paper (e.g., Maximum Induced Forest is not expressible in the first-order logic, while Maximum Independent Set consisting of vertices belonging to triangles is expressible in the first-order logic but does not fall into the scope of the current paper).
Finally, it is worth mentioning that our results only apply to maximization problems. We were able to extend the previous uses of fractional treewidth-fragility by giving a way to handle dependencies over any bounded distance. However, for the minimization problems, we do not know whether fractional treewidth-fragility is sufficient even for the distance- problems. For a simple example, consider the Minimum Vertex Cover problem in fractionally treewidth-fragile graphs, or more generally in hereditary classes with sublinear separators. While the unweighted version can be dealt with by the local search method , we do not know whether there exists a for the weighted version of this problem.
2 Paths and orientations in graphs with bounded expansion
For , a graph is an -shallow minor of a graph if can be obtained from a subgraph of by contracting pairwise vertex-disjoint connected subgraphs, each of radius at most . For a function , a class of graphs has expansion bounded by if for all non-negative integers , all -shallow minors of graphs from have average degree at most . A class has bounded expansion if its expansion is bounded by some function . The theory of graph classes with bounded expansion has been developed in the last 15 years, and the concept has found many algorithmic and structural applications; see  for an overview. Crucially for us, this theory includes a number of tools for dealing with short paths. Moreover, as we have pointed out before, all hereditary graph classes with sublinear separators  as well as all fractionally treewidth-fragile classes  have bounded expansion.
Let be an orientation of a graph , i.e, is an edge of if and only if the directed graph contains at least one of the directed edges and ; note that we allow to contain both of them at the same time, and thus for the edge to be oriented in both directions. We say that a directed graph with the same vertex set is a -step fraternal augmentation of if , for all distinct edges , either or is an edge of , and for each edge , there exists a vertex such that . That is, to obtain from , for each pair of edges we add an edge between and in one of the two possible directions (we do not specify the direction, but in practice we would choose directions of the added edges that minimize the maximum outdegree of the resulting directed graph). For an integer , we say is an -step fraternal augmentation of if there exists a sequence where for , is a -step fraternal augmentation of . We say is an -step fraternal augmentation of an undirected graph if is an -step fraternal augmentation of some orientation of . A key property of graph classes with bounded expansion is the existence of fraternal augmentations with bounded outdegrees. Let us remark that whenever we speak about an algorithm returning an -step fraternal augmentation or taking one as an input, this implicitly includes outputing or taking as an input the whole sequence of -step fraternal augmentations ending in .
Lemma 5 (Nešetřil and Ossona de Mendez ).
For every class with bounded expansion, there exists a function such that for each and each non-negative integer , the graph has an -step fraternal augmentation of maximum outdegree at most . Moreover, such an augmentation can be found in time .
As shown already in , fraternal augmentations can be used to succintly represent distances between vertices of the graph. For the purposes of this paper, we need a more explicit representation by an orientation of the original graph (without the additional augmentation edges). By a walk in a directed graph , we mean a sequence such that for , or ; that is, the walk does not have to respect the orientation of the edges. The walk is inward directed if for some , we have for and for . For a positive integer , an orientation of a graph represents -distances if for each and each , the distance between and in is at most if and only if contains an inward-directed walk of length at most between and . Note that given such an orientation with bounded maximum outdegree for a fixed , we can determine the distance between and (up to distance ) by enumerating all (constantly many) walks of length at most directed away from and away from and inspecting their intersections.
Our goal now is to show that graphs from classes with bounded expansion admit orientations with bounded maximum outdegree that represent -distances. Let us define a more general notion used in the proof of this claim, adding to the fraternal augmentations the information about the lengths of the walks in the original graph represented by the added edges. A directed graph with -length sets is a pair , where is a directed graph and is a function assigning a subset of to each unordered pair of vertices of , such that if neither nor is an edge of , then . We say that is an orientation of a graph if is the underlying undirected graph of and for each . We say that is an -augmentation of if , for each we have , and for each and there exists a walk of length from to in . Let be another directed graph with -length sets. We say is a 1-step fraternal augmentation of if is a -step fraternal augmentation of and for all distinct and , we have if and only if or there exist , , and such that and . Note that a -step fraternal augmentation of an -augmentation of a graph is again an -augmentation of . The notion of an -step fraternal augmentation of a graph is then defined in the natural way, by starting with an orientation of and peforming the -step fraternal augmentation operation -times. Let us now restate Lemma 5 in these terms (we just need to maintain the edge length sets, which can be done with overhead per operation).
Let be a class of graphs with bounded expansion, and let be the function from Lemma 5. For each and each non-negative integer , we can in time construct a directed graph with -length sets of maximum outdegree at most such that is an -step fraternal augmentation of .
Let be an -augmentation of a graph . For , a length walk in is a tuple , where is a walk in , for , and . Note that if there exists a length walk from to in , then there also exists a walk of length from to in . We say that represents -distances in if for all vertices at distance from one another, contains an inward-directed length walk between and . Next, we show that this property always holds for sufficient fraternal augmentations.
Let be a graph and a positive integer and let be a directed graph with -length sets. If is obtained as an -step fraternal augmentation of , then it represents -distances in .
For , consider any length walk in an -augmentation of , and let be a -step augmentation of . Note that is also a length walk between and in . Suppose that is not inward-directed in , and thus there exists such that . By the definition of -step fraternal augmentation, this implies , and thus is a length walk from to in .
Let , …, be a sequence of -augmentations of , where is an orientation of , , and for , is a -step fraternal augmentation of . Let and be any vertices at distance in , and let be a shortest path between them. Then naturally corresponds to a length walk in . For , if is inward-directed, then let , otherwise let be a length walk in obtained from as described in the previous paragaph. Since each application of the operation decreases the number of vertices of the walk, we conclude that is an inward-directed length walk between and in . Hence, represents -distances in . ∎
Next, let us propagate this property back through the fraternal augmentations by orienting some of the edges in both directions. We say that is an -step fraternal superaugmentation of a graph if there exists an -step fraternal augmentation of such that , and for each , we have . We say that is a support of .
Let be a graph and a positive integer and let be an -augmentation of of maximum outdegree representing -distances. For , suppose that is an -step fraternal superaugmentation of . Then we can in time obtain an -step fraternal superaugmentation of representing -distances, of maximum outdegree at most .
Let be an -step fraternal augmentation of forming a support of , obtained as a -step fraternal augmentation of an -step fraternal augmentation of . Let be the -step fraternal superaugmentation of obtained from as follows:
For all distinct vertices such that , , and , we add the edge .
For each edge and integer , we choose a vertex such that and for some and , and add the edge . Note that such a vertex and integers and exist, since was added to when was obtained from as a -step fraternal augmentation.
Each edge arises from an edge leaving and an element , and each such pair contributes at most one edge leaving . Hence, the maximum outdegree of is at most .
Consider a length inwards-directed walk in , for any . Then contains a length inwards-directed walk from to obtained by natural edge replacements: For any edge of this walk and , the construction described above ensures that if or , then there exists such that and for some and , and we can replace the edge in the walk by the edges and of . Since represents -distances in , this transformation shows that so does . ∎
We are now ready to prove the main result of this section.
For any class with bounded expansion, there exists a function such that for each and each positive integer , the graph has an orientation with maximum outdegree at most that represents -distances in . Moreover, such an orientation can be found in time .
Let be the function from Lemma 5, and let . By Lemma 6, we obtain an -step fraternal augmentation of of maximum outdegree at most . By Lemma 7, represents -distances in . Repeatedly applying Lemma 8, we obtain a -step fraternal superaugmentation of of maximum outdegree at most representing -distances. Clearly, is an orientation of of maximum outdegree at most representing -distances. ∎
3 Approximation schemes
Let us now prove Theorem 2. To this end, let us start with a lemma to be applied to the sets arising from fractional treewidth-fragility.
Let be an orientation of a graph with maximum outdegree . Let be a set of vertices of and for a positive integer , let be a system of subsets of such that each vertex belongs to at most of the subsets. For and a positive integer , let be the union of the sets for all vertices such that contains a walk from to of length at most directed away from . For a positive integer , let , …, be a system of subsets of such that each vertex belongs to at most of the subsets. For any assignment of non-negative weights to vertices of , there exists such that .
For a vertex , let be the set of vertices reachable in from vertices such that by walks of length at most directed away from . Note that and that for each , we have if and only if .
Suppose for a contradiction that for each we have , and thus . Then
which is a contradiction. ∎
Next, let us derive a lemma on admissibility for -distance determined problems.
For a positive integer , let be an orientation of a graph representing -distances. For a set , let be the set of vertices such that contains a walk from to of length at most directed away from . For any -distance determined problem, a set is admissible in if and only if it is admissible in .
Since the problem is -distance determined, it suffices to show that holds for all . Clearly, , and thus it suffices to show that if the distance between and is is , then contains a walk of length between and . Since represents -distances, there exists an inward-directed walk of length between and in . Since , we have , and thus is also a walk of length between and in . ∎
We are now ready to prove the main result.
Proof of Theorem 2.
Let be the function from Lemma 9 for the class . Let us define . The algorithm is as follows. Since is -efficiently fractionally treewidth--fragile, in time we can find sets such that each vertex belongs to at most of them, and for each , a tree decomposition of of width at most . Clearly, . Next, using Lemma 9, we find an orientation of that represents -distances. Let be defined as in the statement of Lemma 11. Since the problem is -tw-tractable problem, for each we can in time find a subset of admissible in of largest weight. By Lemma 11, each of these sets is admissible in ; the algorithm return the heaviest of the sets , …, .
As the returned set is admissible in , it suffices to argue about its weight. Let be a heaviest admissible set in . Let be the system of subsets from the definition of -near-monotonicity, and let be defined as in the statement of Lemma 10. By the definition of -near-monotonicity, for each the set is admissible in . Since for each , we have , and thus by Lemma 11, is also admissible in , and by the choice of , we have . By Lemma 10, we conclude that for at least one , we have , as required. ∎
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