1 Plotkin-Rao-Smith algorithm and its consequences
A quite general argument for obtaining small balanced separators (which gives another proof that proper minor-closed classes have strongly sublinear separators) was obtained by Plotkin, Rao, and Smith . Let us state their result in a slightly reformulated way, seen to hold by an inspection of their proof (we do not provide details, as we anyway prove more general Theorem 5 later). A model of a clique in a graph is a system of pairwise-disjoint subsets of such that is connected for and contains an edge with one end in and the other end in for . The support of the model is . The model has depth if each of the subgraphs , …, has radius at most . Let denote the maximum integer such that contains a model of of depth . For , a model of depth is -bounded if for , and in particular the support of the model has size at most .
Theorem 1 (Plotkin, Rao, and Smith ).
There exists a non-decreasing function satisfying and a polynomial-time algorithm that, given an -vertex graph and integers , returns either an -bounded model of of depth in , or disjoint sets such that
is a balanced separator in ,
for some , is the support of a -bounded model of of depth in .
Note that the separator constructed in the previous theorem (with , so that the first conclusion does not apply) has order at most ; if is from a proper minor-closed class, then is bounded by a constant, and thus we obtain a separator of order by setting .
The fact that Theorem 1 only restricts minors of bounded depth connects strongly sublinear separators to another concept: bounded expansion. For an integer , let denote the maximum of average degrees of graphs that can be obtained from subgraphs of by contracting vertex-disjoint subgraphs of radius at most (clearly, ). For a function , we say that a class of graphs has expansion bounded by if for every integer and every , ; we say that has -expansion bounded by if for all such and , . If such a function exists, then has bounded expansion or is nowhere-dense, respectively. We say that has polynomial expansion or polynomial -expansion, respectively, if this is the case for some polynomial .
Starting with their introduction by Nešetřil and Ossona de Mendez [14, 15, 16], the notions of bounded expansion and nowhere-density played important roles as models of sparse graph classes due to their numerous structural and algorithmic properties. We refer the reader to their book  for a detailed treatment of the theory. Relevantly to us, Dvořák and Norin  proved that a class has strongly sublinear separators if and only if it has polynomial expansion. Theorem 1 also implies this is equivalent to having polynomial -expansion.
For any class , the following holds.
If has -expansion bounded by a function for some real number , then for some , .
If for some , then for every , has expansion bounded by a function .
In particular, the following claims are equivalent.
has polynomial expansion.
has polynomial -expansion.
has strongly sublinear separators.
To prove (a), we give an argument analogous to the one used in Dvořák and Norin [6, Corollary 2]. Without loss of generality, we can assume that is closed under subgraphs, and thus it suffices to show that graphs in have small balanced separators. Consider any -vertex graph and apply Theorem 1 with and , so that . Since has -expansion bounded by , we have and
which gives the desired bound on the size of the balanced separator.
The fact that polynomial expansion and polynomial -expansion coincide is noteworthy, since for classes whose -expansion grows faster, this is not the case. In particular, there exist nowhere-dense classes that do not have bounded expansion, and Lemma 2 shows that their -expansion must grow superpolynomially.
For all and , there exists a polynomial-time algorithm that given an -vertex graph returns a balanced separator of order .
Let us remark that the approximation algorithm of Feige et al.  could be used instead to give the same result (actually even with stronger guarantees on the size of the separator). However, the idea of Corollary 3 also gives the following surprising fact: if all small subgraphs of a graph have strongly sublinear separators, then itself does as well.
For all and , there exists as follows. Suppose that is an -vertex graph. If all subgraphs of with at most vertices belong to , then belongs to .
By Lemma 2(b), the -expansion of is bounded by a non-decreasing function such that . Let be the function from the statement of Theorem 1. Choose such that . Apply the algorithm of Theorem 1 with . Note that if is a support of an -bounded model of depth of for some , then by the choice of . By the assumptions, we have . Consequently , and thus the first outcome of Theorem 1 does not apply. Hence, we obtain a balanced separator of of order at most
as required. ∎
2 Weighted separators and fractional treewidth-fragility
The cornerstone of this paper is the following weighted strengthening of Theorem 1. For a function (where are the nonnegative rational numbers) and a set , let us define .
There exists a non-decreasing function satisfying and a polynomial-time algorithm that, given an -vertex graph , integers , and an assignment of nonnegative costs to vertices of , returns either an -bounded model of of depth in , or disjoint sets such that
is a balanced separator in ,
for some , is the support of a -bounded model of of depth in .
Note that the last condition implies . The proof of this theorem is given in Section 4.
Before seeing its applications, let us give a few remarks on Theorem 5.
There is another way how to introduce weights into Theorem 1: the weights could influence what “balanced” means. This was already done by Plotkin, Rao, and Smith , and Theorem 5 could be modified in this way (adding a weight function , and requiring that each component of satisfies instead of being a balanced separator) without any significant changes to its proof. Since we do not need this device, we opted not to complicate the statement of Theorem 5.
The fact that the cost is not restricted cannot be entirely avoided. Consider e.g. the case that is the star with cost given to the center of the star and cost spread uniformly across the rays. For any balanced separator , either contains the center of the star, or at least of the rays, and thus ; hence, for larger than , not counting some part of the separator towards its cost is necessary.
For graphs from a class with polynomial -expansion, the bound on the size of in Theorem 5 depends polylogarithmically on the number of vertices of . I conjecture that there actually always exists a balanced separator of with for some set of constant size (dependent on the class and , but not on ).
Iterating Theorem 5, we can break-up the graph into small pieces combined in a tree-like fashion; i.e., forming a subgraph of small treewidth. A tree decomposition of a graph is a pair , where is a tree and assigns to each vertex of a subset of vertices of , such that for every there exists satisfying , and for every the set induces a non-empty connected subtree of . The width of the tree decomposition is , and the treewidth of is the minimum width over all tree decompositions of . The importance of bounded treewidth in the context of classes with strongly sublinear separators stems from the fact that graphs of treewidth at most have balanced separators of order at most .
There exist non-decreasing functions and satisfying and , and a polynomial-time algorithm that, given an -vertex graph , integers , and an assignment of costs to vertices of , not identically , returns either an -bounded model of of depth in , or a set with and a tree decomposition of of width at most .
Let . Let be the function from the statement of Theorem 5 and let us define . Let be a rooted tree and , , and functions mapping to subsets of obtained as follows. For the root of , let . Considering a vertex , let us apply the algorithm of Theorem 5 to , with and restricted to . If the outcome is an -bounded model of of depth in , we return this model. Otherwise, let and be as in the statement of Theorem 5. We set , , and for each component of , we create a child of with . We repeat this construction to obtain , , , and .
Let . For , let be the union of the sets over all ancestors of in (including itself). Then is a tree decomposition of . Note that has depth at most . Since for all , we conclude that the width of the decomposition is , and we define the function according to this bound.
For , let denote the set of vertices of at distance from the root. Note that for distinct , the sets and are disjoint. We conclude that , and since , we have , as required. ∎
, we say that a probability distribution onis -thin if a set chosen at random from this distribution satisfies for all . The support of this distribution consists of the elements of with non-zero probability. For , let denote the collection of all sets such that . We say that a class is fractionally -treewidth-fragile if for each graph there exists an -thin probability distribution on . A class is fractionally treewidth-fragile if for all there exists such that is fractionally -treewidth-fragile.
As an example, consider a connected planar graph . Let be any vertex of , and for , let be the set of vertices of at distance exactly from . For , let . A result of Robertson and Seymour  implies that the graph has treewidth at most , and thus . Let us assign each of the sets for probability , and all other sets in probability . Since the sets , …, are pairwise disjoint, if a set is chosen at random from this distribution, we have for all . Hence, the class of planar graphs is fractionally -treewidth-fragile. Consequently, for every , the class of planar graphs is fractionally -treewidth-fragile, and thus the class of planar graphs is fractionally treewidth-fragile.
Let us remark that the word “fractional” in the definition of fractional treewidth-fragility refers to the fact that the sets in the support of the distribution do not need to be pairwise disjoint; hence, planar graphs are actually examples of classes that are “treewidth-fragile”, in a non-fractional sense. Three-dimensional grids are among examples of natural graph classes that are fractionally treewidth-fragile, but not treewidth-fragile in the non-fractional sense; see [4, 2] for more details.
Fractional treewidth-fragility has applications in algorithmic design, especially regarding approximation algorithms (essentially, a set sampled from an -thin probability distribution on will likely intersect an optimal solution in only a small fraction of vertices, and one is often able to efficiently recover the large part of the solution contained in using the fact that this graph has small treewidth); see  for more details. Also, fractional treewidth-fragility implies sublinear separators (see [4, Lemma 14]), and in , I conjectured a converse: strongly sublinear separators imply fractional treewidth-fragility. If in Corollary 6, the treewidth of did not depend on the number of vertices of , this would imply this conjecture. As it stands, Corollary 6 only implies a weakening of the claim we now describe.
There exist non-decreasing functions and satisfying and , and a polynomial-time algorithm that, given an -vertex graph and integers , returns either an -bounded model of of depth in , or a -thin probability distribution on with support of size at most , where .
The proof of this theorem is given in Section 4.
In particular, for classes of graphs with strongly sublinear separators, Lemma 2 and Theorem 7 give the following. Let be a non-decreasing function. We say that an -vertex graph is fractionally -treewidth-fragile if for , there exists a -thin probability distribution on .
Let and be real numbers. There exists a non-decreasing function satisfying such that all graphs in are fractionally -treewidth-fragile. Furthermore, there exists a polynomial-time algorithm that for each -vertex graph and an integer either returns a -thin probability distribution on with support of size at most , or shows that .
By Lemma 2(b), the -expansion of is bounded by a non-decreasing function such that . For a given -vertex graph and an integer , let us apply the algorithm of Theorem 7 with . If the algorithm returns a model of of depth in , then , and thus . Otherwise, the algorithm returns a -thin probability distribution on with support of size at most , where
We can choose the function accordingly. ∎
Fractional -treewidth-fragility for the function from Corollary 8 does not necessarily imply strongly sublinear separators. However, it does imply existence of small separators in subgraphs of at least polylogarithmic size.
For a real number and a non-decreasing function satisfying , there exists a real number as follows. Suppose that an -vertex graph is fractionally -treewidth-fragile. If is a subgraph of with vertices, then has a balanced separator of order at most .
Let . For chosen at random from a -thin probability distribution, the expected value of is at most . Hence, there exists such that and has treewidth at most . Consequently, also has treewidth at most , and thus has a balanced separator of order at most , using the assumption that . We conclude that is a balanced separator in of order at most . ∎
3 Certification of strongly sublinear separators
Many of the polynomial-time algorithms for fractionally treewidth-fragile classes from  become only pseudopolynomial (with time complexity ) or worse when used for fractionally -treewidth-fragile graphs with . An exception is the following subgraph testing algorithm.
Let be a real number and let be a non-decreasing function satisfying . Let be an -vertex fractionally -treewidth-fragile graph, and suppose a -thin probability distribution on with support of size at most is given for all . Then it is possible to decide whether an -vertex graph is a subgraph of in time .
Clearly, we can assume that , as if , then the trivial algorithm testing all bijections from to suffices, and if , then the answer is always no.
Let . Let for some be the support of the -thin probability distribution on which we are given. If is a subgraph of , then for chosen at random from this distribution the expected value of is at most , and thus the probability that is non-zero. Hence, is a subgraph of if and only if is a subgraph of one of the graphs , …, . Since these graphs have treewidth at most , we can determine whether is a subgraph in each of them in time using a standard dynamic programming algorithm. ∎
Finally, we turn attention to the question of testing whether a graph belongs to a class with strongly sublinear separators. Testing exact membership in a class for some given is likely hard (determining the smallest size of a balanced separator is NP-hard , but this is a slightly different problem). It is possible to approximate the smallest size of a balanced separator , however it is not clear whether this is helpful, as to test the (approximate) membership of a graph in , one needs to verify that all (exponentially many) subgraphs of have small separators. Hence, the following result is of interest.
For every , there exist and a polynomial-time algorithm that for each input graph , determines either that , or that .
Let . First, we run the algorithm of Corollary 8 for . This either shows that , or gives us -thin probability distributions on with supports of size at most for , where . By Lemma 9, this shows that each subgraph of with vertices has a balanced separator of order .
Next, we test whether all subgraphs of with vertices belong to . Note that there are only non-isomorphic graphs with at most vertices; for each such graph , we can test whether it belongs to by brute-force testing all its induced subgraphs and subsets of their vertices in time ; and we can test whether in time according to Lemma 10. Hence, this part can be carried out in total time .
Now, if is a subgraph of with vertices, then according to the previous paragraph, all its subgraphs with at most vertices belong to , and by Corollary 4, has a balanced separator of order .
Consequently, each -vertex subgraph of has a balanced separator of order or depending on whether or not, and thus for some . ∎
The proof of Theorem 5 essentially follows the argument of Plotkin, Rao, and Smith , with a few minor modifications. Let us start with the key lemma, showing that either a graph contains a small cost separator, or it has bounded radius.
There exists a polynomial-time algorithm that, given a graph with at most vertices, integers , and an an assignment of positive costs to vertices of such that for every , returns either a vertex such that each other vertex is at distance at most from , or a partition of to parts , and such that there are no edges between and , , and .
If is not connected, then we can let , let be the vertex set of a component of and let . If , then we can return the only vertex of as . Hence, assume that is connected and has at least two vertices.
Let be a vertex of of maximum cost. For any integer , let , , and be the set of vertices of at distance exactly , less than , and more than from . Let be the maximum index such that . If there exists such that and , then we can return , and . Hence, we can assume that for , we have either or . In the former case, and . In the latter case, and .
Since and , the number of indices such that and satisfies . Since , we conclude . Since and , the number of indices such that and satisfies . Since , it follows that . Consequently, , and each vertex is at distance at most from ; hence, the algorithm can return . ∎
We are now ready to prove the theorem.
Proof of Theorem 5.
Let us define . Clearly, we can assume , since otherwise and we can return and equal to the vertex set of the largest component of (forming the support of a -bounded model of of depth ).
Let be the set of vertices such that . Clearly, . Let . The algorithm maintains a partition of the vertices of to sets , , and satisfying the following invariants.
if is the set of vertices of with a neighbor in , then , and
for some integer , is the support of a -bounded model of of depth in .
Initially, we have . We repeat the following steps.
If , we can return and stop. Hence, suppose that .
If , then we can return the subsets and of , which clearly satisfy the requirements of the theorem. Hence, suppose that , and thus and .
Let , …, be the parts of the partition forming the model of with support . If there exists such that and has no neighbors in , then let and . Since , this clearly preserves the invariants.
Hence, we can assume that for , the set of neighbors of in is non-empty. Let us apply Lemma 12 to with and . If the result is a vertex at distance at most from all other vertices of , then let be the union of the vertex sets of the shortest paths from to , …, and . Clearly, this preserves the invariants.
Finally, suppose that the result of invocation of the algorithm from Lemma 12 is a partition of into sets , , and such that , and , and there are no edges between and . By symmetry, we can assume that . We let . Note that the set of neighbors of this new set in is a subset of , and thus its cost is at most . Furthermore, denoting by the old set , we have , since . Hence, the invariants are again preserved.
Since in each step we either increase or decrease , and never decreases and never increases, this algorithm terminates in at most iterations. ∎
Proof of Theorem 7.
Let and be as in Corollary 6. Let us form a linear program as follows. For each , let us introduce a variable . Additionally, let be a variable. For each vertex , we have the following constraint:
An -thin probability distribution on exists if and only if the minimum subject to these constraints is at most .
Consider the dual to this linear program, with variables for and another variable :
Given any assignment of values such that , the algorithm of Corollary 6 either returns an -bounded model of of depth in (in which case we can return this model and stop), or a set such that . Run the ellipsoid method algorithm for the polytope defined by inequalities
using the algorithm of Corollary 6 as a separation oracle. Unless at some point we stop due to the discovery of an -bounded model of , the conclusion will necessarily be that this polytope is empty. During the run of the ellipsoid method algorithm, the algorithm of Corollary 6 will return polynomially many sets . Let . Clearly, the polytope defined by