1 Introduction and previous work
In this note, we initiate the study of Steiner tree instances that are stable to multiplicative perturbations to the distances in the underlying metric. Our analysis lies in the Bilu-Linial stability  setting, which provides a way to study tractable instances of NP-hard problems.
Instances that are -stable in the Bilu-Linial model have the property that the structure of the optimal solution is not only unique, but also does not change even when the underlying distances among the input points are perturbed by a multiplicative factor . In their original paper, Bilu and Linial analyzed MAX-CUT clustering, and since their seminal work, other problems have been analyzed including center-based clustering [4, 6, 7], multi-way cut problems , and metric TSP .111Bilu-Linial stability is one among other notions of data stability studied in the literature [1, 5]. This is in contrast to notions of of algorithmic stability, which focus on properties algorithms as opposed to data, see e.g. [2, 8, 11, 14].
Here, we look at the metric Steiner tree problemand also the more restricted Euclidean version. For general metrics, the Steiner tree problem is known to be APX-hard in the worst case . For the Euclidean metric, a PTAS is known .
In this paper we begin by providing strong geometric structural properties that need to be satisfied by stable instances. These point to the existence of algorithms for non-trivial families. We then make use of, and strengthen, these geometric properties to show that -stable instances of Euclidean Steiner trees are polynomial-time solvable. Finally, we discuss the connections between certain approximation algorithms and Bilu-Linial stability for Steiner trees.
2 Model and definitions
In this section, we recall the relevant definitions. Fist we define the Steiner tree problem, which is among Karp’s 21 original NP-hard problems . It has various applications including in network design, circuit layouts, and phylogenetic tree reconstruction.
Definition 1 (the Steiner tree problem).
For an undirected graph with edge weights for every edge , and a set of terminals. A Steiner tree is a tree in the graph that spans all terminal vertices and may contain some of the non-terminals (also called Steiner points). The goal is to find such a tree of lowest weight, which we call
We can assume without loss of generality222For any graph with distances specified on edges, a metric can be formed by taking the vertices to be points and considering the shortest path distances in the graph between pairs of points vertices. Solving (or approximating) the Steiner tree problem on a metric formed in this matter solves (or approximates) the problem on the original graph. See Vazirani  for further discussion of this issue. that the vertices are points in a metric space and the weights of the edges are given by the distance function – when the input is in the form of a metric, we call this the metric Steiner tree problem. Our results use properties of metric spaces, but move freely between the metric space and graph representations of the problem. When the metric is Euclidean, this is called the Euclidean Steiner tree problem.
Now we move on to defining Bilu-Linial stability for the Steiner tree problem on metrics.
Definition 2 (Bilu-Linial -stabile instances).
Let be an instance of a metric Steiner tree problem and . is -stable if for any function such that ,
the optimal Steiner tree under is equal to the optimal Steiner tree under .
We note that the perturbations can be such that instances originally satisfying the metric or Euclidean properties no longer have to satisfy these properties after perturbation. We also note that due to the triangle inequality, no instances have stability or greater in the metric setting.
Notation: For a graph , is the weight of edge in . We abbreviate and . Let denote the minimum weight Steiner tree of , let denote the weight of the Steiner tree.
3 Structural properties in general metrics
In this section, we work in the context of a general metric space, and we develop interesting restrictions on the types of problems with -stable solutions, for various values of
The techniques of this section do not give, in complete generality, an efficient algorithm for finding the optimal Steiner tree for any value of less than a problem we leave open. However, when more information about the metric space is available, one can use the structural results here to give restrictions on the arrangements of Steiner points which does yield a definitive solution. In particular,
More generally, in the case that no two Steiner points are adjacent in the optimal solution, Lemma 10 together with the other results of the section can be used to give an efficient and very simple algorithm to find the minimal weight Steiner tree. Other more general situations can be efficiently handled via only slightly more elaborate arguments - e.g. if one has a bound on the length of the longest path of Steiner points in the optimal solution.
The degree of any Steiner point in the optimal solution is greater than .
Consider a Steiner node in the optimal solution, that is connected to other points, . Let , and let and be such that . Let be obtained by perturbing each edge by a factor of . Let
Clearly, is also a Steiner tree. Using the fact that , we have
Using the fact that , we have
Rearranging, we have
Now we state some additional structural properties of optimal Steiner trees in -stable instances. These are not used in Section 4. Nevertheless, we hope that they are of independent interest.
If are nearest neighbors in the graph, then the edge is in the optimal solution.
Suppose , then
Assume w.l.o.g. . Suppose that , let be obtained by perturbing by a factor of . Then is also a Steiner tree in of weight contradicting stability. This completes the proof of
The proof of follows from and the fact that .
Let be obtained by perturbing by a factor of . Then is also a Steiner tree of weight
On the other hand, stability gives us that
Repeating the same argument but swapping for gives us .
Let be a subgraph of with at least one edge. Let Fix any vertex satisfying ; then we have .
If , then adding the edge to produces a cycle which includes edge Suppose that the cycle also includes . Let be obtained by perturbing by a factor of . Then is a Steiner tree of weight at most , contradicting stability.
If the cycle does not include , it includes some edge other than which has endpoint at . This edge, call it , is in . By Lemma 5, . Let be obtained by perturbing by a factor of . We have . Then is a Steiner tree of weight less than , again contradicting stability. ∎
Let Let , a subgraph of Suppose that is a vertex with , then .
Let Note that is some real number larger than If , then by part of Lemma 5, we must have
On the other hand,
We now have a contradiction as long as . The function is decreasing for and for any . So, we have that as desired. ∎
Let be a subgraph of with at least one edge. Suppose that and suppose that with . Then we must have and
Assume that . Assume that is a subgraph of with at least two vertices. Let Fix any vertex . Then we have if and only if .
Let be vertices (either terminal or Steiner points). We denote by the tree on vertex set in which is connected to each element of Let the average weight of be
Suppose that is a subgraph of . We call a terminal component fan relative to if is a Steiner point and are all terminals or vertices in distinct connected components of each with at least two vertices. We call the collection of components of together with the terminals not in the terminal components of .
Let and suppose that is a subgraph of and in the optimal solution, no two Steiner points are adjacent. Suppose that with is a terminal component fan such that:
the average weight of is less than all edges not in which connect two terminal components of ,
the average weight of is minimal among all terminal component fans,
the edges of are all within a factor of of each other.
Then is a subgraph of .
Suppose that the fan is not in Then for some subset of the edges of are not in - the components of that contain each are connected. Specifically, if there are edges of which are not in , then there are at least edges of such that in we may remove these edges and still have a Steiner tree.333In the case that , there may be only such edges, as may not be in , but the argument works identically in that case. Moreover, since no two Steiner points are adjacent, these edges are either
terminal to terminal edges, or
part of a terminal component fan.
In the first case, the terminal to terminal edges have weight at least In this case perturb this edge by a factor of , and swap it with one edge of the terminal component fan . Since the edges of are within a factor of of each other and their average weight is , this swap decreases of the weight of the resulting Steiner tree after the perturbation.
Similarly in the case that one of the edges is in another terminal component fan, , the average weight of edges in that fan is at least , and applying part of Lemma 5, the minimal weight edge in is at least Now, perturb such an edge by a factor of to make the weight at least which is larger than the weight of the largest weight edge of , which is a most because
Performing any of these swaps yields a lower weight Steiner tree than under the above perturbations, contradicting -stability. ∎
4 Euclidean Steiner trees
In this section, we consider the restriction of the Steiner tree problem to the Euclidean metric.
Under the assumption of stability the min angle between two terminal points can be defined as a function of .
Definition 11 (angle).
Let be points on a Euclidean metric. Then we call the angle between .
For a -stable instance of a Euclidean Steiner tree, the angle between two terminal points with respect to their common Steiner neighbor in the tree should be greater than .
Lets assume, for a -stable instances of Steiner tree, the angle between two terminal points , and at a Steiner point is . Without loss of generality, let . Clearly , since otherwise, perturbing edge by a factor of allows one to replace by in a minimal Steiner tree, contradicting stability. Let us use to denote the angle . Clearly, . Thus by the sine rule, we have
Rearranging, we have
as desired. ∎
Thus we immediately get the following Corollary.
For a -stable instances of Steiner tree, if then the angle between two terminal points is .
If there are points in such that the angle between every pair with respect to a point is at least , then .
Let and let
be unit vectors insuch that . Consider the matrix whose columns are the s. We know that by construction is positive semi-definite. But if , then the sum of every row is negative, which contradicts the positive semidefiniteness of , and so it must be the case that . ∎
For the degree of a Steiner node in the optimal solution is at most .
From Lemma 12 we have
and so or . Since , we have
When , the optimal Steiner tree for a -stable instance does not have Steiner nodes.
By solving the above equation for we get , which is bounded from above by . ∎
This geometric property implies that for -stable instances, Steiner points will not be used in the optimal solution. Hence, an MST algorithm on just the terminal points will give the answer in polynomial time.
Finally, we point to the existence of Gilbert and Pollak’s the Steiner ratio conjecture , which states that in the Euclidean plane, there always exists an MST within a cost of of the minimum Steiner tree,and the behavior of this ratio for higher dimensions is yet unknown. Assuming this conjecture, in certain cases it may imply some limitations on the stability of Euclidean instances, especially in low dimensions, using the idea that even if the Steiner tree distances are “blown up” by more than the Steiner ratio, one could instead use the MST instead and get a cheaper solution. Unfortunately, because the MST may overlap with the Steiner tree, we cannot give a concrete statement.
5 Using approximation algorithms to solve stable instances
In this section we give a general argument about how strong approximation algorithms for Steiner tree problems give stability guarantees. We note that it is known that an FPTAS for the Steiner tree would imply P=NP , so there is no hope to use the result below in the general metric case. But if at some future point an FPTAS for the Euclidean variant of the Steiner tree problem is developed (currently, only a PTAS is known to exist ), then this would immediately imply the existence of polynomial-time algorithms for stable instances for any constant .
An FPTAS for the Steiner tree problem gives a polynomial time algorithm for optimally solving any -stable Steiner tree problem in time poly. In particular, this gives a polynomial-time algorithm for any constant .
Assume we are given an FPTAS for the Steiner tree problem. This means that we have an algorithm that runs in time on instances of size to give -approximations to the optimum Steiner tree. Now consider a -stable instance for constant . We run our FPTAS on that instance with to get an Steiner tree with weight within . We now claim that must contain every edge in OPT whose weight is at least . Suppose it doesn’t – then we could perturb such an edge by and increase the weight of the optimal solution to and would become cheaper than , thereby violating -stability.
By the fractional pigeonhole principle, the most expensive edge of the FPTAS satisfies the desired property above and is therefore in . Hence, we can contract this edge into a new vertex and get a new instance with vertices at -stability. We can continue this process, getting one new edge of the optimal in each iteration, until we have a constant-size problem that we can brute-force. ∎
We note that the above technique could be used to convert event slightly weaker (than FPTAS) approximation algorithms to nontrivial stability guarantees.
M. Ackerman and S. Ben-David.
Clusterability: A theoretical study.
In D. A. V. Dyk and M. Welling, editors,
Proceedings of the Twelfth International Conference on Artificial Intelligence and Statistics, AISTATS 2009, Clearwater Beach, Florida, USA, April 16-18, 2009, volume 5 of JMLR Proceedings, pages 1–8. JMLR.org, 2009.
-  I. M. Alabdulmohsin. Algorithmic stability and uniform generalization. In C. Cortes, N. D. Lawrence, D. D. Lee, M. Sugiyama, and R. Garnett, editors, Advances in Neural Information Processing Systems 28: Annual Conference on Neural Information Processing Systems 2015, December 7-12, 2015, Montreal, Quebec, Canada, pages 19–27, 2015.
-  S. Arora. Polynomial time approximation schemes for euclidean traveling salesman and other geometric problems. J. ACM, 45(5):753–782, 1998.
-  P. Awasthi, A. Blum, and O. Sheffet. Center-based clustering under perturbation stability. Inf. Process. Lett., 112(1-2):49–54, 2012.
-  M. Balcan, A. Blum, and A. Gupta. Approximate clustering without the approximation. In C. Mathieu, editor, Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2009, New York, NY, USA, January 4-6, 2009, pages 1068–1077. SIAM, 2009.
-  M. Balcan and Y. Liang. Clustering under perturbation resilience. SIAM J. Comput., 45(1):102–155, 2016.
-  S. Ben-David and L. Reyzin. Data stability in clustering: A closer look. Theor. Comput. Sci., 558:51–61, 2014.
-  S. Ben-David, U. von Luxburg, and D. Pál. A sober look at clustering stability. In G. Lugosi and H. U. Simon, editors, Learning Theory, 19th Annual Conference on Learning Theory, COLT 2006, Pittsburgh, PA, USA, June 22-25, 2006, Proceedings, volume 4005 of Lecture Notes in Computer Science, pages 5–19. Springer, 2006.
-  Y. Bilu and N. Linial. Are stable instances easy? Comb. Probab. Comput., 21(5):643–660, 2012.
-  M. Chlebík and J. Chlebíková. The steiner tree problem on graphs: Inapproximability results. Theor. Comput. Sci., 406(3):207–214, 2008.
B. Fan, D. Ihara, N. Mohammadi, F. Sgherzi, A. Sidiropoulos, and M. Valizadeh.
Learning lines with ordinal constraints.
Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2020.
-  E. N. Gilbert and H. O. Pollak. Steiner minimal trees. SIAM Journal on Applied Mathematics, 16(1):1–29, 1968.
-  R. M. Karp. Reducibility among Combinatorial Problems, pages 85–103. Springer US, Boston, MA, 1972.
T. Liu, G. Lugosi, G. Neu, and D. Tao.
Algorithmic stability and hypothesis complexity.
In D. Precup and Y. W. Teh, editors,
Proceedings of the 34th International Conference on Machine Learning, ICML 2017, Sydney, NSW, Australia, 6-11 August 2017, volume 70 of Proceedings of Machine Learning Research, pages 2159–2167. PMLR, 2017.
-  K. Makarychev, Y. Makarychev, and A. Vijayaraghavan. Bilu-linial stable instances of max cut and minimum multiway cut. In C. Chekuri, editor, Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 890–906. SIAM, 2014.
-  M. Mihalák, M. Schöngens, R. Srámek, and P. Widmayer. On the complexity of the metric TSP under stability considerations. In I. Cerná, T. Gyimóthy, J. Hromkovic, K. G. Jeffery, R. Královic, M. Vukolic, and S. Wolf, editors, SOFSEM 2011: Theory and Practice of Computer Science - 37th Conference on Current Trends in Theory and Practice of Computer Science, Nový Smokovec, Slovakia, January 22-28, 2011. Proceedings, volume 6543 of Lecture Notes in Computer Science, pages 382–393. Springer, 2011.
-  V. V. Vazirani. Approximation Algorithms. Springer-Verlag, Berlin, Heidelberg, 2001.