Bilu-Linial stability, certified algorithms and the Independent Set problem

Bilu-Linial stability, certified algorithms and the Independent Set problem

Abstract

We study the notion of stability introduced by Bilu and Linial (2010) in the context of the classical Independent Set problem. A weighted instance of Independent Set is -stable if it has a unique optimal solution that remains the unique optimal solution under multiplicative perturbations of the weights by a factor of at most . In this work, we design algorithms for stable instances of Independent Set on several classes of graphs; we give algorithms for -stable instances on graphs of maximum degree , for -stable instances on -colorable graphs and for -stable instances on planar graphs, using standard LP relaxations and the Sherali-Adams hierarchy. We also prove a general structural result that shows that the integrality gap of convex relaxations of several maximization problems reduces dramatically on stable instances. For general graphs, we give an algorithm for -stable instances, for any fixed , and, on the negative side, we show that there are no efficient algorithms for -stable instances of Independent Set, assuming the planted clique conjecture. As a side note, we also exploit the connection between Vertex Cover (a problem equivalent to Independent Set with respect to exact solvability) and Node Multiway Cut, and give the first results about stable instances of Node Multiway Cut.

Moreover, we initiate the study of certified algorithms for Independent Set. The notion of a -certified algorithm was introduced very recently by Makarychev and Makarychev (2018), and is a class of -approximation algorithms that satisfy one crucial property: the solution returned is optimal for a perturbation of the original instance, where perturbations are again multiplicative up to a factor of (hence, such algorithms solve -stable instances optimally). Using the results of Makarychev and Makarychev (2018) about convex relaxations, as well as combinatorial techniques, we obtain -certified algorithms for Independent Set on graphs of maximum degree , and -certified algorithms on planar graphs. Finally, we analyze the algorithm of Berman and Fürer (1994) and prove that it is a -certified algorithm for Independent Set on graphs of maximum degree where all weights are equal to 1.

1 Introduction

The Maximum Independent Set problem is a central problem in theoretical computer science and has been the subject of numerous works over the last few decades. As a result we now have a thorough understanding of the worst-case behavior of the problem. In general graphs, the problem is -hard to approximate, assuming that  [34, 46], and -hard to approximate, assuming that  [37]. On the positive side, the current best algorithm is due to Feige [26] achieving a -approximation1. In order to circumvent the strong lower bounds, many works have focused on special classes of graphs, such as bounded-degree graphs (see e.g. [32, 1, 31, 33, 9, 10, 4, 20]), planar graphs [6] etc. In this work, we continue this long line of research and study the Maximum Weighted Independent Set problem (which, from now on, we denote as MIS) within the beyond worst-case analysis framework introduced by Bilu and Linial [16].

In an attempt to capture real-life instances of combinatorial optimization problems, Bilu and Linial [16] proposed a notion of stability, which we now define in the context of MIS.

Definition 1 (-perturbation [16]).

Let , , be an instance of MIS. An instance is a -perturbation of , for some parameter , if for every we have .

Definition 2 (-stability [16]).

Let , , be an instance of MIS. The instance is -stable, for some parameter , if:

  1. it has a unique maximum independent set ,

  2. every -perturbation of has a unique maximum independent set equal to .

This definition of stability is motivated by the observation that in many real-life instances, the optimal solution stands out from the rest of the solution space, and thus is not sensitive to small perturbations of the parameters. This suggests that the optimal solution does not change (structurally) if the parameters of the instance are perturbed (even adversarially). Observe that the larger the so-called stability threshold is, the more severe the restrictions imposed on the instance are. Thus, the main goal in this framework is to recover (exactly and in polynomial time) the unique optimal solution for as small as possible. We note that perturbations are scale-invariant, and thus, it is sufficient to only consider perturbations that scale up and never down. We also observe that an algorithm that solves -stable instances of MIS also solves -stable instances of minimum Vertex Cover, as the two problems are equivalent with respect to exact solvability and the notion of Bilu-Linial stability.

Although stability was introduced in the context of Max Cut [16], the authors note that it naturally extends to many optimization problems, such as the MIS problem, and, moreover, they observe that the greedy algorithm for MIS solves -stable instances of graphs of maximum degree . The work of Bilu and Linial has inspired a sequence of works about stable instances of problems such as Max Cut [15, 41], Edge Multiway Cut [41, 3], TSP [44], and clustering problems such as -median, -means, and -center [5, 8, 7, 3, 23, 25, 29].

Prior works on stability have also studied robust algorithms for stable instances [41, 3]; these are algorithms that either output an optimal solution or provide a polynomial-time verifiable certificate that the instance is not -stable; see Section 2 for a formal definition. Motivated by the notion of stability above, in a recent work, Makarychev and Makarychev [40] introduced an intriguing class of algorithms, namely -certified algorithms.

Definition 3 (-certified algorithm [40]).

An algorithm for MIS is called -certified, for some parameter , if for every instance , , it computes

  1. a feasible independent set of ,

  2. a -perturbation of such that is a maximum independent set of .

We highlight here that a certified algorithm works for every instance of the problem. Note that if the instance is -stable, then the solution returned will be optimal for the original instance, while if it is not stable, the solution will be within a -factor of optimal for the original instance, and, so, a -certified algorithm is also a -approximation algorithm.

In this work our focus is on understanding the complexity of stable instances of MIS, with an emphasis on designing robust and certified algorithms. From a practical point of view, designing algorithms for -stable instances for small values of is highly desirable. Unfortunately, for general stable instances of MIS this is not always possible. For example, the recent work of [3] shows that there is no robust algorithm for -stable instances of MIS on general graphs (unbounded degree), assuming that . In this work, we go a step further and prove a lower bound that holds for any algorithm and not just robust algorithms. In particular, our lower bound follows from the fact that a random graph drawn from with a -sized planted independent set will be -stable with high probability. Hence, assuming the planted clique conjecture, we do not expect polynomial-time algorithms for -stable instances of the problem on general graphs. To the best of our knowledge, this is only the second case of a lower bound for stable instances of a graph optimization problem that applies to any polynomial-time algorithm as opposed to only robust algorithms [41, 3] (the first being the very strong lower bound for Max -Cut [41]). As a result, our focus is on special classes of graphs, such as bounded-degree graphs, small chromatic number graphs and planar graphs, where we prove that one can indeed handle small values of the stability parameter. Nevertheless, we do provide an algorithm for stable instances of MIS on general graphs as well.

Motivation. Stability is especially natural to consider for optimization problems where the given objective function may be a proxy for a true goal of identifying a hidden correct solution. For the Independent Set problem, a natural such scenario is applying a machine learning algorithm in the presence of pairwise constraints. Consider, for instance, an algorithm that scans news articles on the web and aims to extract events such as “athlete X won the Olympic gold medal in Y”. For each such statement, the algorithm gives a confidence score (e.g., it might be more confident if it saw this listed in a table rather than inferring it from a free-text sentence that the algorithm might have misunderstood). But in addition, the algorithm might also know logical constraints such as “at most one person can win a gold medal in any given event”. These logical constraints would then become edges in a graph, and the goal of finding the most likely combination of events would become a maximum Independent Set problem. Stability would be natural to assume in such a setting since the exact confidence weights are somewhat heuristic, and the goal is to recover an underlying ground truth. It is also easy to see the usefulness of a certified algorithm in this setting. Given a certified algorithm for Independent Set that outputs a -perturbation, the user of the machine learning algorithm can further test and debug the system by trying to gather evidence for events on which the perturbation puts higher weight.

Related Work. As mentioned, there have been many works about the worst-case complexity of MIS and the best known approximation algorithm due to Feige [26] achieves a factor of . For degree- graphs, Halperin [33] designed an -approximation algorithm. The MIS problem has also been studied from the lens of beyond worst-case analysis. In the case of random graphs with a planted independent set, the problem is equivalent to the classic planted clique problem. Inspired by semi-random models of [17], Feige and Killian [27] designed SDP-based algorithms for computing large independent sets in semi-random graphs.

The notion of stability introduced by [16] goes beyond random and semi-random models and instead proposes looking at deterministic conditions on the graph that give rise to non worst-case, real-life instances. The study of this notion has led to insights into the complexity of many different problems in optimization and machine learning. In the context of the MIS problem, Bilu [14] analyzed the greedy algorithm and showed that it recovers the optimal solution for -stable instances of graphs of maximum degree . The same result is also a corollary of a general theorem about the greedy algorithm and -extendible independence systems proved by Chatziafratis et al. [22]. Finally, we would like to mention that there has also been work on studying MIS under adversarial perturbations to the graph [39, 21, 11].

Our results. In this work, we explore the notion of stability in the context of MIS and significantly improve our understanding of the problem’s behavior on stable instances. In particular, using both combinatorial and LP-based methods, we design several algorithms for stable instances of MIS for different classes of graphs. Moreover, we initiate the systematic study of certified algorithms for MIS. More concretely, we obtain the following results.

  • Independent set on planar graphs: We show that on planar graphs, any constant stability suffices to solve the problem exactly in polynomial time. More precisely, we provide robust algorithms for -stable instances of planar MIS, as well as -certified algorithms, for any fixed .

  • Independent set on graphs of bounded degree or small chromatic number: We provide a robust algorithm for solving -stable instances of MIS on graphs of maximum degree , as well as a robust algorithm for -stable instances of MIS on -colorable graphs. While these results are based on convex relaxations, we also show that the standard greedy algorithm is a -certified algorithm for weighted MIS and the algorithm of Berman and Fürer (1994) is a -certified algorithm for unweighted instances of MIS, both on graphs of maximum degree . Finally, we design a practical greedy -certified algorithm for MIS on graphs of maximum degree , slightly improving upon the standard greedy algorithm, especially for small values of .

  • Independent set on general graphs: For general graphs, we present an algorithm for -stable instances of MIS on vertices, for any fixed . Furthermore we show that solving -stable instances of MIS is hard assuming the hardness of finding maximum cliques in a random graph. This is in contrast with most existing lower bounds for stable instances of graph problems that only apply to robust algorithms [41, 3].

  • Convex relaxations and stability: We present a structural result about the integrality gap of convex relaxations of several maximization problems on stable instances: if the integrality gap of a convex relaxation is at most , then it is at most for -stable instances, for any . This implies -estimation algorithms for -stable instances.

  • Node Multiway Cut: We present the first results on stable instances of the Node Multiway Cut problem, a strict generalization of the well-studied Edge Multiway Cut problem, and a problem intimately related to Vertex Cover (and thus, to MIS). In particular, we give a robust algorithm for -stable instances, where is the number of terminals, and also show that all known negative results on stable instances of MIS directly apply to Node Multiway Cut as well.

Organization of material. In Section 2, we formally define the related notions. In Section 3, we present robust algorithms for stable instances of MIS on special classes of graphs, such as bounded-degree graphs, planar graphs, and graphs with small chromatic number. For general graphs we give a (non-robust) algorithm for stable instances in Section 4. In Section 5, we prove that the integrality gap of convex relaxations of several maximization problems drops dramatically on stable instances, thus implying estimation algorithms for such instances. In Section 6, we discuss a strong lower bound about solving stable instances of MIS with any kind (robust or non-robust) of polynomial-time algorithm. In Section 7, we discuss our results for the Node Multiway Cut problem. In Section 8 we discuss various certified algorithms for MIS. We conclude with some discussion and open problems in Section 9.

2 Preliminaries and definitions

2.1 Perturbations and Bilu-Linial stability

The definition of Bilu-Linial stability was already given in Definition 2. We now give an equivalent characterization of -stability, first given in [16]. We introduce the notation for any subset and any weight function .

Lemma 4 ([16]).

Let , , and let be a maximum independent set of . The instance is -stable, for some parameter , iff for every feasible independent set .

For completeness, we include the proof in Appendix A.1. Given a -stable instance, our goal is to design polynomial-time algorithms that recover the unique optimal solution, for as small as possible. A special class of such algorithms that is of particular interest is the class of robust algorithms. The notion was introduced by Makarychev et al. [41].

Definition 5 (robust algorithm [41]).

Let , , be an instance of MIS. An algorithm is a robust algorithm for -stable instances if:

  1. it always returns the unique optimal solution of , when is -stable,

  2. it either returns an optimal solution of or reports that is not stable, when is not -stable.

Note that a robust algorithm is not allowed to err, while a non-robust algorithm is allowed to return a suboptimal solution, if the instance is not -stable. As already mentioned, strong lower bounds are known for the existence of robust algorithms for MIS on general graphs [3].

Finally, we define the notion of weak stability [41], which generalizes the notion of stability. An instance of MIS is weakly-stable if there is a unique optimal solution and a set of feasible solutions with , such that remains strictly better in every -perturbation than any independent set . The set can be thought of as a neighborhood of feasible solutions of , and the definition in that case implies that the optimal solution might change, but not too much (i.e. it will be some element of ). The algorithmic task here is to find a solution .

Definition 6 (weak stability [41]).

Let be an instance of MIS with a unique optimal solution . Let be a set of feasible independent sets of such that , and let . The instance is -weakly stable if for every -perturbation , we have , for every independent set . Equivalently, the instance is -weakly stable if for every independent set .

Observe that a -stable instance of MIS whose optimal solution is is -weakly stable.

2.2 Certified algorithms

The notion of certified algorithms (Definition 3) was very recently introduced by Makarychev and Makarychev [40]. It is easy to observe the following.

Observation 7 ([40]).

A -certified algorithm returns the unique optimal solution, when run on a -stable instance.

We would like to stress here that not all algorithms for stable instances are certified, so there is no equivalence between the two notions. Some examples that demonstrate this fact (communicated to us by Yury Makarychev [42]) include the algorithm for stable instances of TSP [44], the Goemans-Williamson rounding for the Max Cut SDP (strengthened with triangle inequalities) that solves stable instances, and the algorithms for stable instances of various clustering objectives. In all these cases, the algorithms solve stable instances optimally but are not certified. Thus, designing a certified algorithm is, potentially, a harder task than designing an algorithm for stable instances.

Observation 8 ([40]).

A -certified algorithm for MIS is also a -certified algorithm for Vertex Cover, and vice versa.

An implication of Definition 3 is that a -certified algorithm for MIS is a -approximation algorithm for MIS and Vertex Cover.

Corollary 9 ([40]).

Let be a -certified algorithm for MIS. Then, is a -approximation algorithm for MIS and Vertex Cover.

The proof can be found in Appendix A.1. We now present an equivalent characterization for a -certified algorithm. From now on, if an algorithm for MIS does not return a -perturbation but only a feasible solution , it can always be assumed to be a “candidate” -certified algorithm that returns , where with for and , otherwise.

Lemma 10 ([40]).

Let be an instance of MIS. Let be the independent set returned by an algorithm . Then, is -certified iff for every independent set of .

The proof can be found in Appendix A.1. Finally, it is easy to prove that a -certified algorithm solves -weakly stable instances, and so, the certified algorithms of Section 8 can also be used to solve weakly-stable instances.

Observation 11 ([40]).

A -certified algorithm for MIS returns a solution , when run on a -weakly stable instance. The algorithm does not know .

3 Robust algorithms for stable instances of Mis

In the next few sections, we obtain robust algorithms for stable instances of MIS by using the standard LP relaxation and the Sherali-Adams hierarchy. Since there are strong lower bounds for robust algorithms on general graphs [3], we focus on special classes of graphs, such as graphs with small chromatic number, bounded-degree graphs and planar graphs.

3.1 Convex relaxations and robust algorithms

In order to design robust algorithms, we use convex relaxations of MIS. An important component is the structural result of Makarychev et al. [41], that gives sufficient conditions for the integrality of convex relaxations on stable instances. We now introduce a definition and then restate their theorem in the setting of MIS.

Definition 12 (-rounding).

Let be a feasible fractional solution of a convex relaxation of MIS whose objective value for an instance is . A randomized rounding scheme for is an -rounding, for some parameters , if it always returns a feasible independent set , such that the following two properties hold for every vertex :

  1. ,

  2. .

Theorem 13 ([41]).

Let be an optimal fractional solution of a convex relaxation of MIS whose objective value for an instance is . Suppose that there exists an -rounding for , for some . Then, is integral for -stable instances.

For completeness, the proof of Theorem 13 is given in Appendix A.2. The theorem suggests a simple robust algorithm: solve the relaxation, and if the solution is integral, report it, otherwise report that the instance is not stable (observe that the rounding scheme is used only in the analysis).

In the next section, we study a simple rounding scheme for the standard LP for MIS, and prove that it satisfies the properties of the theorem. The standard LP for MIS for a graph has an indicator variable for each vertex , and is given in Figure 1. It is well known that the polytope is half-integral [45], and thus there always exists an optimal solution (that can be found efficiently) that satisfies for every vertex . This will prove very useful for designing -rounding schemes, since it allows us to consider randomized combinatorial algorithms and present them as rounding schemes, as long as they “respect” the integral part of the LP (i.e. they never pick a vertex if and they always pick a vertex if ).

s.t.:
Figure 1: The standard LP relaxation for MIS.

3.2 A robust algorithm for -stable instances of Mis on -colorable graphs

In this section, we give a robust algorithm for -stable instances of MIS on -colorable graphs. The crucial observation that we make is that, since the rounding scheme in Theorem 13 is only used in the analysis and is not part of the algorithm, it can be an exponential-time rounding scheme. Let be a -colorable graph, and let be an optimal half-integral solution. Let , and . We consider the following rounding scheme of Hochbaum [35] (see Algorithm 1). We use the notation .

  • 1.

    Let be the induced graph on the set .

  • 2.

    Compute a -coloring of .

  • 3.

    Pick uniformly at random from the set , and set .

  • 4.

    Return .

  • Algorithm 1 Hochbaum’s -colorable rounding scheme
    Theorem 14.

    Let be a -colorable graph. Given an optimal half-integral solution , the rounding scheme of Algorithm 1 is a -rounding for .

    Proof.

    It is easy to see that the rounding scheme always returns a feasible solution, since there is no edge between and and is a valid coloring. For , we have and . Similarly, for , we have and . Let . We have and . The result follows. ∎

    Theorems 13 and 14 now imply the following theorem.

    Theorem 15.

    The standard LP for MIS is integral for -stable instances of -colorable graphs.

    It is easy to see that this result is tight. For that, we fix and for any , we consider a clique of vertices , and vertices that are of degree 1, and whose only neighbor is . We set , for every , and for every . It is easy to see that the instance is -colorable, -stable and its unique optimal solution is . We now consider the fractional solution that assigns to every vertex; its cost is and thus, the integrality gap is strictly larger than 1.

    It is a well-known fact that the chromatic number of a graph of maximum degree is at most . Thus, the above result implies a robust algorithm for -stable instances of graphs of maximum degree , giving a robust analog of the result of Bilu [14] about the greedy algorithm. We will now see how we can slightly improve upon it by using Theorem 15 and Brook’s theorem [19].

    Theorem 16 (Brook’s theorem [19]).

    The chromatic number of a graph is at most the maximum degree , unless the graph is complete or an odd cycle, in which case it is .

    MIS is easy to compute on cliques and cycles. Thus, by Brook’s theorem, every interesting instance of maximum degree is -colorable. More formally, we can state the following theorem.

    Theorem 17.

    There exists a robust algorithm for -stable instances of MIS, where is the maximum degree.

    Proof.

    The algorithm is very simple. If , the graph is a collection of paths and cycles, and we can find the optimal solution in polynomial time. Let’s assume that . In this case, we first separately solve all disjoint components, if any (we pick the heaviest vertex of each ), and then solve the standard LP on the remaining graph (whose stability is the same as the stability of the whole graph). By Brook’s theorem, the remaining graph is -colorable. If the LP is integral, we return the solution for the whole graph, otherwise we report that the instance is not stable. ∎

    3.3 Robust algorithms for -stable instances of Mis on planar graphs

    In this section, we design a robust algorithm for -stable instances of MIS on planar graphs. We note that Theorem 15 already implies a robust algorithm for -stable instances of planar MIS, but we will use the Sherali-Adams hierarchy (which we denote as SA from now on) to reduce this threshold down to , for any fixed . In particular, we show that rounds of SA suffice to optimally solve -stable instances of MIS on planar graphs. We will not introduce the SA hierarchy formally, and we refer the reader to the many available surveys about LP/SDP hierarchies (see e.g. [24]). The -th level of the SA relaxation for MIS has a variable for every subset of vertices of size at most , whose intended value is , where is the indicator variable of whether belongs to the independent set. The relaxation has size , and thus can be solved in time . For completeness, we give the relaxation in Figure 2.

    s.t.:
    Figure 2: The Sherali-Adams relaxation for Independent Set.

    Our starting point is the work of Magen and Moharrami [39], which gives a SA-based PTAS for MIS on planar graphs, inspired by Baker’s technique [6]. In particular, [39] gives a rounding scheme for the -th round of SA that returns a -approximation. In this section, we slightly modify and analyze their rounding scheme, and prove that it satisfies the conditions of Theorem 13. For that, we need a theorem of Bienstock and Ozbay [13]. For any subgraph of a graph , let denote the set of vertices contained in .

    Theorem 18 ([13]).

    Let and be a feasible vector for the -th level SA relaxation of the standard Independent Set LP for a graph . Then, for any subgraph of of treewidth at most , the vector is a convex combination of independent sets of .

    The above theorem implies that the -th level SA polytope is equal to the convex hull of all independent sets of the graph, when the graph has treewidth at most .

    The rounding scheme of Magen and Moharrami [39]. Let be a planar graph and be an optimal -th level solution of SA. We denote as , for any . We first fix a planar embedding of . can then be naturally partitioned into sets , for some , where is the set of vertices in the boundary of the outerface, is the set of vertices in the boundary of the outerface after is removed, and so on. Note that for any edge , we have and with . We will assume that , since, otherwise, the graph is at most -outerplanar and the problem can then be solved optimally [6].

    Following [6], we fix a parameter , and for every , we define . We now pick an index uniformly at random. Let , and for , , where for a subset , is the induced subgraph on . Observe that every edge and vertex of appears in one or two of the subgraphs , and every vertex appears in exactly one .

    Magen and Moharrami observe that for every subgraph , the set of vectors is a feasible solution for the -th level SA relaxation of the graph . This is easy to see, as the Independent Set LP associated with is weaker than the LP associated with (on all common variables), since is a subgraph of , and this extends to SA as well. We need one more observation. In [18], it is proved that a -outerplanar graph has treewidth at most . By construction, each graph is a -outerplanar graph. Thus, by setting , Theorem 18 implies that the vector (we remind the reader that ) can be written as a convex combination of independent sets of .

    Let be the corresponding distribution of independent sets of , implied by the fractional solution . We now consider the following rounding scheme, which always returns a feasible independent set of the whole graph. For each , we (independently) sample an independent set of according to the distribution . Each vertex belongs to exactly one graph and is included in the final independent set if . A vertex might belong to two different graphs , and so, it is included in the final independent set only if . The algorithm then returns .

    Before analyzing the algorithm, we note that standard tree-decomposition based arguments show that the rounding is constructive (i.e. polynomial-time; this fact is not needed for the algorithm for stable instances of planar MIS, but will be used when designing certified algorithms).

    Theorem 19.

    The above randomized rounding scheme always returns a feasible independent set , such that for every vertex ,

    1. ,

    2. .

    Proof.

    It is easy to see that is always a feasible independent set. We now compute the corresponding probabilities. Since the marginal probability of on a vertex is , for any fixed , for every vertex , we have , and for every vertex , we have . Since is picked uniformly at random, each vertex belongs to with probability exactly equal to . Thus, we conclude that for every vertex , we have , and . ∎

    The above theorem implies that the rounding scheme is a -rounding. The following theorem now is a direct consequence of Theorems 13 and 19.

    Theorem 20.

    For every , the SA relaxation of rounds is integral for -stable instances of MIS on planar graphs.

    Proof.

    For any given , by Theorem 19, the rounding scheme always returns a feasible independent set of that satisfies and for every vertex . By Theorem 13, this means that must be integral for -stable instances. For any fixed , by setting , we get that rounds of Sherali-Adams return an integral solution for -stable instances of MIS on planar graphs. ∎

    4 Algorithms for stable instances of Mis on general graphs

    In this section, we use our algorithm for -stable instances of -colorable graphs and the standard greedy algorithm as subroutines to solve -stable instances on graphs of vertices, in time . Thus, from now on we assume that is a fixed constant. Before presenting our algorithm, we will prove a few lemmas. For any graph , let be its chromatic number.

    Lemma 21 (Welsh-Powell coloring).

    Let be a graph, with , and let be the sequence of its degrees in non-increasing order. Then, .

    The above lemma states a well-known fact; for completeness we present its proof in Appendix A.4.

    Lemma 22.

    Let be a graph, with . Then, for any natural number , one of the following two properties is true:

    1. , or

    2. there are at least vertices in whose degree is at least .

    Proof.

    Suppose that . Let be the vertices of , with corresponding degrees . It is easy to see that . We now observe that if , then we would have , and thus, by Lemma 21, we would get that , which is a contradiction. We conclude that we must have , which, since the vertices are ordered in decreasing order of their degrees, implies that there are at least vertices whose degree is at least . ∎

    Lemma 23.

    Let be -stable instance of MIS whose optimal independent set is . Then, is -stable, for any set .

    Proof.

    Fix a subset . It is easy to see that any independent set of is an independent set of the original graph . Let’s assume that is not -stable, i.e. there exists a -perturbation such that is a maximum independent set of . This means that . By extending the perturbation to the whole vertex set (simply by not perturbing the weights of the vertices of ), we get a valid -perturbation for the original graph such that is at least as large as . Thus, we get a contradiction. ∎

    Since we use the standard greedy algorithm as a subroutine, we state the algorithm here (see Algorithm 2). From now on, we denote the neighborhood of a vertex of a graph as .

  • 1.

    Let and .

  • 2.

    while :

  •        Pick .
           Set and .
  • 3.

    Return .

  • Algorithm 2 The greedy algorithm for MIS

    Bilu [14] proved the following theorem.

    Theorem 24 ([14]).

    The greedy algorithm (see Algorithm 2) solves -stable instances of MIS on graphs of maximum degree .

    We will now present an algorithm for -stable instances of graphs with vertices, for any natural number , that runs in time . Thus, by setting , for any , we can solve -stable instances of MIS with vertices, in total time . Let be a -stable instance of MIS, where . The algorithm is defined recursively (see Algorithm 3).

    Unbounded-Degree-Alg:
  • 1.

    If , run greedy algorithm (Algorithm 2) on , report solution and exit.

  • 2.

    Solve standard LP relaxation for and obtain (fractional) solution .

  • 3.

    If is integral, report solution and exit.

  • 4.

    Let , and for each , let .

  • 5.

    For each , run Unbounded-Degree-Alg and obtain solution .

  • Set .
  • 6.

    Let and run Unbounded-Degree-Alg to obtain solution .

  • 7.

    Return the maximum independent set among and .

  • Algorithm 3 The algorithm for -stable instances of MIS
    Theorem 25.

    Algorithm 3 solves -stable instances of MIS on graphs of vertices in time .

    Proof.

    We will prove the theorem using induction on . Let , , be a -stable instance whose optimal independent set is . If , Theorem 24 shows that the greedy algorithm computes the optimal solution (by setting ), and thus our algorithm is correct.

    Let , and let’s assume that the algorithm correctly solve -stable instances of graphs with vertices, for any . We will show that it also correctly solves -stable instances. Lemma 22 implies that either the chromatic number of is at most , or there are at least whose degree is at least . If the chromatic number is at most , then, Theorem 15 implies that the standard LP relaxation is integral if is -stable. We have . Thus, in this case, the LP will be integral and the algorithm will terminate at step (3), returning the optimal solution.

    So, let’s assume that the LP is not integral for , which means that the chromatic number of the graph is strictly larger than . Thus, the set of vertices has size at least . Fix a vertex . If , then, by Lemma 29, we get that is -stable, and moreover, , where is the optimal independent set of . Note that has at most vertices. It is easy to verify that , which implies that is a -stable instance with vertices. Thus, by the inductive hypothesis, the algorithm will compute its optimal independent set .

    There is only one case remaining, and this is the case where . In this case, by Lemma 23, we get that is -stable. There are at most vertices in , and so, by a similar argument as above, the graph is a -stable instance with vertices; by the inductive hypothesis, the algorithm will compute its optimal independent set.

    It is clear now that, since the algorithm always picks the best possible independent set, then at step (7) it will return the optimal independent set of . This concludes the induction and shows that our algorithm is correct. Regarding the running time, it is quite easy to see that we have at most levels of recursion, and at any level, each subproblem gives rise to at most new subproblems. Thus, the total running time is bounded by . ∎

    5 Stability and integrality gaps of convex relaxations

    In this section, we state a general theorem about the integrality gap of convex relaxations of maximization problems on stable instances. As already stated, Theorem 13 ([41]) was the first result that analyzed the performance of convex relaxations on stable instances, and gave sufficient conditions for the integrality gap to be 1. Here, we show that, even if the conditions of Theorem 13 are not satisfied, the integrality gap still significantly decreases as stability increases.

    Theorem 26.

    Consider a convex relaxation for MIS that assigns a value to every vertex of a graph , such that its objective function is . Let be its integrality gap, for some . Then, the relaxation has integrality gap at most for -stable instances, for any .

    Proof.

    Let be an -stable instance, let denote its (unique) optimal independent set and be its cost. We assume that is such that (otherwise the statement is trivial), which holds for . Let be the optimal value of the relaxation, and let’s assume that .

    We claim that . To see this, suppose that . We have , which implies that . This contradicts our assumption, and so we conclude that . This implies that .

    We now consider the induced graph . Let be an optimal independent set of . We observe that the restriction of the fractional solution to the vertices of is a feasible solution for the corresponding relaxation for . Since the integrality gap is always at most , we have . Finally, we observe that is a feasible independent set of the graph . By Lemma 4, , which gives . Combining the above inequalities, we conclude that . Thus, we get a contradiction. ∎

    The above result is inherently non-constructive. Nevertheless, it suggests estimation algorithms for stable instances of MIS, such as the following.

    Corollary 27 (Bansal et al. [10] + Theorem 26).

    For any fixed , the Lovasz -function SDP relaxation has integrality gap at most on -stable instances of MIS of maximum degree , where the notation hides some factors.

    We note that the theorem naturally extends to many other maximization graph problems, and is particularly interesting for relaxations that require super-constant stability for the recovery of the optimal solution (e.g. the Max Cut SDP has integrality gap for -stable instances although the integrality gap drops to exactly 1 for -stable instances).

    In general, such a theorem is not expected to hold for minimization problems, but, in our case, MIS gives rise to its complementary minimization problem, the minimum Vertex Cover problem, and it turns out that we can prove a very similar result for Vertex Cover as well.

    Theorem 28.

    Suppose that there exists a convex relaxation of MIS whose objective function is and its integrality gap (w.r.t. MIS) is . Then, there exists a -estimation algorithm for -stable instances of Vertex Cover, for any .

    For its proof, we need the following lemma.

    Lemma 29.

    Let be a -stable instance of MIS whose optimal independent set is . Let . Then, the instance is -stable, and its maximum independent set is .

    Proof.

    It is easy to see that is a maximum independent set of . We will now prove that the instance is -stable. Let’s assume that there exists a perturbation of such that is a maximum independent set of . This means that . We now extend to the whole vertex set by setting for every . It is easy to verify that is now a -perturbation for . Observe that is a feasible independent set of , and we now have . Thus, we get a contradiction. ∎

    The proof of Theorem 28 uses the above lemma and a standard trick that exploits the half-integrality property and can be found in Appendix A.3. We can now easily obtain the following corollary.

    Corollary 30 (Bansal et al. [10] + Theorem 28).