Lasserre Integrality Gaps for Graph Spanners and Related Problems

05/17/2019
by   Michael Dinitz, et al.
0

There has been significant recent progress on algorithms for approximating graph spanners, i.e., algorithms which approximate the best spanner for a given input graph. Essentially all of these algorithms use the same basic LP relaxation, so a variety of papers have studied the limitations of this approach and proved integrality gaps for this LP in a variety of settings. We extend these results by showing that even the strongest lift-and-project methods cannot help significantly, by proving polynomial integrality gaps even for n^Ω(ϵ) levels of the Lasserre hierarchy, for both the directed and undirected spanner problems. We also extend these integrality gaps to related problems, notably Directed Steiner Network and Shallow-Light Steiner Network.

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1 Introduction

A spanner is a subgraph which approximately preserves distances: formally, a -spanner of a graph is a subgraph such that for all (where and denote shortest-path distances in and respectively). Since is a subgraph it is also the case that , and thus a -spanner preserves all distances up to a multiplicative factor of , which is known as the stretch. Graph spanners originally appeared in the context of distributed computing [27, 28], but have since been used as fundamental building blocks in applications ranging from routing in computer networks [33] to property testing of functions [6] to parallel algorithms [20].

Most work on graph spanners has focused on tradeoffs between various parameters, particularly the size (number of edges) and the stretch. Most notably, a seminal result of Althöfer et al. [1] is that every graph admits a -spanner with at most edges, for every integer . This tradeoff is also known to be tight, assuming the Erdős girth conjecture [19], but extensions to this fundamental result have resulted in an enormous literature on graph spanners.

Alongside this work on tradeoffs, there has been a line of work on optimizing spanners. In this line of work, we are usually given a graph and a value , and are asked to find the -spanner of with the fewest number of edges. If is undirected then this is known as Basic -Spanner, while if is directed then this is known as Directed -Spanner. The best known approximation for Directed -Spanner is an -approximation [4], while for Basic -Spanner the best known approximations are when  [4] and when  [17], and when . Note that this approximation for is directly from the result of [1] by using the trivial fact that the optimal solution is always at least (in a connected graph), and so is in a sense “generic” as both the upper bound and the lower bound are universal, rather than applying to the particular input graph.

One feature of the algorithms of [4, 17], as well as earlier work [15] and extensions to related settings (such as approximating fault-tolerant spanners [16, 17] and minimizing the maximum degree [9]), is that they all use some variant of the same basic LP: a flow-based relaxation originally introduced for spanners by [15]. The result of [4] uses a slightly different LP (based on cuts rather than flows), but it is easy to show that the LP of [4] is no stronger than the LP of [15].

The fact that for Basic -Spanner we cannot do better than the “generic” bound when , as well as the common use of a standard LP relaxation, naturally gives rise to a few questions. Is it possible to do better than the generic bound when ? Can this be achieved with the basic LP? Can we analyze this LP better to get improvements for Directed -Spanner? In other words: what is the power of convex relaxations for spanner problems? It seems particularly promising to use lift-and-project methods to try for stronger LP relaxations, since one of the very few spanner approximations that uses a different LP relaxation was the use of the Sherali-Adams hierarchy to give an approximation algorithm for the Lowest Degree -Spanner problem [13].

It has been known since [18, 22] that Directed -Spanner does not admit an approximation better than for any constant , and it was more recently shown in [14] that Basic -Spanner cannot be approximated any better than for any constant . Thus no convex relaxation, and in particular the basic LP, can do better than these bounds. But it is possible to prove stronger integrality gaps: it was shown in [15] that the integrality gap of the basic LP for Directed -Spanner is at least , while in [17] it was shown that the basic LP for Basic -Spanner has an integrality gap of at least , which nearly matches the generic upper bound (particularly for large ).

But this left open a tantalizing prospect: perhaps there are stronger relaxations which could be used to get improved approximation bounds. Of course, the hardness of approximation results prove a limit to this. But even with the known hardness results and integrality gaps, it is possible that there is, say, an -approximation for Directed -Spanner and an -approximation for Basic -Spanner that uses more advanced relaxations.

1.1 Our Results and Techniques

This is the problem which we investigate: can we design stronger relaxations for spanners and related problems? While we cannot rule out all possible relaxations, we show that an extremely powerful lift-and-project technique, the Lasserre hierarchy [23], does not give relaxations which are massively better than the basic LP. This is true despite the fact that Lasserre is an SDP hierarchy rather than an LP hierarchy, and despite the fact that we allow a polynomial number of levels in the hierarchy even though it can only be efficiently solved for a constant number of levels. And since the Lasserre hierarchy is at least as strong as other hierarchies such as the Sherali-Adams hierarchy [32] and the Lovasz-Schrijver hierarchy [25], our results also imply integrality gaps for these hierarchies.

Slightly more formally, we first rewrite the basic LP in a way that is similar to [4] but is equivalent to the stronger original formulation [15]. This makes the Lasserre lifts of the LP easier to reason about, thanks to the new structure of this formulation. We then consider the Lasserre hierarchy applied to to this LP, and prove the following theorems.

Theorem 1.1.

For every constant and sufficiently large , the integrality gap of the -th level Lasserre SDP for Directed -Spanner is at least .

Theorem 1.2.

For every constant and sufficiently large , the integrality gap of the -th level Lasserre SDP for Basic -Spanner is at least .

Note that, while the constant in the exponent is different, Theorem 1.2 is similar to [17] in that it shows that the integrality gap “tracks” the trivial approximation from [1] as a function of . Thus for undirected spanners, even using the Lasserre hierarchy cannot give too substantial an improvement over the trivial greedy algorithm.

At a very high level, we follow the approach to building spanner integrality gaps of [15, 17]. They started with random instances of the Unique Games problem, which could be shown probabilistically to not admit any good solutions. They then used these Unique Game instances to build spanner instances with the property that every spanner had to be large (or else the Unique Games instance would have had a good solution), but by “splitting flow” the LP could be very small.

In order to apply this framework to the Lasserre hierarchy, we need to make a number of changes. First, since Unique Games can be solved reasonably well by Lasserre [3, 7], starting with a random instance of Unique Games will not work. Instead, we start with a more complicated problem known as Projection Games (the special case of Label Cover in which all the edge relations are functions). An integrality gap for the Lasserre hierarchy for Projection Games was recently given by [10, 26] (based on an integrality gap for CSPs from [34]), so we can use this as our starting point and try to plug it into the integrality gap framework of [15, 17] to get an instance of either directed or undirected spanners. Unfortunately, the parameters and structure that we get from this are different enough from the parameters used in the integrality gap of the basic LP that we cannot use [15, 17] as a black box. We need to reanalyze the instance using different techniques, even for the “easy” direction of showing that there are no good integral solutions. In order to do this, we also need some additional properties of the gap instance for Projection Games from [10] which were not stated in their original analysis. So we cannot even use [10] as a black box.

The main technical difficulty, though, is verifying that there is a “low-cost” fractional solution to the SDP that we get out of this reduction. For the basic LP this is straightforward, but for Lasserre we need to show that the associated slack moment matrices are all PSD. This turns out to be surprisingly tricky, but by decomposing these matrices carefully we can show that each matrix in the decomposition is PSD, and thus the slack moment matrices are PSD. At a high level, we decompose the slack moment matrices as a summation of several matrices in a way that allows us to use the consistency properties of the feasible solution to the

Projection Games instance in [10] to show that the overall sum is PSD.

Doing this requires us to use some nice properties of the feasible fractional solution provided by [10], some of which we need to prove as they were not relevant in the original setting. In particular, one important property which makes our task much easier is that their fractional solution actually satisfies all of the edges in the Projection Games instance. That is, their integrality gap is in a particular “place”: the fractional solution has value while every integral solution has much smaller value. Because spanners and the other network design problems we consider are minimization problems (where we need to satisfy all demands in a cheap way), this is enormously useful, as it essentially allows us to use “the same” fractional solution (as it will also be feasible for the minimization version since it satisfies all edges). Technically, we end up combining this fact about the fractional solution of [10] with several properties of the Lasserre hierarchy to infer some more refined structural properties of the derived fractional solution for spanners, allowing us to argue that they are feasible for the Lasserre lifts.

Extensions.

A number of other network design problems exhibit behavior that is similar to spanners, and we can extend our integrality gaps to these problems. In particular, we give a new integrality gap for Lasserre for Directed Steiner Network (DSN) (also called Directed Steiner Forest) and Shallow-Light Steiner Network (SLSN) [2]. In DSN we are given a directed graph (possibly with weights) and a collection of pairs , and are asked to find the cheapest subgraph such that there is a to path for all . In SLSN the graph is undirected, but each and is required to be connected within a global distance bound . The best known approximation for DSN is an -approximation for arbitrarily small constant  [12], which uses a standard flow-based LP relaxation. We can use the ideas we developed for spanners to also give integrality gaps for the Lasserre lifts of these problems. We provide the theorems here; details and proofs can be found in Appendix E.

Theorem 1.3.

For every constant and sufficiently large , the integrality gap of the -th level Lasserre SDP for Directed Steiner Network is at least .

Theorem 1.4.

For every constant and sufficiently large , the integrality gap of the -th level Lasserre SDP for Shallow-Light Steiner Network is at least .

Lift-and-Project for Network Design.

Lift and project methods such as Sherali-Adams [32] and Lasserre [23] have been studied and used extensively for approximation algorithms. For example, strong results are known about their performance on CSPs [34, 31], independent set in hypergraphs [8], graph coloring [11], and Densest -Subgraph [5, 13]. However, there is surprisingly little known about the power of these hierarchies for network design problems (the main exception being Directed Steiner Tree [21, 29]). We begin to address this gap by providing Lasserre integrality gaps for a variety of difficult network design problems (Basic -Spanner, Directed -Spanner, Directed Steiner Network, and Shallow Light Steiner Network). Our results can be seen as general framework for proving Lasserre integrality gaps for these types of hard network design problems.

2 Preliminaries: Lasserre Hierarchy

The Lasserre hierarchy is a way of lifting a polytope to a higher dimensional space, and then optionally projecting this left back to the original space in order to get tighter relaxations. The standard characterization for Lasserre is as follows [23, 24, 30]:

Definition 2.1 (Lasserre Hierarchy).

Let and , and define the polytope . The -th level of the Lasserre hierarchy

consists of the set of vectors

where means the power set, and they satisfy the following constraints:

The matrix is called the moment matrix, and the matrices are called the slack moment matrices.

Let us review (see, e.g., [30]) multiple helpful properties that we will use later. We include proofs in Appendix A for completeness.

Claim 2.2.

If , , and , then for all .

Claim 2.3.

If , , and , then for all .

Lemma 2.4.

If then for any we have and .

3 Projection Games: Background and Previous Work

In this section we discuss the Projection Games problem, its Lasserre relaxation, and the integrality gap that was recently developed for it [10] which form the basis of our integrality gaps for spanners and related problems. We begin with the problem definition.

Definition 3.1 (Projection Games).

Given a bipartite graph , where is the (label) alphabet set and for each , the objective is to find a label assignment that maximizes (i.e. the number of edges where , which we refer to as satisfied edges).

We will sometimes use relation notation for the functions , w.g., we will talk about . Note that Projection Games is the famous Label Cover problem but where the relation for every edge is required to be a function (and hence we inherit the relation notation when useful). Similarly if we further restrict every function to be a bijection then we have the Unique Games problem. So Projection Games lies “between” Unique Games and Label Cover.

The basis of our integrality gaps is the integrality gap instance recently shown by [10] for Lasserre relaxations of Projection Games. We first formally define this SDP. For every we will have a variable . Then the -th level Lasserre SDP for Projection Games is the following.

It is worth noting that this is not the original presentation of this SDP given by [10]: they wrote it using a vector inner product representation. But it can be shown that these representations are equivalent, and in particular we prove the important direction of this in Appendix B.1: any feasible solution to their version gives an equivalent feasible solution to , and thus their fractional solutions are also fractional solutions to .

[10] gives a Projection Games instance with following properties. One of the properties is not proven in their paper, but is essentially trivial. We give a proof of this property, as well as a discussion of how the other properties follow from their construction, in Appendix B.2.

Lemma 3.2.

For any constant , there exists a Projection Games instance with the following properties:

  1. , , , where .

  2. There exists a feasible solution for the -th level , such that for all , we have .

  3. At most edges can be satisfied.

  4. The degree of vertices in is , and the degree of vertices in is at most .

We also define if .

4 Lasserre Integrality Gap for Directed -Spanner

In this section we prove our main result for the Directed -Spanner problem: a polynomial integrality gap for polynomial levels of the Lasserre hierarchy. We begin by discussing the base LP that we will use and its Lasserre lifts, then define the instance of Directed -Spanner that we will analyze (based on the integrality gap instance for Projection Games in Lemma 3.2), and then analyze this instance.

4.1 Spanner LPs and their Lasserre lifts

The standard flow-based LP for spanners (including both the directed and basic -spanner problems) was introduced by [15], and has subsequently been used in many other spanner problems [4, 16, 9]. Let denote the set of all stretch- paths from to .

While this LP is extremely large (the number of variables can be exponential if there are general lengths on the edges, or if all lengths are unit but is large enough), it was shown in [15] that it can be solved in polynomial time. However, for the purposes of studying its behavior in the Lasserre hierarchy, is a bit awkward. Since it has (potentially) exponential size, so do its Lasserre lifts. And from a more “intuitive” point of view, since there are two different “types” of variables, the lifts become somewhat difficult to reason about.

Since the variables do not appear in the objective function, we can project the polytope defined by onto the variables and use the same objective function to get an equivalent LP but with only the variables. More formally, let be the polytope bounded by for all and for all . Then it is not hard to see that if we project the polytope defined by onto just the variables, we get precisely the following LP (this can also be seen via the duality between stretch- flows and fractional cuts against stretch- paths):

While as written there are an infinite number of constraints, it is easy to see by convexity that we need to only include the (exponentially many) constraints corresponding to vectors that are vertices in the polytope , for each . Thus there are only an exponential number of constraints, but for simplicity we will analyze this LP as if there were constraints for all possible . This LP is completely equivalent to , in the sense that a vector is feasible for if any only if there exist variables so that is feasible for . The proof of the following theorem is included in Appendix C.1

Theorem 4.1.

is feasible for if and only if there is some for each and so that is feasible for .

From Definition 2.1, the -th level Lasserre SDP of is:

This SDP is the basic object of study in this paper, and is what we will prove integrality gaps about.

4.2 Spanner Instance

In this section we formally define the instance of Directed -Spanner that we will analyze to prove the integrality gap. We basically follow the framework of [15], who showed how to use the hardness framework of [18, 22] to prove integrality gaps for the basic flow LP. We start with a different instance (integrality gaps instances for Projection Games rather than random instances of Unique Games), and also slightly change the reduction in order to obtain a better dependency on .

Roughly speaking, given a Projection Games instance, we start with the “label-extended” graph. For each original vertex in the projection game, we create a group of vertices in the spanner instance of size . So each vertex in the group can be thought as a label assignment for the Projection Games vertex. We then add paths between these groups corresponding to each function (we add a path if the associated assignment satisfies the Projection Games edges). We add many copies of the Projection Games graph itself as the “outer edges”, and then connect each Projection Games vertex to the group associated with it. The key point is to prove that any integral solution must contain either many outer edges or many “connection edges” (in order to span the outer edges), while the fractional solution can buy connection and inner edges fractionally and simultaneously span all of the outer edges.

More formally, given the Projection Games instance from Lemma 3.2, we create a directed -spanner instance as follows (note that is the degree of the vertices in ):

For every , we create vertices: for all and for all . We also create edges for each and . We call this edge set .

For every , we create vertices: for and for . We also create edge for each and . We call this edge set .

For every , we create edges for each . We call this edge set .

For each and , we also create vertices for and edges . We call this edge set .

Finally, for technical reasons we needs some other edges and inside groups of and , which will be defined later.

To be more specific, , , such that:

Note that if , then there is no vertex set , but only edge set , which directly connect and for each and . To get some intuition, a schematic version of this graph is given as Figure 1 in Appendix C.

4.3 Fractional Solution

In this section, we provide a low-cost feasible vector solution for the -th level Lasserre lift of the spanner instance described above. Slightly more formally, we define values and show that they form a feasible solution for the -th level Lasserre lift , and show that the objective value is . We do this by starting with a feasible solution to the -th level Lasserre lift for the Projection Games instance (based on Lemma 3.2) we used to construct our directed spanner instance, and adapting it for the spanner context. Before defining , we define a function (where indicates the power set) as follows.

We then extend the definition of to by setting .

Next, we define the solution . For any set containing any edge in , we define , otherwise, let . Note that based on how we defined the function , for all edges in we have . In other words, these edges will be picked integrally in our feasible solution. At a very high level, what we are doing is fractionally buying edges in and integrally buying edges in in order to span edges in . The edges in and are used to span edges in and . We first argue that our fractional solution has cost only . The proof is included in Appendix C.3

Lemma 4.2.

The objective value of is .

4.3.1 Feasibility

In this section we show that the described vector solution is feasible for the -th level of Lasserre, i.e., that all the moment matrices defined in are PSD. This is the most technically complex part of the analysis, particularly for the slack moment matrices for edges in . So we start with the easier matrices, working our way up to the more complicated ones. In particular, we first use the fact that the base moment matrix in is PSD for the Projection Games solution to show in Theorem 4.3 that the the base moment matrix of is PSD for solution .

Theorem 4.3.

The moment matrix is positive semidefinite, so does .

Proof.

We know that the moment matrix , since is a solution of . Now for each principal submatrix of , we consider three cases. In the first case, suppose that has an index that contains an edge in . Then the whole row and whole column of this index is , so the determinant is . In the second case, includes two distinct indices of , such that . In this case is not full rank, and thus the determinant is also . Otherwise, does not include any indices that contain edges in , and no two indices have the same value. Then is by definition a principal submatrix of , since each row/column index of can be converted to a different index of based on function . Now, since all the principal submatrices of have non-negative determinant, by definition is PSD. ∎

Showing that the slack moment matrices of our spanner solution are all PSD is more subtle and requires a case by case analysis, combined with several properties of the Lasserre hierarchy. We divide this argument into three parts. First we show (in Theorem 4.4) that this is true for slack moment matrices corresponding to pairs for which we assigned . Then we show (Theorem 4.5) the same for edges in and . Finally, we handle the most difficult case of slack moment matrices corresponding to edges in (Theorem 4.6).

Theorem 4.4.

The slack moment matrix is PSD for all and .

Proof.

Recall that for every we set . So basic properties of Lasserre (Claim 2.2) imply that for all . Thus

Here the third equality follows from the fact and itself is a path connecting and , and thus . In the last equality we use the fact that for all , and is a principal submatrix of , which is positive semidefinite according to Theorem 4.3. ∎

Now we prove a similar theorem for the edges in and , which is a bit more complex since these edges are only bought fractionally in our solution.

Theorem 4.5.

The slack moment matrix is PSD for every and .

Proof.

The proof for is similar to the proof for , so without loss of generality we focus on the case. Recall that each edge in can be represented as , where . Now we consider two cases based on how the edge is spanned: or . When , let , and for every , let . When , let , , and for every , let .

In the case, because paths are disjoint, we can partition the slack moment matrix as following:

The first term in the final sum is PSD because for all , and is a principal submatrix of , which is positive semidefinite by Theorem 4.3. The second term in the final sum is PSD because either

which makes the matrix a zero matrix, or

and we can use Lemma 2.4 to prove the matrix is PSD. The third term in the final sum is PSD because for all (since ) together with the fact that the matrix is a principal submatrix of . The fourth term in the final sum is the zero matrix because the slack moment constraint of . Thus when .

In the case, all the paths share the common edge . Thus we can partition the sum in the slack moment matrix as follows: