Polynomial integrality gaps for strong SDP relaxations of Densest k-subgraph

Polynomial integrality gaps for strong SDP relaxations of Densest -subgraph

Aditya Bhaskara    Moses Charikar    Aravindan Vijayaraghavan    Aditya Bhaskara Department of Computer Science, Princeton University, and Center for Computational Intractability. Supported by Supported in part by NSF under CCF 0832797 and AF 0916218. Email: bhaskara@cs.princeton.edu.    Moses Charikar Department of Computer Science, Princeton University, and Center for Computational Intractability. Supported in part by NSF under CCF 0832797 and AF 0916218. Email: moses@cs.princeton.edu.    Venkatesan Guruswami Computer Science Department, Carnegie Mellon University. Research supported in part by a Packard Fellowship and NSF CCF 1115525. Email: guruswami@cmu.edu.    Aravindan Vijayaraghavan Department of Computer Science, Princeton University, and Center for Computational Intractability. Supported in part by NSF under CCF 0832797 and AF 0916218. Email: aravindv@cs.princeton.edu.    Yuan Zhou Computer Science Department, Carnegie Mellon University. Supported in part by US-Israel BSF grant 2008293, MSR-CMU Center for Computational Thinking, and NSF CCF 1115525. Email: yuanzhou@cs.cmu.edu.
Abstract

The Densest -subgraph  problem (i.e. find a size subgraph with maximum number of edges), is one of the notorious problems in approximation algorithms. There is a significant gap between known upper and lower bounds for Densest -subgraph: the current best algorithm gives an approximation, while even showing a small constant factor hardness requires significantly stronger assumptions than . In addition to interest in designing better algorithms, a number of recent results have exploited the conjectured hardness of Densest -subgraph  and its variants. Thus, understanding the approximability of Densest -subgraph  is an important challenge.

In this work, we give evidence for the hardness of approximating Densest -subgraph within polynomial factors. Specifically, we expose the limitations of strong semidefinite programs from SDP hierarchies in solving Densest -subgraph. Our results include:

  • A lower bound of on the integrality gap for rounds of the Sherali-Adams relaxation for Densest -subgraph. This also holds for the relaxation obtained from Sherali-Adams with an added SDP constraint. Our gap instances are in fact Erdös-Renyi random graphs.

  • For every , a lower bound of on the integrality gap of rounds of the Lasserre SDP relaxation for Densest -subgraph, and an gap for rounds. Our construction proceeds via a reduction from random instances of a certain Max-CSP over large domains.

In the absence of inapproximability results for Densest -subgraph, our results show that beating a factor of is a barrier for even the most powerful SDPs, and in fact even beating the best known factor is a barrier for current techniques.

Our results indicate that approximating Densest -subgraph  within a polynomial factor might be a harder problem than Unique Games or Small Set Expansion, since these problems were recently shown to be solvable using rounds of the Lasserre hierarchy, where is the completeness parameter in Unique Games and Small Set Expansion.

1 Introduction

The densest -subgraph problem takes as input a graph on vertices and a parameter , and asks for a subgraph of on at most vertices having the maximum number of edges. While it is a fundamental graph optimization problem and arises in several applications (community detection in social networks, identifying protein families and molecular complexes in protein-protein interaction networks, etc), there is a huge gap between the best approximation algorithm and the known inapproximability results. The current best approximation algorithm due to [BCC10] gives -factor approximation algorithm which runs in time for any constant . On the inapproximability side, [Fei02] initially showed a small constant factor inapproximability for Densest -subgraph using the random 3-SAT assumption. [Kho04] used quasi-random PCPs to rule out a PTAS. More recently, [RS10, AAM10] used more non-standard assumptions to rule out any constant factor approximation algorithms.

While only constant factor approximations have been ruled out, it is commonly believed that Densest -subgraph is much harder to approximate even on average (for a natural distribution on hard instances). Recently, average-case hardness assumptions based on the hardness of “planted” versions of Densest -subgraph were used for public key cryptography [ABW08] and in showing that financial derivates can be fraudulently priced without detection [ABBG10]. Given the interest in Densest -subgraph from both the algorithms and the complexity point of view, developing a better understanding of the problem is an important challenge for the field.

In this work, we study lift-and-project relaxations for Densest -subgraph. Lift-and-project methods are systematic iterative procedures to obtain sequences of increasingly stronger mathematical programming relaxations for an integer optimization problem (e.g. Lovász-Schrijver [LS91], Sherali-Adams [SA90] and Lasserre [Las01]. See the survey by Laurent [Lau03] for a comparison). Typically, the relaxation obtained after levels of these strengthenings can be solved in time. A number of recent papers have studied the strength and limitations of such relaxations as a basis for designing approximation algorithms for various problems [ABLT06, CMM09, Chl06, CS06, dVM07, GMPT07, GMT09, KS09, RS09, Sch08, STT07a, STT07b, Tul09] (see the recent survey by Chlamtac and Tulsiani [CT11]). In most cases of approximation algorithms that use strengthened LP and SDP relaxations, such relaxations can be obtained from a few levels of such lift-and-project procedures. In fact, the approximation algorithm of [BCC10] for Densest -subgraph  uses a linear programming relaxation which is weaker than that obtained from levels of the Sherali-Adams hierarchy.111[BCC10] also gives a purely combinatorial algorithm that does not use a linear program. [BCC10] also show that the integrality gap becomes after levels of the Sherali Adams LP hierarchy.

Our results.

In this paper, we first study lift-and-project relaxations for Densest -subgraph  obtained from the Sherali-Adams hierarchy. We show that even levels of the Sherali-Adams relaxation have an integrality gap of . Then, we turn to the Lasserre hierarchy for Densest -subgraph. We show an integrality gap of polynomial ratio (, for small enough constant ) for almost linear () levels of the Lasserre relaxation. If we only aim at an integrality gap for polynomial () levels of the Lasserre relaxation, the ratio of the gap can be as large as .

Our gap instances are actually (Erdös - Renyi) random graph instances and random bipartite graphs under a special distribution – hence, we show that natural distributions of instances are integrality gap instances with high probability.

We note that prior results exhibiting gap instances for lift-and-project relaxations do so for problems that are already known to be hard to approximate under some suitable assumption; based on this hardness result, one would expect lift-and-project relaxations to have an integrality gap that matches the inapproximability factor.

Our gap constructions for Densest -subgraph in this paper are a rare exception to this trend, as the integrality gaps we show are substantially stronger than the (very weak) hardness bounds known for the problem. In fact, we are only aware of the following examples where a polynomial-round Lasserre integrality gap stronger than the corresponding NP-hardness result is known : Max -CSP, -coloring [Tul09], Balanced Separator and Uniform Sparsest Cut [GSZ11]. In the first two cases, NP-hardness results that are not that far from the gaps are known [ST00, Kho02] and for Max -CSP a matching Unique-Games hardness is also known [ST09]. For the other two problems, constant factor integrality gaps were shown for linear number of rounds of Lasserre hierarchy [GSZ11]. Again, while these problems are not known to be APX-hard, under the conjectured intractability of Small Set Expansion, they are known to be hard to approximate within any constant factor.

In the absence of inapproximability results for Densest -subgraph, our results show that beating a factor of is a barrier for even the most powerful SDPs, and in fact even beating the best known factor is a barrier for current techniques. These results are perhaps indicative of the hardness of approximating Densest -subgraph  within factors.

Relation to the Small Set Expansion and Unique Games.

A problem related to Densest -subgraph  is the Small Set Expansion (SSE) problem, which has received a lot of recent attention due to strong connections to the Unique Games conjecture [RS10]. One way to state the SSE conjecture [RS10] (which is known to imply the Unique Games conjecture) is as follows: for all , there exists (think of as a constant), such that the following problem is not polynomial-time solvable:

Definition 1.1 (The Gap-SSE problem).

Given a -regular instance with , the Gap-SSE problem is to distinguish between the following two cases.

  • Yes case. There exists a subgraph of vertices with average degree at least .

  • No case. All subgraphs of vertices have average degree at most .

Clearly, Densest -subgraph  is hard to approximate within any constant factor, assuming the Small Set Expansion conjecture. On the other hand, our results indicate that approximating Densest -subgraph  even within a polynomial factor may be a harder problem than Unique Games or Small Set Expansion, because these problems were recently shown to be solvable using rounds of the Lasserre hierarchy, where is the completeness parameter in Unique Games and Small Set Expansion [BRS11, GS11].

2 Preliminaries

2.1 Notation.

We introduce some notation which will be used throughout the paper. refers to a graph which is an instance of the Densest -subgraph  problem on vertices, and refers to the size of the subgraph we are required to output. For an induced subgraph , we denote by the average degree (or density of ). For a vertex in subgraph , we will denote by the set of neighbors of in (the suffix will be dropped when ).

The phrase “with high probability” will mean: with probability , for any polynomial .

2.2 The relaxation hierarchies for Densest -subgraph.

We will be concerned with the SDP relaxations derived from the Sherali-Adams and Lasserre hierarchies for Densest -subgraph. As in other lift-and-project schemes, a feasible solution to levels of these hierarchies satisfies the condition that for any set of vertices, it defines a valid distribution over integral solutions for these vertices – in particular, the integrality gap becomes after levels. Further, the relaxations given by levels of the Sherali-Adams and Lasserre hierarchies can be solved in time. We are interested in the integrality gap of levels of these relaxations for Densest -subgraph. Refer to [CT11] for a more comprehensive comparison of these relaxation hierarchies.

2.2.1 The Sherali-Adams LP hierarchy.

The Sherali-Adams hierarchy starts with a simple LP relaxation of a integer program, and obtains a sequence of successively tighter relaxations with more levels. The natural LP relaxation for Densest -subgraph  (LP1 in Figure 1) [SW98, FS97] has variables to denote if vertex belongs to the solution, and edge variables to denote if both are in the subgraph. This LP has an integrality gap of ([FKP01, FS97]).

Natural LP (LP1): Min. degree LP (LP2):
s.t.
s.t.
s.t.
Figure 1: Two Linear Programming relaxations for Densest -subgraph

For our integrality gaps, we will in fact start with a stronger basic (first-level) linear program (LP2 in Figure 1) which is equivalent upto a factor of (see [BCC10]). Intuitively, it tries to find a -subgraph where the minimum degree is maximized. An LP hierarchy obtained from this min. degree LP (LP2) was in fact used by [BCC10] to obtain their approximation algorithm. 222While the program as stated is not linear, we guess the degree and consider the feasibility linear program that is obtained.

Let us consider strengthening this LP by considering levels of the Sherali-Adams hierarchy (, shown in Figure 2). In the lifted LP, the variable is supposed to capture whether every vertex in belongs to the chosen -subgraph (i.e., ). Further if we take two sets of vertices, the local distributions induced by a feasible solution (using the inclusion-exclusion constraints), agree on the variables in the intersection . We follow the notation established in [CT11] while defining the hierarchy.

2.2.2 The mixed hierarchy (Sherali-Adams + SDP).

The mixed hierarchy (also refered to as SA+) imposes an additional SDP constraint on top of the Sherali-Adams LP relaxation. In particular, it asks for the values to come from vector inner products i.e. the matrix is p.s.d. Most known algorithms which proceed by rounding a relaxation obtained from an SDP hierarchy [Chl06, CS06, BRS11] work with this mixed hierarchy 333[GS11] is an exception and seems to need a relaxation given by the Lasserre hierarchy..[RS09, KS09] and [GMT09] considered this hierarchy and obtained integrality gaps for Unique Games and approximation-resistant CSPs.

(1)
(2)
(3)
Figure 2: Sherali-Adams LP relaxation ( levels) for Densest -subgraph:

One level of the mixed hierarchy for Densest -subgraph  gives the SDP relaxation introduced in [FS97, SW98]. [BCC10] show that the mixed hierarchy performs better than log-density based arguments (which are captured by just the LP hierarchy) in a planted model.444In particular, the problem of detecting if dense -subgraph is planted in a random graph or not, in the parameter range . It is interesting in this light to obtain integrality gaps for mixed hierarchy.

2.2.3 The Lasserre hierarchy.

The Lasserre hierarchy produces a sequence of SDP relaxations which are stronger than the Sherali-Adams and the mixed hierarchies. As in [CT11], the -level Lasserre SDP for Densest -subgraph  introduces a vector for each subset with (Figure 3).

max
such that
Figure 3: Lasserre hierarchy ( levels) for Densest -subgraph

The intended solution sets if every vertex in belongs to the densest -subgraph, and otherwise. The vector lengths correspond to valid LP values for the Sherali-Adams relaxation presented above.

Remark 2.1.

As in Section 2.2.1, we can write an SDP which tries to find the -subgraph of largest induced minimum degree . This can be captured by the SDP constraint (analogous to (2))

(4)

However, we show in Section 4.3 that our integrality gaps also hold for the Lasserre hierarchy defined by this SDP. We refer to the SDP with constraint (4) as the Min degree Lasserre SDP .

3 Integrality Gap for the Sherali-Adams hierarchy

In what follows will denote the number of levels of the hierarchy we will consider.

Theorem 3.1.

Let . The integrality gap of is at least .

To prove Theorem 3.1, we present instances where the relaxation has a solution with value , while the integer optimum, i.e., the largest density of a -subgraph in is only . It will be notationally convenient to construct gaps for levels.

3.1 The instance.

We in fact give a distribution over instances, and prove that the desired gap holds with high probability. The instances we consider are random graphs with (thus the expected degree of each vertex is ). The parameter is chosen to be . An easy calculation shows that in any subgraph, the density (and hence the min-degree) is at most (see full version or [BCC10, FKP01]). The meat of the argument is thus to show that there exists an LP solution to (Equations (1)-(3)) of value even for of the order .

The following are the properties of the distribution (with above parameters) we will truly be using [see Section A for proofs]. Any graph with these properties admits the solution to which we describe.

  1. Every vertex has degree between and .

  2. Any two vertices have at least one common neighbor and has at most common neighbours.

3.2 Feasible solution.

Before formally giving the values, we give intuition as to what they ought to be. First, we start out setting (equal for all vertices, since and no vertex is special). Next, suppose with and think of . Now (2) implies that . Further from (1), we obtain . Thus we conclude that must be roughly for , while for , it should be only . Now consider which span a tree: we could imagine starting with one vertex and adding vertices one by one (each added vertex is a neighbour of the previous ones), and thus conclude that is roughly (since to begin with). Now let be an arbitrary set of vertices and consider a tree : by monotonicity (a corollary of (3)), , and since this is true for every such , we need to set to be at least , where is the number of vertices (size) in the minimum Steiner tree of .

These, with additional ‘dampening’ factors (-terms), are precisely the values we will set. More precisely we consider the solution

(5)

where , as above, is the size of the minimum Steiner tree of . Thus for instance , while when and otherwise (the latter is because there is a path of length-2 between any with high probability).

Let us fix . We now show that the LP solution presented above is feasible for with high probability. The following lemma is useful in simplifying the analysis: it implies that we need to only consider while showing that the LP solution satisfies constraints (1) and (2). This is where the ’dampening’ factors come into play.

Lemma 3.2.

Let be disjoint subsets of of size at most and be the solution described above. Then

Proof.

One property of the assignment (5) is that for . Further all the are , and thus in the sum above, the term corresponding to contibutes positively when is even and negatively otherwise. Hence,

A similar proof shows the upper bound, since the terms for dominate the contributions of for . ∎

Corollary 3.3.

In checking feasibility, it suffices to check (1) and (2) with .

Proof.

Lemma 3.2 allows us to ‘remove’ the on both sides of the equations (and set ) by losing a factor of 2. Since we allow constant slack, the claim follows. ∎

We refer to the constraints (1) and (2) as the size and the density constraints respectively, because the former says that we should pick only a -subgraph, and the latter says the minimum degree (density) is at least . The assignment we described allows us to prove the density constraint easily.

Lemma 3.4.

(Density Constraint) The described above satisfy constraints (2).

Proof.

Let and . We need to check that . It is easy to see that for every , , and thus (the term is due to the dependence on in (5)). Since there are at least terms in the LHS, the inequality follows. ∎

3.3 The Size Constraint and Minimum Steiner trees in .

By the above corollary, it suffices to check (noting ) that

(6)

We show this by proving that for most , in particular we bound the number of exceptions (lemmas below state the precise bounds). This then implies that (6) holds.

We start with some basic facts (and notation) about Minimum Steiner trees (minST) of in , with our parameters. We will refer to the vertices in as the terminals, and the rest of the vertices in a minST as the non-terminals. First, the minST must have all its leaves to be terminals. Further, since every two vertices in have a path of length two, we must have for all . This helps us bound the number of tree structures the minST of can have. We define this formally.

Given , a tree structure for is a tree along with a mapping which is one-one (not necessarily onto). The vertices in without an inverse image in are called internal vertices and the rest are also called fixed vertices. A tree structure for is valid if it is possible to ‘fill in’ the internal vertices with distinct vertices from such that all the edges in the tree are also present in . [The relation to Steiner trees is apparent – the internal vertices are the Steiner vertices]. Given an internal vertex in , the vertices of which take that position in some valid ‘filling in’ are called the set of candidates for that position.

Before we get to the lemmas, we note that the number of tree structures for of size is at most (this is just by a naïve bound using the number of trees). Let us now bound the number of for which .

Lemma 3.5.

Let and be a tree structure for a min Steiner tree of (so the leaves of are elements of ). Then the number of candidates for each of the positions in is at most .

Proof.

The proof is by induction on the size of . The base case is trivial. Assume the result for all tree structures of sets of size . Now consider . We may assume that has at least one non-terminal, as otherwise there is nothing to prove.

First, note that there exists a vertex which is adjacent to at most one non-leaf vertex in . This is because deleting all the leaves in gives a tree (which is not empty as there is at least one non-terminal in ), and a leaf in this tree our required . If is a terminal, we could remove the leaves attached to (thus obtaining a subset of the terminals), and the remaining tree structure would be a valid min Steiner tree for . Further, the set of non-terminals is precisely the same, and thus the inductive hypothesis implies the claim for . Thus suppose is a non-terminal.

If the degree of (in ) is 2, then has precisely one leaf attached to it (call it ). Consider the tree obtained by removing , and let . Now is a min Steiner tree for (if not, we could consider use this smaller tree for along with a path of length to to obtain a smaller minimum steiner tree for ). If is the vertex in attached to , there are at most candidates for , by Induction Hypothesis. For each candidate , the number of candidate is only , and since is a terminal. Thus the number of candidates for is at most . The rest of the non-terminals in are also present in , and this gives the result.

If degree, then there are at least two leaves attached to , thus the number of candidates for is only . Consider one candidate for . Let be the tree obtained by removing all the leaves attached to (thus is now a leaf), and be minus the set of leaves attached to . Now is a min Steiner tree structure for (otherwise we can obtain a smaller tree for ). Thus by the inductive hypothesis, the number of candidates for any internal vertex in is at most . Since there are only of the ’s, it follows that the total #(candidates) for an internal vertex is at most .

This completes the proof, by induction. ∎

An easy corollary is the following.

Corollary 3.6.

Let . There are at most vertices such that .

Proof.

Each such must be the internal vertex of some min Steiner tree for , and there are at most tree structures. Lemma 3.5 now implies the claim. ∎

Lemma 3.7.

Let . There are at most vertices such that .

Proof.

Let be such a vertex. First, note that if there exists a min Steiner tree for with as a leaf, we are done. This is because removing gives a min Steiner tree for , and thus is a neighbour of an internal vertex in a min Steiner tree for . Thus by Corollary 3.6 there are only such .

Thus suppose that the min Steiner tree for has as an internal vertex. We will prove the bound as follows: we consider a tree structure of size with leaves being terminals from ; then we show that the number of candidates for any fixed position in is at most . This suffices, because the number of choices of tree structures adds an additional factor of .

Figure 4: An example

Let us consider a structure as above, and a position . Since is not a leaf, it has degree at least . Let the degree be , and let be the subtrees of formed by removing (see figure …). Now if for some , is the min Steiner tree for the terminals in , we are done, because then, each candidate for must be neighbour of an internal vertex in the tree, and by Corollary 3.6 there are only candidates. Thus for each , must have a strictly smaller tree . Let the vertex in connected to be called . Now construct a new tree as follows: leave intact, and replace by ; connect to using paths of length . The number of edges in the new tree is now at most . The first term is the original cost, followed by removal of , followed by the decrease by using as opposed to , followed by the cost of adding length-2 paths.

Thus the new tree has cost at most , and thus it is optimal for ! Further, is adjacent to which is an internal vertex, and thus the number of candidates is bounded by the desired quantity. ∎

Putting things together.

Consider the sum . Corollary 3.6 implies that there are at most terms which contribute a value . Lemma 3.7 implies that there are at most terms which contribute a value . Thus if we pick , we have the bound that the sum is at most , as desired.

Thus we have verified each of the constraints (1)-(3). This completes the proof of Theorem 3.1.

3.4 Gaps for the mixed hierarchy (SA+).

Consider the relaxation described in (1)-(2), along with the constraint: . The solution considered earlier (Equation (5)) turns out to also satisfy this PSD condition with high probability. The entries of are

Thus we have

where is the adjacency matrix of . Now is a matrix with . Thus the least eigenvalue is at least with high probability (by the Semicircle law). This is at least . Thus we have . Using the fact that , we obtain that .

This shows that adding an SDP constraint at the first level does not give us any additional power – the relaxation obtained after levels also has an integrality gap of .

Question 3.8.

Does Theorem 3.1 hold for ?

We conjecture that even levels does not reduce the integrality gap substantially. We need a different approach (involving a better argument for bounding the number of trees) to extend the arguments above to this range of .

4 Integrality Gap for the Lasserre hierarchy

In this section, we show a gap instance with arbitrary large constant ratio for linear-round Lasserre relaxation, and a gap instance with ratio for -round Lasserre relaxation (Theorem 4.7). We also aim at maximizing the ratio of a polynomial-round Lasserre gap instance, getting a ratio of (Theorem 4.8).

Our construction is based on a variant of Tulsiani’s gap instance for Max -CSP  [Tul09] – we extend the parameter range of Tulsiani’s instance. Then we convert the Max -CSP  instance to a constraint-variable graph and duplicate the variable vertices, which is our gap instance for Densest -subgraph. Note that the gap for Max -CSP  problem is indeed a set of random instances. The vector solution from Lasserre gap for Max -CSP  will help us exhibit a good Lasserre vector solution for Densest -subgraph. We finally use the structure of random instances of Max -CSP  to show the soundness holds with high probability.

Now, let us proceed to the first step, the gap instance for Max -CSP.

4.1 Lasserre Gap for Max -Csp from [Tul09].

We start by defining the Max -CSP  problem.

Definition 4.1.

Let be a -ary linear code of block length .

  1. An instance of Max -CSP is a set of constraints where each constraint is over a -tuple , and is of the form for some .

  2. A random instance of Max -CSP is sampled by choosing each constraint independently, where we sample variables without replacement from to get and is chosen from uniformly.

The following theorem is an extension of the main theorem in [Tul09], showing that polynomial-round Lasserre relaxation cannot refute random Max -CSP  with high probability.

Theorem 4.2.

If is the dual code of a distance code (in terms of number of coordinates, not fractional distance), for every , if for some , then for large enough , a random instance of Max -CSP over constraints and variables, with probability , admits a perfect solution for the SDP relaxation obtained by rounds of the Lasserre hierarchy, i.e. there are vectors for all with and all , such that

  • the value of the solution is perfect: ;

  • for all ;

  • if ;

  • for all and ;

  • and for all .

Note that Theorem 4.2 extends the original theorem of [Tul09] to the regime where might be superconstant (even ). The proof of Theorem 4.2 follows the proof in Tulsiani’s paper, with the following changes.

Observation 4.3.

By the first property of the solution given in Theorem 4.2, we know that for every , we have , and therefore .

Recall that Tulsiani showed that, if the constraint-variable graph of a Max -CSP instance has very high left-expansion, then the Lasserre SDP admits a perfect solution for it. Formally, the following lemma is (implicitly) shown in [Tul09].

Lemma 4.4 ([Tul09]).

Given a Max -CSP instance, if every set of constraints of cardinality involves more than variables (where is the distance of the dual code of ), and if , then there is a perfect solution for the SDP relaxation obtained by rounds of the Lasserre hierarchy.

Hence, we only need to prove the following lemma which shows that the constraint-variable graph still has very high left-expansion, even when a constraint might involve superconstant many variables (i.e. the left degree might be superconstant).

Lemma 4.5.

Given as in Theorem 4.2, with probability , for all , every set of constraints involves more than variables.

A similar lemma can be found in [Tul09] (Lemma A.1), which only deals with constant . We need a more refined argument for superconstant , which is in Section 4.4.

4.2 The Lasserre gap for Densest -subgraph.

The gap instance is reduced from the gap instance for Max -CSP in Theorem 4.2. Let be the dual code of a code as used in Theorem 4.2, where is the block length, is the dimension, and is the distance of the code. Such a code has size , and is very sparse for small enough . For and , we let , and do the following reduction.

Given a Max -CSP instance with constraints and variables. Let be the bipartite graph with left vertices and right vertices. For every constraint and every partial assignment to variables in the corresponding tuple which satisfies the constraint , we introduce a left vertex. For every variable and its corresponding assignment, we introduce a right vertex. Formally,

We connect a left vertex and right vertex when and is consistent with , i.e.

Now we define the final graph in which we want to find a dense -subgraph where . We take copies of the right vertices in to get . To get , we connect a left vertex and a right vertex if is connected to ’s corresponding vertex in in . The graph has vertices.

In our analysis of the reduction, we need a -ary linear code that has a small constant distance (but no less than ), small block length (but more than ), and very high dimension. Thus, we instantiate the code with Generalized BCH codes given by the following.

Lemma 4.6 (Generalized BCH Codes).

For every prime tower , and integer , there are -ary linear codes of block length , dimension , and distance at least .

We include a simple proof of Lemma 4.6 as follows.

Proof.

Let be a primitive element of . Let for notational ease. We construct the following code

We first show that the distance of is at least . Since is a linear code, we only need to show that every non-zero codeword has weight at least .

We show the contrapositive statement : the only codeword of weight at most is . For every codeword of weight at most , suppose the non-zero entries are in the set , we have

Note that the coefficients form a Vandermonde matrix (which has full rank). Therefore we have , i.e. the codeword is .

Now we show that the dimension of is at least . Note that each constraint can be implemented by linear constraints in (since ), while the constraint is indeed a linear constraint in . Therefore, we need at most linear constraints for , i.e. the dimension of is at least .

Finally, if the dimension of is more than , we can take a linear subspace of of dimension , while the distance of the subspace code is no less than the distance of . ∎

4.3 Analysis.

We get a family of gap instances parameterized by and (using Lemma 4.6). We obtain our two main results of this section by picking appropriate parameters for code as follows. To get lasserre integrality gaps for levels , we show the following by setting the distance .

Theorem 4.7.

For every (where is an absolute small constant), there is a gap instance of ratio for -level Lasserre SDP. The same construction also works for the Min degree Lasserre SDP, when and .

We now aim at getting a gap instance of ratio for polynomial-round Lasserre SDP, where is maximized. By setting for some small constant , the distance , and optimizing the other parameters, we obtain the following (refer to section 4.3 for details)

Theorem 4.8.

For small enough , there is a gap instance of ratio for the -round Min degree Lasserre SDP.

The two theorems follow because of Theorem 4.2, Lemma 4.9, Lemma 4.11 (completeness) and Lemma 4.12 (soundness). In the completeness case, we will use our -level Lasserre solution for Max -CSP  to show that the Lasserre SDP after levels of the hierarchy has value at least . In the soundness case, we show that with probability , the graph does not have any -subgraph of value more than times the SDP value (Lemma 4.12). Therefore, the graph is a gap instance of ratio for -round Lasserre SDP. We proceed by first proving these lemmas.

Completeness.

Lemma 4.9.

If the Max -CSP instance admits a perfect solution for -round Lasserre SDP relaxation, then the -round Lasserre SDP relaxation for the Densest -subgraph instance has a solution of value .

Proof.

For any set , suppose the left vertices included in are

and the right vertices included in are

where . Let

We have