# Linear Level Lasserre Lower Bounds for Certain k-CSPs

@article{Schoenebeck2008LinearLL, title={Linear Level Lasserre Lower Bounds for Certain k-CSPs}, author={Grant Robert Schoenebeck}, journal={2008 49th Annual IEEE Symposium on Foundations of Computer Science}, year={2008}, pages={593-602} }

We show that for kges3 even the Omega(n) level of the Lasserre hierarchy cannot disprove a random k-CSP instance over any predicate type implied by k-XOR constraints, for example k-SAT or k-XOR. (One constant is said to imply another if the latter is true whenever the former is. For example k-XOR constraints imply k-CNF constraints.) As a result the Omega(n) level Lasserre relaxation fails to approximate such CSPs betterthan the trivial, random algorithm. As corollaries, we obtain Omega(n…

## 253 Citations

### Approximating rectangles by juntas and weakly-exponential lower bounds for LP relaxations of CSPs

- Computer ScienceSTOC
- 2017

For constraint satisfaction problems (CSPs), sub-exponential size linear programming relaxations are as powerful as nΩ(1)-rounds of the Sherali-Adams linear programming hierarchy and lower bounds are obtained by exploiting and extending the recent progress in communication complexity for "lifting" query lower bounds to communication problems.

### Optimal Sherali-Adams Gaps from Pairwise Independence

- Computer Science, MathematicsAPPROX-RANDOM
- 2009

This work shows that if the set of assignments accepted by P contains the support of a balanced pairwise independent distribution over the domain of the inputs, then such a problem on n variables cannot be approximated better than the trivial (random) approximation, even using ***(n ) levels of the Sherali-Adams LP hierarchy.

### Size-degree trade-offs for sums-of-squares and positivstellensatz proofs

- Mathematics, Computer ScienceComputational Complexity Conference
- 2019

We show that if a system of degree-k polynomial constraints on n Boolean variables has a Sums-of-Squares (SOS) proof of unsatisfiability with at most s many monomials, then it also has one whose…

### CSP gaps and reductions in the lasserre hierarchy

- Mathematics, Computer ScienceSTOC '09
- 2009

The results for CSPs provide the first examples of Ω(n) round integrality gaps matching hardness results known only under the Unique Games Conjecture, which allow for gaps for in case of Independent Set and Chromatic Number which are stronger than the NP-hardness results known even under the unique gamesconjecture.

### Hardness of Maximum Constraint Satisfaction

- Mathematics, Computer Science
- 2013

The main ingredient is a new gap-amplification technique inspired by XOR-lemmas, which improves the NP-hardness of approximating Independent-Set on bounded-degree graphs, Almost-Coloring, Two-Prover-One-Round-Game, and various other problems.

### On the Power of Lasserre SDP Hierarchy

- Computer Science, Mathematics
- 2015

This work shows that Lasserre/Sum-of-Squares SDP solution achieves the best possible approximation ratio for all Max CSPs among all symmetric SDP relaxations of similar size.

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

- Computer Science, MathematicsSODA
- 2012

The results indicate that approximating Densest k-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 neΩ(1) rounds of the Lasserre hierarchy, where e is the completeness parameter in Unique Games and Small Set expansion.

### SDP Gaps from Pairwise Independence

- Computer Science, MathematicsTheory Comput.
- 2012

It is shown that for any promising predicate P, the integrality gap remains the same as the approximation ratio achieved by a random assignment, even after W(n) levels of the Sherali-Adams hierarchy.

### Sum of squares lower bounds for refuting any CSP

- Computer ScienceSTOC
- 2017

This work shows that if P is δ-close to supporting a t-wise uniform distribution on satisfying assignments, then the degree-Θ(n/Δ2/(t - 1) logΔ) SOS algorithm cannot (δ+o(1))-refute a random instance of CSP(P).

### Constant Factor Lasserre Integrality Gaps for Graph Partitioning Problems

- Computer ScienceSIAM J. Optim.
- 2014

It is proved that for balanced separator and uniform sparsest cut, semidefinite programs from the Lasserre hierarchy have an integrality gap bounded away from 1, even for $\Omega(n)$ levels of the hierarchy.

## References

SHOWING 1-10 OF 22 REFERENCES

### New Lower Bounds for Vertex Cover in the Lovasz-Schrijver Hierarchy

- Computer Science, Mathematics21st Annual IEEE Conference on Computational Complexity (CCC'06)
- 2006

It is conjecture that the new technique introduced to prove the lower bound for LS0, LS and LS+, the "fence" method, may lead to linear or nearly linear LS round lower bounds for VERTEX COVER, which are conjecture to be as large as the girth of the input graph for interesting graphs.

### Towards Strong Nonapproximability Results in the Lovász-Schrijver Hierarchy

- Computer Science, MathematicsSTOC '05
- 2005

It is shown that the relaxations produced by as many as Ω(n) rounds of the LS+ procedure do not allow nontrivial approximation, thus ruling out the possibility that the LS- approach gives even slightly subexponential approximation algorithms for well-known problems such as max-3sat, hypergraph vertex cover and set cover.

### Approximation Algorithms Using Hierarchies of Semidefinite Programming Relaxations

- Computer Science, Mathematics48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)
- 2007

A framework for studying semidefiniie programming (SOP) relaxations based on the Lasserre hierarchy in the context of approximation algorithms for combinatorial problems is introduced and improved approximation algorithms are given for two problems.

### Rank bounds and integrality gaps for cutting planes procedures

- Mathematics, Computer Science44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings.
- 2003

This work proves near-optimal rank bounds for Cutting Planes and Lovasz-Schrijver proofs for several prominent unsatisfiable CNF examples, including random kCNF formulas and the Tseitin graph formulas.

### Random 3CNF formulas elude the Lovasz theta function

- Computer Science, MathematicsElectron. Colloquium Comput. Complex.
- 2006

A simple nonconstructive argument shows that when m is sufficiently large, most 3CNF formulas are not satisfiable, and shows that for random formulas with m < n 3/2 o(1) clauses, the above approach fails.

### A Linear Round Lower Bound for Lovasz-Schrijver SDP Relaxations of Vertex Cover

- MathematicsTwenty-Second Annual IEEE Conference on Computational Complexity (CCC'07)
- 2007

It is proved that the integrality gap remains at least 7/6 - epsiv after cepivn rounds, where n is the number of vertices and cepsiv > 0 is a constant that depends only on epsIV.

### Linear programming relaxations of maxcut

- MathematicsSODA '07
- 2007

It is well-known that the integrality gap of the usual linear programming relaxation for Maxcut is 2 - ε. For general graphs, we prove that for any ε and any fixed bound k, adding linear constraints…

### The Lovász Theta Function and a Semidefinite Programming Relaxation of Vertex Cover

- MathematicsSIAM J. Discret. Math.
- 1998

It is shown that vc(G) can be more than $(2 - \varepsilon)$ times $|V|- \vartheta (G)$ for any $\varpsilon > 0$.

### On semidefinite programming relaxations for graph coloring and vertex cover

- Mathematics, Computer ScienceSODA '02
- 2002

We investigate the power of a strengthened SDP relaxation for graph coloring whose value is equal to a variant of the Lovász ϑ-function. We show families of graphs where the value of the relaxation…

### Tight integrality gaps for Lovasz-Schrijver LP relaxations of vertex cover and max cut

- Computer ScienceSTOC '07
- 2007

It is proved that the integrality gap of Vertex Cover remains at least 2-ε after Ω<sub>ε</sub> (n) rounds, and that the integration gap of Max Cut remains at most 1/2 +ε after â‚¬ afterΩ(sub)ε(n), where n is the number ofvertices.