125 research outputs found
Hardness of robust graph isomorphism, Lasserre gaps, and asymmetry of random graphs
Building on work of Cai, F\"urer, and Immerman \cite{CFI92}, we show two
hardness results for the Graph Isomorphism problem. First, we show that there
are pairs of nonisomorphic -vertex graphs and such that any
sum-of-squares (SOS) proof of nonisomorphism requires degree . In
other words, we show an -round integrality gap for the Lasserre SDP
relaxation. In fact, we show this for pairs and which are not even
-isomorphic. (Here we say that two -vertex, -edge graphs
and are -isomorphic if there is a bijection between their
vertices which preserves at least edges.) Our second result is that
under the {\sc R3XOR} Hypothesis \cite{Fei02} (and also any of a class of
hypotheses which generalize the {\sc R3XOR} Hypothesis), the \emph{robust}
Graph Isomorphism problem is hard. I.e.\ for every , there is no
efficient algorithm which can distinguish graph pairs which are
-isomorphic from pairs which are not even
-isomorphic for some universal constant . Along the
way we prove a robust asymmetry result for random graphs and hypergraphs which
may be of independent interest
A Birthday Repetition Theorem and Complexity of Approximating Dense CSPs
A -birthday repetition of a
two-prover game is a game in which the two provers are sent
random sets of questions from of sizes and respectively.
These two sets are sampled independently uniformly among all sets of questions
of those particular sizes. We prove the following birthday repetition theorem:
when satisfies some mild conditions, decreases exponentially in where is the total number of
questions. Our result positively resolves an open question posted by Aaronson,
Impagliazzo and Moshkovitz (CCC 2014).
As an application of our birthday repetition theorem, we obtain new
fine-grained hardness of approximation results for dense CSPs. Specifically, we
establish a tight trade-off between running time and approximation ratio for
dense CSPs by showing conditional lower bounds, integrality gaps and
approximation algorithms. In particular, for any sufficiently large and for
every , we show the following results:
- We exhibit an -approximation algorithm for dense Max -CSPs
with alphabet size via -level of Sherali-Adams relaxation.
- Through our birthday repetition theorem, we obtain an integrality gap of
for -level Lasserre relaxation for fully-dense Max
-CSP.
- Assuming that there is a constant such that Max 3SAT cannot
be approximated to within of the optimal in sub-exponential
time, our birthday repetition theorem implies that any algorithm that
approximates fully-dense Max -CSP to within a factor takes
time, almost tightly matching the algorithmic
result based on Sherali-Adams relaxation.Comment: 45 page
Mildly Exponential Time Approximation Algorithms for Vertex Cover, Balanced Separator and Uniform Sparsest Cut
In this work, we study the trade-off between the running time of approximation algorithms and their approximation guarantees. By leveraging a structure of the "hard" instances of the Arora-Rao-Vazirani lemma [Sanjeev Arora et al., 2009; James R. Lee, 2005], we show that the Sum-of-Squares hierarchy can be adapted to provide "fast", but still exponential time, approximation algorithms for several problems in the regime where they are believed to be NP-hard. Specifically, our framework yields the following algorithms; here n denote the number of vertices of the graph and r can be any positive real number greater than 1 (possibly depending on n).
- A (2 - 1/(O(r)))-approximation algorithm for Vertex Cover that runs in exp (n/(2^{r^2)})n^{O(1)} time.
- An O(r)-approximation algorithms for Uniform Sparsest Cut and Balanced Separator that runs in exp (n/(2^{r^2)})n^{O(1)} time.
Our algorithm for Vertex Cover improves upon Bansal et al.\u27s algorithm [Nikhil Bansal et al., 2017] which achieves (2 - 1/(O(r)))-approximation in time exp (n/(r^r))n^{O(1)}. For Uniform Sparsest Cut and Balanced Separator, our algorithms improve upon O(r)-approximation exp (n/(2^r))n^{O(1)}-time algorithms that follow from a work of Charikar et al. [Moses Charikar et al., 2010]
Definability of semidefinite programming and lasserre lower bounds for CSPs
We show that the ellipsoid method for solving semidefinite
programs (SDPs) can be expressed in fixed-point logic
with counting (FPC). This generalizes an earlier result that the
optimal value of a linear program can be expressed in this logic.
As an application, we establish lower bounds on the number
of levels of the Lasserre hierarchy required to solve many
optimization problems, namely those that can be expressed
as finite-valued constraint satisfaction problems (VCSPs). In
particular, we establish a dichotomy on the number of levels
of the Lasserre hierarchy that are required to solve the problem
exactly. We show that if a finite-valued constraint problem is not
solved exactly by its basic linear programming relaxation, it is
also not solved exactly by any sub-linear number of levels of the
Lasserre hierarchy.
The lower bounds are established through logical undefinability
results. We show that the SDP corresponding to any
fixed level of the Lasserre hierarchy is interpretable in a VCSP
instance by means of FPC formulas. Our definability result of
the ellipsoid method then implies that the solution of this SDP
can be expressed in this logic. Together, these results give a way
of translating lower bounds on the number of variables required
in counting logic to express a VCSP into lower bounds on the
number of levels required in the Lasserre hierarchy to eliminate
the integrality gap.
As a special case, we obtain the same dichotomy for the class of
MAXCSP problems, generalizing earlier Lasserre lower bound
results by Schoenebeck [17]. Recently, and independently of the
work reported here, a similar linear lower bound in the Lasserre
hierarchy for general-valued CSPs has also been announced by
Thapper and Zivny [20], using different techniques
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