7 research outputs found
Approximating Dense Max 2-CSPs
In this paper, we present a polynomial-time algorithm that approximates
sufficiently high-value Max 2-CSPs on sufficiently dense graphs to within
approximation ratio for any constant .
Using this algorithm, we also achieve similar results for free games,
projection games on sufficiently dense random graphs, and the Densest
-Subgraph problem with sufficiently dense optimal solution. Note, however,
that algorithms with similar guarantees to the last algorithm were in fact
discovered prior to our work by Feige et al. and Suzuki and Tokuyama.
In addition, our idea for the above algorithms yields the following
by-product: a quasi-polynomial time approximation scheme (QPTAS) for
satisfiable dense Max 2-CSPs with better running time than the known
algorithms
The Strongish Planted Clique Hypothesis and Its Consequences
We formulate a new hardness assumption, the Strongish Planted Clique Hypothesis (SPCH), which postulates that any algorithm for planted clique must run in time n^?(log n) (so that the state-of-the-art running time of n^O(log n) is optimal up to a constant in the exponent).
We provide two sets of applications of the new hypothesis. First, we show that SPCH implies (nearly) tight inapproximability results for the following well-studied problems in terms of the parameter k: Densest k-Subgraph, Smallest k-Edge Subgraph, Densest k-Subhypergraph, Steiner k-Forest, and Directed Steiner Network with k terminal pairs. For example, we show, under SPCH, that no polynomial time algorithm achieves o(k)-approximation for Densest k-Subgraph. This inapproximability ratio improves upon the previous best k^o(1) factor from (Chalermsook et al., FOCS 2017). Furthermore, our lower bounds hold even against fixed-parameter tractable algorithms with parameter k.
Our second application focuses on the complexity of graph pattern detection. For both induced and non-induced graph pattern detection, we prove hardness results under SPCH, improving the running time lower bounds obtained by (Dalirrooyfard et al., STOC 2019) under the Exponential Time Hypothesis
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
Pliability and approximating max-CSPs
We identify a sufficient condition, treewidth-pliability, that gives a polynomial-time
algorithm for an arbitrarily good approximation of the optimal value in a large class of
Max-2-CSPs parameterised by the class of allowed constraint graphs (with arbitrary constraints on an unbounded alphabet). Our result applies more generally to the maximum
homomorphism problem between two rational-valued structures.
The condition unifies the two main approaches for designing a polynomial-time approximation scheme. One is Baker’s layering technique, which applies to sparse graphs
such as planar or excluded-minor graphs. The other is based on Szemer´edi’s regularity
lemma and applies to dense graphs. We extend the applicability of both techniques to
new classes of Max-CSPs. On the other hand, we prove that the condition cannot be used
to find solutions (as opposed to approximating the optimal value) in general.
Treewidth-pliability turns out to be a robust notion that can be defined in several
equivalent ways, including characterisations via size, treedepth, or the Hadwiger number.
We show connections to the notions of fractional-treewidth-fragility from structural graph
theory, hyperfiniteness from the area of property testing, and regularity partitions from
the theory of dense graph limits. These may be of independent interest. In particular
we show that a monotone class of graphs is hyperfinite if and only if it is fractionallytreewidth-fragile and has bounded degree
The Complexity of Finding Dense Subgraphs in Graphs with Large Cliques
The GapDensest-k-Subgraph(d) problem (GapDkS(d)) is defined as follows: given a graph G and parameters k,d, distinguish between the case that G contains a k-clique, and the case that every k-subgraph of G has density at most d.
GapDkS(d) is a natural relaxation of the standard Clique problem, which is known to be NP-complete. For d very close to 1, the GapDkS(d) problem is equivalent to the Clique problem, and when d is very close to 0 the GapDkS(d) problem can easily be solved in polynomial time. However, despite much work on both the algorithmic and hardness front, the exact k and d parameter values for which GapDkS(d) can be solved in polynomial time are still unknown. In particular, the best polynomial-time algorithms can solve GapDkS(d) when d is an inverse polynomial in the number of vertices n, but there have been no NP-hardness results beyond the trivial result.
This thesis attempts to understand the GapDkS(d) problem better by studying the case when k is restricted to be linear in n (where n is the number of vertices in G). In particular, we survey the GapDkS(d) algorithms and hardness results that can be best applied to this restriction in an attempt to determine the threshold for when the problem becomes NP-hard. With some modifications to the algorithms and proofs, we produce algorithms and hardness results for the GapDkS(d) problem with k linear in n.
In addition, we study the connection between GapDkS(d) and MaxClique, and show that despite having strong hardness results for MaxClique, reductions from MaxClique do not give strong hardness bounds for GapDkS(d)
Treewidth-Pliability and PTAS for Max-CSPs
We identify a sufficient condition, treewidth-pliability, that gives a
polynomial-time approximation scheme (PTAS) for a large class of Max-2-CSPs
parametrised by the class of allowed constraint graphs (with arbitrary
constraints on an unbounded alphabet). Our result applies more generally to the
maximum homomorphism problem between two rational-valued structures.
The condition unifies the two main approaches for designing PTASes. One is
Baker's layering technique, which applies to sparse graphs such as planar or
excluded-minor graphs. The other is based on Szemer\'{e}di's regularity lemma
and applies to dense graphs. We extend the applicability of both techniques to
new classes of Max-CSPs.
Treewidth-pliability turns out to be a robust notion that can be defined in
several equivalent ways, including characterisations via size, treedepth, or
the Hadwiger number. We show connections to the notions of
fractional-treewidth-fragility from structural graph theory, hyperfiniteness
from the area of property testing, and regularity partitions from the theory of
dense graph limits. These may be of independent interest. In particular we show
that a monotone class of graphs is hyperfinite if and only if it is
fractionally-treewidth-fragile and has bounded degree