2,058 research outputs found
Sub-exponential Approximation Schemes for CSPs: From Dense to Almost Sparse
It has long been known, since the classical work of (Arora, Karger, Karpinski, JCSS\u2799), that MAX-CUT admits a PTAS on dense graphs, and more generally, MAX-k-CSP admits a PTAS on "dense" instances with Omega(n^k) constraints. In this paper we extend and generalize their exhaustive sampling approach, presenting a framework for (1-epsilon)-approximating any MAX-k-CSP problem in sub-exponential time while significantly relaxing the denseness requirement on the input instance.
Specifically, we prove that for any constants delta in (0, 1] and epsilon > 0, we can approximate MAX-k-CSP problems with Omega(n^{k-1+delta}) constraints within a factor of (1-epsilon) in time 2^{O(n^{1-delta}*ln(n) / epsilon^3)}. The framework is quite general and includes classical optimization problems, such as MAX-CUT, MAX-DICUT, MAX-k-SAT, and (with a slight extension) k-DENSEST SUBGRAPH, as special cases. For MAX-CUT in particular (where k=2), it gives an approximation scheme that runs in time sub-exponential in n even for "almost-sparse" instances (graphs with n^{1+delta} edges).
We prove that our results are essentially best possible, assuming the ETH. First, the density requirement cannot be relaxed further: there exists a constant r 0, MAX-k-SAT instances with O(n^{k-1}) clauses cannot be approximated within a ratio better than r in time 2^{O(n^{1-delta})}. Second, the running time of our algorithm is almost tight for all densities. Even for MAX-CUT there exists r delta >0, MAX-CUT instances with n^{1+delta} edges cannot be approximated within a ratio better than r in time 2^{n^{1-delta\u27}}
Product-state approximations to quantum ground states
The local Hamiltonian problem consists of estimating the ground-state energy (given by the minimum eigenvalue) of a local quantum Hamiltonian. It can be considered as a quantum generalization of constraint satisfaction problems (CSPs) and has a key role in quantum complexity theory, being the first and most natural QMA-complete problem known. An interesting regime for the local Hamiltonian problem is that of extensive error, where one is interested in estimating the mean ground-state energy to constant accuracy. The problem is NP-hard by the PCP theorem, but whether it is QMA-hard is an important open question in quantum complexity theory. A positive solution would represent a quantum analogue of the PCP theorem. A key feature that distinguishes quantum Hamiltonians from classical CSPs is that the solutions may involve complicated entangled states. In this paper, we demonstrate several large classes of Hamiltonians for which product (i.e. unentangled) states can approximate the ground state energy to within a small extensive error.
First, we show the mere existence of a good product-state approximation for the ground-state energy of 2-local Hamiltonians with one of more of the following properties: (1) super-constant degree, (2) small expansion, or (3) a ground state with sublinear entanglement with respect to some partition into small pieces. The approximation based on degree is a new and surprising difference between quantum Hamiltonians and classical CSPs, since in the classical setting, higher degree is usually associated with harder CSPs. The approximation based on expansion is not new, but the approximation based on low entanglement was previously known only in the regime where the entanglement was close to zero. Since the existence of a low-energy product state can be checked in NP, this implies that any Hamiltonian used for a quantum PCP theorem should have: (1) constant degree, (2) constant expansion, (3) a ``volume law'' for entanglement with respect to any partition into small parts.
Second, we show that in several cases, good product-state approximations not only exist, but can be found in deterministic polynomial time: (1) 2-local Hamiltonians on any planar graph, solving an open problem of Bansal, Bravyi, and Terhal, (2) dense k-local Hamiltonians for any constant k, solving an open problem of Gharibian and Kempe, and (3) 2-local Hamiltonians on graphs with low threshold rank, via a quantum generalization of a recent result of Barak, Raghavendra and Steurer.
Our work involves two new tools which may be of independent interest. First, we prove a new quantum version of the de Finetti theorem which does not require the usual assumption of symmetry. Second, we describe a way to analyze the application of the Lasserre/Parrilo SDP hierarchy to local quantum Hamiltonians
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
The algebraic structure of the densification and the sparsification tasks for CSPs
The tractability of certain CSPs for dense or sparse instances is known from
the 90s. Recently, the densification and the sparsification of CSPs were
formulated as computational tasks and the systematical study of their
computational complexity was initiated.
We approach this problem by introducing the densification operator, i.e. the
closure operator that, given an instance of a CSP, outputs all constraints that
are satisfied by all of its solutions. According to the Galois theory of
closure operators, any such operator is related to a certain implicational
system (or, a functional dependency) . We are specifically interested
in those classes of fixed-template CSPs, parameterized by constraint languages
, for which the size of an implicational system is a
polynomial in the number of variables . We show that in the Boolean case,
is of polynomial size if and only if is of bounded width. For
such languages, can be computed in log-space or in a logarithmic time
with a polynomial number of processors. Given an implicational system ,
the densification task is equivalent to the computation of the closure of input
constraints. The sparsification task is equivalent to the computation of the
minimal key. This leads to -algorithm for
the sparsification task where is the number of non-redundant
sparsifications of an original CSP.
Finally, we give a complete classification of constraint languages over the
Boolean domain for which the densification problem is tractable
Exploiting Dense Structures in Parameterized Complexity
Over the past few decades, the study of dense structures from the perspective of approximation algorithms has become a wide area of research. However, from the viewpoint of parameterized algorithm, this area is largely unexplored. In particular, properties of random samples have been successfully deployed to design approximation schemes for a number of fundamental problems on dense structures [Arora et al. FOCS 1995, Goldreich et al. FOCS 1996, Giotis and Guruswami SODA 2006, Karpinksi and Schudy STOC 2009]. In this paper, we fill this gap, and harness the power of random samples as well as structure theory to design kernelization as well as parameterized algorithms on dense structures. In particular, we obtain linear vertex kernels for Edge-Disjoint Paths, Edge Odd Cycle Transversal, Minimum Bisection, d-Way Cut, Multiway Cut and Multicut on everywhere dense graphs. In fact, these kernels are obtained by designing a polynomial-time algorithm when the corresponding parameter is at most ?(n). Additionally, we obtain a cubic kernel for Vertex-Disjoint Paths on everywhere dense graphs. In addition to kernelization results, we obtain randomized subexponential-time parameterized algorithms for Edge Odd Cycle Transversal, Minimum Bisection, and d-Way Cut. Finally, we show how all of our results (as well as EPASes for these problems) can be de-randomized
PTAS for Sparse General-Valued CSPs
We study polynomial-time approximation schemes (PTASes) for constraint
satisfaction problems (CSPs) such as Maximum Independent Set or Minimum Vertex
Cover on sparse graph classes. Baker's approach gives a PTAS on planar graphs,
excluded-minor classes, and beyond. For Max-CSPs, and even more generally,
maximisation finite-valued CSPs (where constraints are arbitrary non-negative
functions), Romero, Wrochna, and \v{Z}ivn\'y [SODA'21] showed that the
Sherali-Adams LP relaxation gives a simple PTAS for all
fractionally-treewidth-fragile classes, which is the most general "sparsity"
condition for which a PTAS is known. We extend these results to general-valued
CSPs, which include "crisp" (or "strict") constraints that have to be satisfied
by every feasible assignment. The only condition on the crisp constraints is
that their domain contains an element which is at least as feasible as all the
others (but possibly less valuable). For minimisation general-valued CSPs with
crisp constraints, we present a PTAS for all Baker graph classes -- a
definition by Dvo\v{r}\'ak [SODA'20] which encompasses all classes where
Baker's technique is known to work, except possibly for
fractionally-treewidth-fragile classes. While this is standard for problems
satisfying a certain monotonicity condition on crisp constraints, we show this
can be relaxed to diagonalisability -- a property of relational structures
connected to logics, statistical physics, and random CSPs
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