1,954 research outputs found
OV Graphs Are (Probably) Hard Instances
© Josh Alman and Virginia Vassilevska Williams. A graph G on n nodes is an Orthogonal Vectors (OV) graph of dimension d if there are vectors v1, . . ., vn ∈ {0, 1}d such that nodes i and j are adjacent in G if and only if hvi, vji = 0 over Z. In this paper, we study a number of basic graph algorithm problems, except where one is given as input the vectors defining an OV graph instead of a general graph. We show that for each of the following problems, an algorithm solving it faster on such OV graphs G of dimension only d = O(log n) than in the general case would refute a plausible conjecture about the time required to solve sparse MAX-k-SAT instances: Determining whether G contains a triangle. More generally, determining whether G contains a directed k-cycle for any k ≥ 3. Computing the square of the adjacency matrix of G over Z or F2. Maintaining the shortest distance between two fixed nodes of G, or whether G has a perfect matching, when G is a dynamically updating OV graph. We also prove some complementary results about OV graphs. We show that any problem which is NP-hard on constant-degree graphs is also NP-hard on OV graphs of dimension O(log n), and we give two problems which can be solved faster on OV graphs than in general: Maximum Clique, and Online Matrix-Vector Multiplication
Syntactic Separation of Subset Satisfiability Problems
Variants of the Exponential Time Hypothesis (ETH) have been used to derive lower bounds on the time complexity for certain problems, so that the hardness results match long-standing algorithmic results. In this paper, we consider a syntactically defined class of problems, and give conditions for when problems in this class require strongly exponential time to approximate to within a factor of (1-epsilon) for some constant epsilon > 0, assuming the Gap Exponential Time Hypothesis (Gap-ETH), versus when they admit a PTAS. Our class includes a rich set of problems from additive combinatorics, computational geometry, and graph theory. Our hardness results also match the best known algorithmic results for these problems
Learning and Testing Variable Partitions
Let be a multivariate function from a product set to an
Abelian group . A -partition of with cost is a partition of
the set of variables into non-empty subsets such that is -close to
for some with
respect to a given error metric. We study algorithms for agnostically learning
partitions and testing -partitionability over various groups and error
metrics given query access to . In particular we show that
Given a function that has a -partition of cost , a partition
of cost can be learned in time
for any .
In contrast, for and learning a partition of cost is NP-hard.
When is real-valued and the error metric is the 2-norm, a
2-partition of cost can be learned in time
.
When is -valued and the error metric is Hamming
weight, -partitionability is testable with one-sided error and
non-adaptive queries. We also show that even
two-sided testers require queries when .
This work was motivated by reinforcement learning control tasks in which the
set of control variables can be partitioned. The partitioning reduces the task
into multiple lower-dimensional ones that are relatively easier to learn. Our
second algorithm empirically increases the scores attained over previous
heuristic partitioning methods applied in this context.Comment: Innovations in Theoretical Computer Science (ITCS) 202
Limits to Non-Malleability
There have been many successes in constructing explicit non-malleable codes for various classes of tampering functions in recent years, and strong existential results are also known. In this work we ask the following question:
When can we rule out the existence of a non-malleable code for a tampering class ??
First, we start with some classes where positive results are well-known, and show that when these classes are extended in a natural way, non-malleable codes are no longer possible. Specifically, we show that no non-malleable codes exist for any of the following tampering classes:
- Functions that change d/2 symbols, where d is the distance of the code;
- Functions where each input symbol affects only a single output symbol;
- Functions where each of the n output bits is a function of n-log n input bits.
Furthermore, we rule out constructions of non-malleable codes for certain classes ? via reductions to the assumption that a distributional problem is hard for ?, that make black-box use of the tampering functions in the proof. In particular, this yields concrete obstacles for the construction of efficient codes for NC, even assuming average-case variants of P ? NC
The power of sum-of-squares for detecting hidden structures
We study planted problems---finding hidden structures in random noisy
inputs---through the lens of the sum-of-squares semidefinite programming
hierarchy (SoS). This family of powerful semidefinite programs has recently
yielded many new algorithms for planted problems, often achieving the best
known polynomial-time guarantees in terms of accuracy of recovered solutions
and robustness to noise. One theme in recent work is the design of spectral
algorithms which match the guarantees of SoS algorithms for planted problems.
Classical spectral algorithms are often unable to accomplish this: the twist in
these new spectral algorithms is the use of spectral structure of matrices
whose entries are low-degree polynomials of the input variables. We prove that
for a wide class of planted problems, including refuting random constraint
satisfaction problems, tensor and sparse PCA, densest-k-subgraph, community
detection in stochastic block models, planted clique, and others, eigenvalues
of degree-d matrix polynomials are as powerful as SoS semidefinite programs of
roughly degree d. For such problems it is therefore always possible to match
the guarantees of SoS without solving a large semidefinite program. Using
related ideas on SoS algorithms and low-degree matrix polynomials (and inspired
by recent work on SoS and the planted clique problem by Barak et al.), we prove
new nearly-tight SoS lower bounds for the tensor and sparse principal component
analysis problems. Our lower bounds for sparse principal component analysis are
the first to suggest that going beyond existing algorithms for this problem may
require sub-exponential time
Bounded Representations of Interval and Proper Interval Graphs
Klavik et al. [arXiv:1207.6960] recently introduced a generalization of
recognition called the bounded representation problem which we study for the
classes of interval and proper interval graphs. The input gives a graph G and
in addition for each vertex v two intervals L_v and R_v called bounds. We ask
whether there exists a bounded representation in which each interval I_v has
its left endpoint in L_v and its right endpoint in R_v. We show that the
problem can be solved in linear time for interval graphs and in quadratic time
for proper interval graphs.
Robert's Theorem states that the classes of proper interval graphs and unit
interval graphs are equal. Surprisingly the bounded representation problem is
polynomially solvable for proper interval graphs and NP-complete for unit
interval graphs [Klav\'{\i}k et al., arxiv:1207.6960]. So unless P = NP, the
proper and unit interval representations behave very differently.
The bounded representation problem belongs to a wider class of restricted
representation problems. These problems are generalizations of the
well-understood recognition problem, and they ask whether there exists a
representation of G satisfying some additional constraints. The bounded
representation problems generalize many of these problems
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