64 research outputs found
Single-Pass Pivot Algorithm for Correlation Clustering. Keep it simple!
We show that a simple single-pass semi-streaming variant of the Pivot
algorithm for Correlation Clustering gives a (3 + {\epsilon})-approximation
using O(n/{\epsilon}) words of memory. This is a slight improvement over the
recent results of Cambus, Kuhn, Lindy, Pai, and Uitto, who gave a (3 +
{\epsilon})-approximation using O(n log n) words of memory, and Behnezhad,
Charikar, Ma, and Tan, who gave a 5-approximation using O(n) words of memory.
One of the main contributions of this paper is that both the algorithm and its
analysis are very simple, and also the algorithm is easy to implement
Mixed-integer convex representability
Motivated by recent advances in solution methods for mixed-integer convex
optimization (MICP), we study the fundamental and open question of which sets
can be represented exactly as feasible regions of MICP problems. We establish
several results in this direction, including the first complete
characterization for the mixed-binary case and a simple necessary condition for
the general case. We use the latter to derive the first non-representability
results for various non-convex sets such as the set of rank-1 matrices and the
set of prime numbers. Finally, in correspondence with the seminal work on
mixed-integer linear representability by Jeroslow and Lowe, we study the
representability question under rationality assumptions. Under these
rationality assumptions, we establish that representable sets obey strong
regularity properties such as periodicity, and we provide a complete
characterization of representable subsets of the natural numbers and of
representable compact sets. Interestingly, in the case of subsets of natural
numbers, our results provide a clear separation between the mathematical
modeling power of mixed-integer linear and mixed-integer convex optimization.
In the case of compact sets, our results imply that using unbounded integer
variables is necessary only for modeling unbounded sets
New Dependencies of Hierarchies in Polynomial Optimization
We compare four key hierarchies for solving Constrained Polynomial
Optimization Problems (CPOP): Sum of Squares (SOS), Sum of Diagonally Dominant
Polynomials (SDSOS), Sum of Nonnegative Circuits (SONC), and the Sherali Adams
(SA) hierarchies. We prove a collection of dependencies among these hierarchies
both for general CPOPs and for optimization problems on the Boolean hypercube.
Key results include for the general case that the SONC and SOS hierarchy are
polynomially incomparable, while SDSOS is contained in SONC. A direct
consequence is the non-existence of a Putinar-like Positivstellensatz for
SDSOS. On the Boolean hypercube, we show as a main result that Schm\"udgen-like
versions of the hierarchies SDSOS*, SONC*, and SA* are polynomially equivalent.
Moreover, we show that SA* is contained in any Schm\"udgen-like hierarchy that
provides a O(n) degree bound.Comment: 26 pages, 4 figure
Robust Sparsification for Matroid Intersection with Applications
Matroid intersection is a classical optimization problem where, given two
matroids over the same ground set, the goal is to find the largest common
independent set. In this paper, we show that there exists a certain
"sparsifer": a subset of elements, of size ,
where denotes the optimal solution, that is guaranteed to contain a
approximation, while guaranteeing certain robustness
properties. We call such a small subset a Density Constrained Subset (DCS),
which is inspired by the Edge-Degree Constrained Subgraph (EDCS) [Bernstein and
Stein, 2015], originally designed for the maximum cardinality matching problem
in a graph. Our proof is constructive and hinges on a greedy decomposition of
matroids, which we call the density-based decomposition. We show that this
sparsifier has certain robustness properties that can be used in one-way
communication and random-order streaming models
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