643 research outputs found
The parallel approximability of a subclass of quadratic programming
In this paper we deal with the parallel approximability of a special class of Quadratic Programming (QP), called Smooth Positive Quadratic Programming. This subclass of QP is obtained by imposing restrictions on the coefficients of the QP instance. The Smoothness condition restricts the magnitudes of the coefficients while the positiveness requires that all the coefficients be non-negative. Interestingly, even with these restrictions several combinatorial problems can be modeled by Smooth QP. We show NC Approximation Schemes for the instances of Smooth Positive QP. This is done by reducing the instance of QP to an instance of Positive Linear Programming, finding in NC an approximate fractional solution to the obtained program, and then rounding the fractional solution to an integer approximate solution for the original problem. Then we show how to extend the result for positive instances of bounded degree to Smooth Integer Programming problems. Finally, we formulate several important combinatorial problems as Positive Quadratic Programs (or Positive Integer Programs) in packing/covering form and show that the techniques presented can be used to obtain NC Approximation Schemes for "dense" instances of such problems.Peer ReviewedPostprint (published version
On the Usefulness of Predicates
Motivated by the pervasiveness of strong inapproximability results for
Max-CSPs, we introduce a relaxed notion of an approximate solution of a
Max-CSP. In this relaxed version, loosely speaking, the algorithm is allowed to
replace the constraints of an instance by some other (possibly real-valued)
constraints, and then only needs to satisfy as many of the new constraints as
possible.
To be more precise, we introduce the following notion of a predicate
being \emph{useful} for a (real-valued) objective : given an almost
satisfiable Max- instance, there is an algorithm that beats a random
assignment on the corresponding Max- instance applied to the same sets of
literals. The standard notion of a nontrivial approximation algorithm for a
Max-CSP with predicate is exactly the same as saying that is useful for
itself.
We say that is useless if it is not useful for any . This turns out to
be equivalent to the following pseudo-randomness property: given an almost
satisfiable instance of Max- it is hard to find an assignment such that the
induced distribution on -bit strings defined by the instance is not
essentially uniform.
Under the Unique Games Conjecture, we give a complete and simple
characterization of useful Max-CSPs defined by a predicate: such a Max-CSP is
useless if and only if there is a pairwise independent distribution supported
on the satisfying assignments of the predicate. It is natural to also consider
the case when no negations are allowed in the CSP instance, and we derive a
similar complete characterization (under the UGC) there as well.
Finally, we also include some results and examples shedding additional light
on the approximability of certain Max-CSPs
Additive Approximation Algorithms for Modularity Maximization
The modularity is a quality function in community detection, which was
introduced by Newman and Girvan (2004). Community detection in graphs is now
often conducted through modularity maximization: given an undirected graph
, we are asked to find a partition of that maximizes
the modularity. Although numerous algorithms have been developed to date, most
of them have no theoretical approximation guarantee. Recently, to overcome this
issue, the design of modularity maximization algorithms with provable
approximation guarantees has attracted significant attention in the computer
science community.
In this study, we further investigate the approximability of modularity
maximization. More specifically, we propose a polynomial-time
-additive approximation algorithm for the
modularity maximization problem. Note here that
holds. This improves the current best additive approximation error of ,
which was recently provided by Dinh, Li, and Thai (2015). Interestingly, our
analysis also demonstrates that the proposed algorithm obtains a nearly-optimal
solution for any instance with a very high modularity value. Moreover, we
propose a polynomial-time -additive approximation algorithm for the
maximum modularity cut problem. It should be noted that this is the first
non-trivial approximability result for the problem. Finally, we demonstrate
that our approximation algorithm can be extended to some related problems.Comment: 23 pages, 4 figure
Beating Random Assignment for Approximating Quantum 2-Local Hamiltonian Problems
The quantum k-Local Hamiltonian problem is a natural generalization of classical constraint satisfaction problems (k-CSP) and is complete for QMA, a quantum analog of NP. Although the complexity of k-Local Hamiltonian problems has been well studied, only a handful of approximation results are known. For Max 2-Local Hamiltonian where each term is a rank 3 projector, a natural quantum generalization of classical Max 2-SAT, the best known approximation algorithm was the trivial random assignment, yielding a 0.75-approximation. We present the first approximation algorithm beating this bound, a classical polynomial-time 0.764-approximation. For strictly quadratic instances, which are maximally entangled instances, we provide a 0.801 approximation algorithm, and numerically demonstrate that our algorithm is likely a 0.821-approximation. We conjecture these are the hardest instances to approximate. We also give improved approximations for quantum generalizations of other related classical 2-CSPs. Finally, we exploit quantum connections to a generalization of the Grothendieck problem to obtain a classical constant-factor approximation for the physically relevant special case of strictly quadratic traceless 2-Local Hamiltonians on bipartite interaction graphs, where a inverse logarithmic approximation was the best previously known (for general interaction graphs). Our work employs recently developed techniques for analyzing classical approximations of CSPs and is intended to be accessible to both quantum information scientists and classical computer scientists
Approximate kernel clustering
In the kernel clustering problem we are given a large positive
semi-definite matrix with and a small
positive semi-definite matrix . The goal is to find a
partition of which maximizes the quantity We study the
computational complexity of this generic clustering problem which originates in
the theory of machine learning. We design a constant factor polynomial time
approximation algorithm for this problem, answering a question posed by Song,
Smola, Gretton and Borgwardt. In some cases we manage to compute the sharp
approximation threshold for this problem assuming the Unique Games Conjecture
(UGC). In particular, when is the identity matrix the UGC
hardness threshold of this problem is exactly . We present
and study a geometric conjecture of independent interest which we show would
imply that the UGC threshold when is the identity matrix is
for every
- …