A Lasserre-based $(1+\varepsilon)$-approximation for $Pm \mid p_j=1, \textrm{prec} \mid C_{\max}$

نویسندگان

  • Elaine Levey
  • Thomas Rothvoss
چکیده

In a classical problem in scheduling, one has n unit size jobs with a precedence order and the goal is to find a schedule of those jobs on m identical machines as to minimize the makespan. It is one of the remaining four open problems from the book of Garey & Johnson whether or not this problem is NP-hard form = 3. We prove that for any fixed ε and m, a Sherali-Adams / Lasserre lift of the timeindex LPwith a slightly super poly-logarithmic number of r = (log(n)) logn) rounds provides a (1+ ε)-approximation. This implies an algorithm that yields a (1+ ε)approximation in time n . The previously best approximation algorithms guarantee a 2− 7 3m+1 -approximation in polynomial time form ≥ 4 and 4 3 form = 3. Our algorithm is based on a recursive scheduling approach where in each step we reduce the correlation in form of long chains. Ourmethod adds to the rather short list of examples where hierarchies are actually useful to obtain better approximation algorithms.

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Efficient Approximation Algorithms for Point-set Diameter in Higher Dimensions

We study the problem of computing the diameter of a  set of $n$ points in $d$-dimensional Euclidean space for a fixed dimension $d$, and propose a new $(1+varepsilon)$-approximation algorithm with $O(n+ 1/varepsilon^{d-1})$ time and $O(n)$ space, where $0 < varepsilonleqslant 1$. We also show that the proposed algorithm can be modified to a $(1+O(varepsilon))$-approximation algorithm with $O(n+...

متن کامل

Deterministic Approximate Counting of Depth-2 Circuits

We describe deterministic algorithms which for a given depth-2 circuit $F$ approximate the probability that on a random input $F$ outputs a specific value $\alpha$. Our approach gives an algorithm which for a given $GF[2]$ multivariate polynomial $p$ and given $\varepsilon >0$ approximates the number of zeros of $p$ within a multiplicative factor $1+ \varepsilon$. The algorithm runs in time $ex...

متن کامل

Mid-IR 111-V Semiconductor Diode Lasers for Trace Gas Monitoring

It is widely recognized that the development of compact and efficient mid-IR (3-12 pm) sources would dramatically enhance chemical sensing capabilities since many gases of technological interest exhibit their fundamental absorption lines in this wavelength range. Examples of these include C& (3.3 pm), C02 (4.2 pm), CO (4.6 pm), and NO (5.3 pm). Though weaker overtone transitions in the 1-2 micr...

متن کامل

Faster Approximate(d) Text-to-Pattern L1 Distance

The problem of finding \emph{distance} between \emph{pattern} of length $m$ and \emph{text} of length $n$ is a typical way of generalizing pattern matching to incorporate dissimilarity score. For both Hamming and $L_1$ distances only a super linear upper bound $\widetilde{O}(n\sqrt{m})$ are known, which prompts the question of relaxing the problem: either by asking for $1 \pm \varepsilon$ appro...

متن کامل

Approximating CSPs with global cardinality constraints using SDP hierarchies

This work is concerned with approximating constraint satisfaction problems (CSPs) with an additional global cardinality constraints. For example, Max Cut is a boolean CSP where the input is a graph G = (V, E) and the goal is to find a cut S ∪ S̄ = V that maximizes the number of crossing edges, |E(S , S̄ )|. The Max Bisection problem is a variant of Max Cut with an additional global constraint tha...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:
  • CoRR

دوره abs/1509.07808  شماره 

صفحات  -

تاریخ انتشار 2015