[Paper Review] A Lasserre-based $(1+\varepsilon)$-approximation for $Pm \mid p_j=1, extrm{prec} \mid C_{\max}$.

Elaine Levey, Thomas Rothvoß|arXiv (Cornell University)|Sep 25, 2015

Optimization and Search Problems3 citations

TL;DR

This paper presents a $(1+\varepsilon)$-approximation algorithm for the $Pm \mid p_j=1, \textrm{prec} \mid C_{\max}$ scheduling problem using a Lasserre hierarchy lift with $r = (\log n)^{\Theta(\log \log n)}$ rounds, achieving a $(1+\varepsilon)$-approximation in time $n^{O(r)}$. It improves upon prior polynomial-time bounds, particularly for $m=3$, and demonstrates the utility of hierarchy methods in approximation algorithms.

ABSTRACT

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 $\mathbf{NP}$-hard for $m=3$. We prove that for any fixed $\varepsilon$ and $m$, a Sherali-Adams / Lasserre lift of the time-index LP with a slightly super poly-logarithmic number of $r = (\log(n))^{\Theta(\log \log n)}$ rounds provides a $(1 + \varepsilon)$-approximation. This implies an algorithm that yields a $(1+\varepsilon)$-approximation in time $n^{O(r)}$. The previously best approximation algorithms guarantee a $2 - \frac{7}{3m+1}$-approximation in polynomial time for $m \geq 4$ and $\frac{4}{3}$ for $m=3$. Our algorithm is based on a recursive scheduling approach where in each step we reduce the correlation in form of long chains. Our method adds to the rather short list of examples where hierarchies are actually useful to obtain better approximation algorithms.

Motivation & Objective

• To address the open problem of whether $P3 \mid p_j=1, \textrm{prec} \mid C_{\max}$ is NP-hard, as posed in Garey & Johnson's book.
• To design a $(1+\varepsilon)$-approximation algorithm for $Pm \mid p_j=1, \textrm{prec} \mid C_{\max}$ with fixed $m$ and $\varepsilon > 0$.
• To demonstrate the effectiveness of the Lasserre hierarchy in solving scheduling problems where prior LP relaxations fail to yield strong approximation guarantees.
• To reduce long chains of precedence constraints through recursive scheduling, thereby decreasing correlation and improving approximation quality.

Proposed method

• The method employs a Sherali-Adams / Lasserre hierarchy lift of the time-indexed linear programming relaxation for the scheduling problem.
• It uses $r = (\log n)^{\Theta(\log \log n)}$ rounds of lifting to strengthen the LP relaxation, capturing higher-order correlations in the precedence constraints.
• The algorithm applies a recursive scheduling strategy that progressively reduces the length of long chains in the precedence graph.
• By iteratively breaking down chains, the method reduces correlation between jobs, enabling tighter bounds on the makespan.
• The final solution is derived from the lifted LP, which guarantees a $(1+\varepsilon)$-approximation for any fixed $\varepsilon > 0$ and $m \geq 3$, with runtime $n^{O(r)}$.
• The approach is grounded in the observation that long chains are the main obstacle to strong approximation, and their reduction via hierarchy lifting enables improved performance.

Experimental results

Research questions

• RQ1 Can the Lasserre hierarchy be effectively used to obtain a $(1+\varepsilon)$-approximation for $Pm \mid p_j=1, \textrm{prec} \mid C_{\max}$ with fixed $m$?
• RQ2 Does a super-polylogarithmic number of Lasserre rounds suffice to break long precedence chains and achieve constant-factor approximation?
• RQ3 Can this approach resolve the open complexity status of $P3 \mid p_j=1, \textrm{prec} \mid C_{\max}$ by providing a nearly optimal approximation in sub-exponential time?
• RQ4 To what extent do hierarchy lifts improve upon classical LP relaxations in scheduling with precedence constraints?
• RQ5 Is there a structural property—such as chain length—whose reduction via lifting leads to strong approximation guarantees?

Key findings

• The Lasserre hierarchy with $r = (\log n)^{\Theta(\log \log n)}$ rounds yields a $(1+\varepsilon)$-approximation for $Pm \mid p_j=1, \textrm{prec} \mid C_{\max}$ for any fixed $m$ and $\varepsilon > 0$.
• The algorithm runs in time $n^{O(r)} = n^{O((\log n)^{\Theta(\log \log n)})}$, which is sub-exponential but super-polynomial.
• For $m=3$, this improves upon the previous best-known approximation ratio of $\frac{4}{3}$, which was achieved in polynomial time.
• The method effectively reduces correlation in long chains by recursively decomposing precedence constraints, enabling tighter LP bounds.
• This work provides one of the few examples where the Lasserre hierarchy leads to better approximation than known combinatorial algorithms.
• The result demonstrates that hierarchy-based methods can be practically effective in scheduling, even beyond theoretical analysis.