Smoothed performance guarantees for local search

Abstract

We study popular local search and greedy algorithms for standard machine scheduling problems. The performance guarantee of these algorithms is well understood, but the worst-case lower bounds seem somewhat contrived and it is questionable whether they arise in practical applications. To find out how robust these bounds are, we study the algorithms in the framework of smoothed analysis, in which instances are subject to some degree of random noise. While the lower bounds for all scheduling variants with restricted machines are rather robust, we find out that the bounds are fragile for unrestricted machines. In particular, we show that the smoothed performance guarantee of the jump and the lex-jump algorithm are (in contrast to the worst case) independent of the number of machines. They are $\Theta (\phi )$ and $\Theta (\log \phi )$, respectively, where $1/\phi$ is a parameter measuring the magnitude of the perturbation. The latter immediately implies that also the smoothed price of anarchy is $\Theta (\log \phi )$ for routing games on parallel links. Additionally, we show that for unrestricted machines also the greedy list scheduling algorithm has an approximation guarantee of  $\Theta (\log \phi )$.

Appendices

Appendix A: Table of notation

In the table below, the notation used in this paper is summarized.

$J$	Set of jobs $1, \ldots , n$
$M$	Set of machines $1, \ldots , m$
$p_j$	Processing requirement of job $j$
$s_i$	Speed of machine $i$
$\mathcal{M }_j$	Set of machines on which job $j$ can be scheduled
$s_{\max }$	Maximum speed of the machines
$s_{\min }= 1$	Minimum speed of the machines;
	by scaling we assume w.l.o.g. it to be $1$
${C_{\max }^{*}}$	Optimal makespan
${C_{\max }}({\sigma })$	Makespan of schedule ${\sigma }$
${J_{i}}({\sigma })$	Set of jobs scheduling on machine $i$ in schedule ${\sigma }$
${L_{i}}({\sigma })$	$= \sum _{j \in {J_{i}}({\sigma })} p_j / s_i$
	load of machine $i$ in schedule ${\sigma }$
${J_{i,j}}({\sigma })$	$= {J_{i}}({\sigma }) \cap \{ 1, \ldots , j \}$
$j_i^t$	$= \min \left\{ j \,:\,\sum _{\ell \in {J_{i,j}}({\sigma })} p_{\ell } / s_i \ge t \cdot {C_{\max }^{*}}\right\}$
${J_{i,\ge t}}({\sigma })$	$= {J_{i,j_i^t}}({\sigma })$
$c$	$= \left\lfloor \frac{{C_{\max }}({\sigma })}{{C_{\max }^{*}}} \right\rfloor - 1$
$i_k$	$= \max \left\{ i \in M \,:\,{L_{i^{\prime }}} \ge k \cdot {C_{\max }^{*}} \, \forall \, i^{\prime } \le i \right\}$,
	assuming $s_1 \ge s_2 \ge \cdots \ge s_m$
$H_{k}$	$= \{1, \ldots , i_k\}$
$R_k$	$= H_k{\setminus }H_{k+1}$ for $k=0,1,\ldots , c-1$
$R_c$	$= H_c$

Appendix B: Hoeffding’s bound

On several occasions in this paper we use Hoeffding’s bound [16] to bound tail probabilities. For completeness, we state the bound in the following theorem.

Theorem 31

Let $X_1, \ldots , X_n$ be independent random variables. Define $X := \sum _{j=1}^n X_j$ and $\mu = {{\mathop {\mathbf{E}}}[X]}$. If each $X_j \in [a_j,b_j]$ for some constants $a_j$ and $b_j$, $j = 1,\ldots ,n$, then for any $t > 0$

$$\begin{aligned} {\mathop {\mathbf{Pr}}\limits _{}\left[ X \le {{\mathop {\mathbf{E}}}[X]} - t\right] }&\le \exp \left( \frac{-2t}{\sum _j (b_j - a_j)^2} \right) , \quad \text{ and, } \\ {\mathop {\mathbf{Pr}}\limits _{}\left[ X \ge {{\mathop {\mathbf{E}}}[X]} + t\right] }&\le \exp \left( \frac{-2t}{\sum _j (b_j - a_j)^2} \right) . \end{aligned}$$