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Dependent rounding with strong negative-correlation, and scheduling on unrelated machines to minimize completion time

David G. Harris

TL;DR

The paper develops a general dependent-rounding framework that achieves strong negative correlation via negatively correlated Exponential variables and applies it to scheduling on unrelated machines. The DepRound algorithm provides margin-preserving, highly negatively correlated rounding on bipartite graphs, enabling a 1.398-approximation for the classic R || w_j C_j problem when combined with an SDP relaxation. Key innovations include a flexible correlation parameterization, a contamination-free contention-resolution scheme, and a rigorous analysis showing the SDP integrality gap is at most 1.398. This yields practical improvements over prior methods and introduces techniques likely applicable to a broader class of combinatorial optimization problems. The results enhance our understanding of how dependent rounding with strong negative correlation can sharpen approximation guarantees for complex scheduling and related tasks.

Abstract

We describe a new dependent-rounding algorithmic framework for bipartite graphs. Given a fractional assignment $\vec x$ of values to edges of graph $G = (U \cup V, E)$, the algorithms return an integral solution $\vec X$ such that each right-node $v \in V$ has at most one neighboring edge $f$ with $X_f = 1$, and the variables $X_e$ also satisfy broad nonpositive-correlation properties. In particular, for any edges $e_1, e_2$ sharing a left-node $u \in U$, the variables $X_{e_1}, X_{e_2}$ have strong negative correlation, i.e. the expectation of $X_{e_1} X_{e_2}$ is significantly below $x_{e_1} x_{e_2}$. This algorithm is based on generating negatively-correlated Exponential random variables and using them in a contention-resolution scheme inspired by an algorithm Im & Shadloo (2020). Our algorithm gives stronger and much more flexible negative correlation properties. Dependent rounding schemes with negative correlation properties have been used for approximation algorithms for job-scheduling on unrelated machines to minimize weighted completion times (Bansal, Srinivasan, & Svensson (2021), Im & Shadloo (2020), Im & Li (2023)). Using our new dependent-rounding algorithm, among other improvements, we obtain a $1.398$-approximation for this problem. This significantly improves over the prior $1.45$-approximation ratio of Im & Li (2023).

Dependent rounding with strong negative-correlation, and scheduling on unrelated machines to minimize completion time

TL;DR

The paper develops a general dependent-rounding framework that achieves strong negative correlation via negatively correlated Exponential variables and applies it to scheduling on unrelated machines. The DepRound algorithm provides margin-preserving, highly negatively correlated rounding on bipartite graphs, enabling a 1.398-approximation for the classic R || w_j C_j problem when combined with an SDP relaxation. Key innovations include a flexible correlation parameterization, a contamination-free contention-resolution scheme, and a rigorous analysis showing the SDP integrality gap is at most 1.398. This yields practical improvements over prior methods and introduces techniques likely applicable to a broader class of combinatorial optimization problems. The results enhance our understanding of how dependent rounding with strong negative correlation can sharpen approximation guarantees for complex scheduling and related tasks.

Abstract

We describe a new dependent-rounding algorithmic framework for bipartite graphs. Given a fractional assignment of values to edges of graph , the algorithms return an integral solution such that each right-node has at most one neighboring edge with , and the variables also satisfy broad nonpositive-correlation properties. In particular, for any edges sharing a left-node , the variables have strong negative correlation, i.e. the expectation of is significantly below . This algorithm is based on generating negatively-correlated Exponential random variables and using them in a contention-resolution scheme inspired by an algorithm Im & Shadloo (2020). Our algorithm gives stronger and much more flexible negative correlation properties. Dependent rounding schemes with negative correlation properties have been used for approximation algorithms for job-scheduling on unrelated machines to minimize weighted completion times (Bansal, Srinivasan, & Svensson (2021), Im & Shadloo (2020), Im & Li (2023)). Using our new dependent-rounding algorithm, among other improvements, we obtain a -approximation for this problem. This significantly improves over the prior -approximation ratio of Im & Li (2023).
Paper Structure (18 sections, 32 theorems, 107 equations, 3 algorithms)

This paper contains 18 sections, 32 theorems, 107 equations, 3 algorithms.

Table of Contents

  1. Introduction
  2. Definitions and results for the bipartite-rounding setting
  3. Machine-scheduling on unrelated machines
  4. Note:
  5. Negatively-correlated Exponential random variables
  6. Bipartite dependent rounding algorithm
  7. Scheduling to minimize completion time
  8. The SDP relaxation
  9. Focusing on a single machine and job.
  10. Determining the approximation ratio
  11. The algorithm upper-bound
  12. The algorithm lower-bound
  13. Marrying the upper and lower bounds
  14. Acknowledgments
  15. Proof of Lemma \ref{['gatt7']}
  16. ...and 3 more sections

Key Result

Theorem 1

Suppose $\sum_{e \in \Gamma(v)} x_v \leq 1$ for all right-nodes $v \in V$, and $\sum_{e \in \Gamma(u)} \rho_u \leq 1$ for all left-nodes $u \in U$. Then the algorithm DepRound satisfies the following properties (A1) --- (A4):

Theorems & Definitions (59)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Theorem 4
  • proof
  • Proposition 6
  • proof
  • Proposition 7
  • proof
  • Lemma 8
  • ...and 49 more