[Paper Review] Approximation-Friendly Discrepancy Rounding

Nikhil Bansal, Viswanath Nagarajan|arXiv (Cornell University)|Dec 7, 2015

Advanced Numerical Analysis Techniques3 citations

TL;DR

This paper introduces an enhanced discrepancy-based rounding algorithm that unifies the strengths of randomized and iterative rounding, enabling multiplicative error bounds and scalability to exponentially large LPs with matroid or laminar constraints. It achieves improved congestion bounds in routing and matroid problems, such as degree-bounded matroid basis and low-congestion routing, with additive violations of $ O( ext{polylog}(n) imes ext{sqrt}(b_e)) $ or $ O( ext{sqrt}( ext{deg}) imes\text{polylog}(n)) $.

ABSTRACT

Rounding linear programs using techniques from discrepancy is a recent approach that has been very successful in certain settings. However this method also has some limitations when compared to approaches such as randomized and iterative rounding. We provide an extension of the discrepancy-based rounding algorithm due to Lovett-Meka that (i) combines the advantages of both randomized and iterated rounding, (ii) makes it applicable to settings with more general combinatorial structure such as matroids. As applications of this approach, we obtain new results for various classical problems such as linear system rounding, degree-bounded matroid basis and low congestion routing.

Motivation & Objective

• To overcome limitations of existing discrepancy-based rounding, such as weak multiplicative error guarantees and poor performance on exponentially large LPs.
• To extend the Lovett-Meka discrepancy rounding framework to support multiplicative error bounds and combinatorial structures like matroids and laminar families.
• To unify the advantages of randomized rounding (multiplicative error) and iterative rounding (handling complex combinatorial constraints) within a single discrepancy-based framework.
• To provide new approximation guarantees for classical problems like degree-bounded matroid basis, low-congestion routing, and linear system rounding with violations.

Proposed method

• Extend the Lovett-Meka (LM) discrepancy rounding algorithm to incorporate Chernoff-type multiplicative error bounds via a modified potential function and iterative rounding steps.
• Introduce a partial rounding procedure that maintains feasibility for matroid and laminar constraints while controlling error per constraint using $ \lambda_j $-dependent bounds.
• Adapt the algorithm to work with exponentially large LPs by exploiting underlying combinatorial structures, such as path sets in routing problems.
• Use a time-expanded network formulation to solve the LP relaxation in polynomial time despite exponential variable count.
• Apply the rounding framework to problems with degree constraints and laminar requirements by modeling them as matroid or laminar constraints on path selections.
• Leverage the fact that the number of tight laminar constraints at any fractional solution is at most half the ground set size, enabling efficient partial rounding.

Experimental results

Research questions

• RQ1 Can discrepancy-based rounding be extended to achieve multiplicative error bounds similar to randomized rounding, while retaining the structural advantages of iterative rounding?
• RQ2 How can discrepancy rounding be adapted to handle exponentially large LPs with complex combinatorial constraints such as matroids or laminar families?
• RQ3 Can a single rounding framework simultaneously achieve guarantees from discrepancy, randomized rounding, and iterative rounding for problems like degree-bounded matroid basis and low-congestion routing?
• RQ4 What is the tightest possible additive violation for edge congestion in routing problems when using a discrepancy-based approach with matroid or laminar constraints?
• RQ5 Is it possible to obtain existential and algorithmic guarantees for multi-budgeted or laminar-constrained routing problems using a unified discrepancy framework?

Key findings

• The proposed algorithm achieves multiplicative error bounds of the form $ O(\sqrt{b_e} \log n + \log^2 n) $ for edge congestion in low-congestion routing, matching the best-known randomized rounding guarantees.
• For degree-bounded matroid basis, the algorithm obtains a solution where each element’s degree is at most $ b_e + O(\sqrt{\Delta} \log n) $, where $ \Delta $ is the maximum path length.
• The method provides a $ O(\sqrt{\Delta} \log n) $-approximation for linear system rounding with violations, improving over prior discrepancy-based results.
• The framework extends to laminar-constrained routing, ensuring that all laminar requirements are met while violating edge capacities by at most $ O(\sqrt{\Delta} \log n) $ or $ O(\sqrt{b_e} \log n + \log^2 n) $.
• The algorithm maintains $ O(\log n) $-factor loss in error bounds compared to ideal randomized rounding, while enabling efficient computation on large-scale LPs.
• The approach yields the first known algorithmic results with such guarantees for multi-budgeted matroid basis and laminar-constrained routing, previously unknown even existentially.