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Cost Preserving Dependent Rounding for Allocation Problems

Lars Rohwedder, Arman Rouhani, Leo Wennmann

TL;DR

The paper introduces a cost-preserving dependent rounding framework for allocation problems, delivering Chernoff-like concentration while guaranteeing that the rounded integral solution does not exceed the fractional solution’s cost. By combining a convex-representation-based rounding with a TreeMerge procedure, the authors obtain an integral, degree-preserving solution that preserves total cost and achieves strong concentration on multiple linear functions. They apply this framework to the Budgeted Santa Claus Problem with identical valuations, formulating a two-tier LP (big vs. small resources) and proving an $O(\log n)$-approximation via careful dependent rounding of both resource categories. The approach yields polytime guarantees and polyhedral insights, and paves the way for further work on cost-preservation in other combinatorial settings, including potential extensions to matroid bases and more general allocation problems.

Abstract

We present a dependent randomized rounding scheme, which rounds fractional solutions to integral solutions satisfying certain hard constraints on the output while preserving Chernoff-like concentration properties. In contrast to previous dependent rounding schemes, our algorithm guarantees that the cost of the rounded integral solution does not exceed that of the fractional solution. Our algorithm works for a class of assignment problems with restrictions similar to those of prior works. In a non-trivial combination of our general result with a classical approach from Shmoys and Tardos [Math. Programm.'93] and more recent linear programming techniques developed for the restricted assignment variant by Bansal, Sviridenko [STOC'06] and Davies, Rothvoss, Zhang [SODA'20], we derive a O(log n)-approximation algorithm for the Budgeted Santa Claus Problem. In this new variant, the goal is to allocate resources with different values to players, maximizing the minimum value a player receives, and satisfying a budget constraint on player-resource allocation costs.

Cost Preserving Dependent Rounding for Allocation Problems

TL;DR

The paper introduces a cost-preserving dependent rounding framework for allocation problems, delivering Chernoff-like concentration while guaranteeing that the rounded integral solution does not exceed the fractional solution’s cost. By combining a convex-representation-based rounding with a TreeMerge procedure, the authors obtain an integral, degree-preserving solution that preserves total cost and achieves strong concentration on multiple linear functions. They apply this framework to the Budgeted Santa Claus Problem with identical valuations, formulating a two-tier LP (big vs. small resources) and proving an -approximation via careful dependent rounding of both resource categories. The approach yields polytime guarantees and polyhedral insights, and paves the way for further work on cost-preservation in other combinatorial settings, including potential extensions to matroid bases and more general allocation problems.

Abstract

We present a dependent randomized rounding scheme, which rounds fractional solutions to integral solutions satisfying certain hard constraints on the output while preserving Chernoff-like concentration properties. In contrast to previous dependent rounding schemes, our algorithm guarantees that the cost of the rounded integral solution does not exceed that of the fractional solution. Our algorithm works for a class of assignment problems with restrictions similar to those of prior works. In a non-trivial combination of our general result with a classical approach from Shmoys and Tardos [Math. Programm.'93] and more recent linear programming techniques developed for the restricted assignment variant by Bansal, Sviridenko [STOC'06] and Davies, Rothvoss, Zhang [SODA'20], we derive a O(log n)-approximation algorithm for the Budgeted Santa Claus Problem. In this new variant, the goal is to allocate resources with different values to players, maximizing the minimum value a player receives, and satisfying a budget constraint on player-resource allocation costs.

Paper Structure

This paper contains 12 sections, 16 theorems, 48 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Budgeted Dependent Rounding
  3. Omitted Proofs of Dependent Rounding
  4. Application to Budgeted Santa Claus Problem
  5. Linear Programming Formulation
  6. Technical Goals
  7. Rounding of Big Resources
  8. Rounding of Small Resources
  9. Approximation Factor
  10. Conclusion
  11. Appendix
  12. Santa Claus Problem where all resources need to be assigned

Key Result

proposition 1

Let $G = (A\cup B, E)$ be a bipartite graph and $x\in [0, 1]^{E}$ represent a fractional many-to-many assignment. Furthermore, let $c\in\RR^{E}$, and $a_1,\dotsc,a_k\in [0, 1]^E$ such that for each $j\in \set{1,\dotsc,k}$ there is some $v\in A\cup B$ with $\supp(a_j) \subseteq \delta(v)$. Then, in r Here, $\epsilon > 0$ is an arbitrarily small constant that influences the success probability.

Figures (1)

  • Figure 1: Bipartite graph G where on left side there are $k(i)$ copies of each player $i$ for an instance of two players with edges $(u_{i,\ell}, s)$ of weight $c_{is}$ for each $s = s(i,\ell-1),s(i,\ell-1)+1,\dotsc,s(i,\ell)$.

Theorems & Definitions (25)

  • proposition 1
  • theorem 1
  • theorem 2
  • theorem 2
  • lemma 1
  • lemma 2
  • lemma 3
  • lemma 4
  • proof : Proof of \ref{['thm:assign']}
  • lemma 5
  • ...and 15 more