[Paper Review] PTAS for Ordered Instances of Resource Allocation Problems with Restrictions on Inclusions

Kamyar Khodamoradi, Ramesh Krishnamurti+2 more|arXiv (Cornell University)|Oct 1, 2016

Supply Chain and Inventory Management4 references3 citations

TL;DR

This paper presents a Polynomial-Time Approximation Scheme (PTAS) for the Max-Min and Min-Max resource allocation problems under ordered and inclusion-free instances, where each player's interest in resources forms consecutive intervals. The approach uses dynamic programming on rounded instance values and remainder graph reconstruction to achieve a (1 + 4/k + 3/k²)-approximation, proving PTAS feasibility for convex bipartite graphs and inclusion-free bipartite permutation graphs.

ABSTRACT

We consider the problem of allocating a set $I$ of $m$ indivisible resources (items) to a set $P$ of $n$ customers (players) competing for the resources. Each resource $j \in I$ has a same value $v_j > 0$ for a subset of customers interested in $j$, and zero value for the remaining customers. The utility received by each customer is the sum of the values of the resources allocated to her. The goal is to find a feasible allocation of the resources to the interested customers such that for the Max-Min allocation problem (Min-Max allocation problem) the minimum of the utilities (maximum of the utilities) received by the customers is maximized (minimized). The Max-Min allocation problem is also known as the extit{Fair Allocation problem}, or the extit{Santa Claus problem}. The Min-Max allocation problem is the problem of Scheduling on Unrelated Parallel Machines, and is also known as the $R \, | \, | C_{\max}$ problem. In this paper, we are interested in instances of the problem that admit a Polynomial Time Approximation Scheme (PTAS). We show that an ordering property on the resources and the customers is important and paves the way for a PTAS. For the Max-Min allocation problem, we start with instances of the problem that can be viewed as a extit{convex bipartite graph}; a bipartite graph for which there exists an ordering of the resources such that each customer is interested in (has a positive evaluation for) a set of extit{consecutive} resources. We demonstrate a PTAS for the inclusion-free cases. This class of instances is equivalent to the class of bipartite permutation graphs. For the Min-Max allocation problem, we also obtain a PTAS for inclusion-free instances. These instances are not only of theoretical interest but also have practical applications.

Motivation & Objective

• To develop a PTAS for the Max-Min and Min-Max resource allocation problems under ordered and inclusion-free instances.
• To identify structural properties—specifically, ordering and inclusion-freeness—that enable the design of a PTAS.
• To establish that convex bipartite graphs and inclusion-free bipartite permutation graphs admit a PTAS for fair allocation and scheduling problems.
• To demonstrate that a dynamic programming approach with value rounding and remainder graph reconstruction yields a (1 + 4/k + 3/k²)-approximation.
• To explore the possibility of a dichotomy classification for PTAS-tractable vs. intractable instances of the resource allocation problem.

Proposed method

• Model the resource allocation problem as a bipartite graph H = (I, P, E), where I is the set of items and P the set of players, with edges indicating positive valuation.
• Use value rounding to discretize item values into rational multiples of 1/k, enabling polynomial-time dynamic programming over rounded vectors.
• Construct a dynamic programming table T of size m × n^{C+1} to track feasible assignments, where each entry stores a bit flag and a pointer to a prior state.
• Apply a forward-phase algorithm to fill the table by checking feasibility of partial assignments using a 'retrieve' function that reconstructs remainder graphs.
• Use backward reconstruction via pointers to recover an actual assignment from the table, ensuring the total load per machine is within (1 + 3/k) of the optimal in the rounded instance.
• Combine alignment error bounds and rounding error analysis to prove the overall approximation ratio of (1 + 4/k + 3/k²) for the original problem.

Experimental results

Research questions

• RQ1 Can a PTAS be designed for the Max-Min and Min-Max resource allocation problems under ordered and inclusion-free instances?
• RQ2 What structural properties of the bipartite graph—specifically ordering and inclusion-freeness—enable the existence of a PTAS?
• RQ3 To what extent do value rounding and remainder graph reconstruction preserve approximation guarantees in resource allocation problems?
• RQ4 Is there a dichotomy that separates bipartite graph classes admitting a PTAS from those that do not?
• RQ5 Can the dynamic programming approach be extended to margined-inclusive interval instances?

Key findings

• A PTAS is established for the Max-Min allocation problem on inclusion-free convex bipartite graphs, equivalent to bipartite permutation graphs.
• A PTAS is also developed for the Min-Max allocation problem (R||C_max) under the same inclusion-free and ordered constraints.
• The algorithm achieves an approximation ratio of (1 + 4/k + 3/k²) for both problems by combining value rounding and remainder graph reconstruction.
• The running time of the algorithm is O((m + n) × n^{2(C+1)}), which is polynomial in the input size for fixed k.
• The correctness of the algorithm is proven via a feasibility lemma showing that a (1 + 1/k)-assignment exists whenever a 1-assignment exists.
• The approach demonstrates that structural ordering and inclusion-freeness are sufficient conditions for PTAS existence in resource allocation problems.