# Stochastic Generalized Assignment Problem In Linear

## Title: Sinkhorn Algorithm for Lifted Assignment Problems

Authors:Yam Kushinsky, Haggai Maron, Nadav Dym, Yaron Lipman

(Submitted on 23 Jul 2017)

Abstract: Recently, Sinkhorn's algorithm was applied for solving regularized linear programs emerging from optimal transport very efficiently. Sinkhorn's algorithm is an efficient method of projecting a positive matrix onto the polytope of doubly-stochastic matrices. It is based on alternating closed-form Bregman projections on the larger polytopes of row-stochastic and column-stochastic matrices.

In this paper we generalize the Sinkhorn projection algorithm to higher dimensional polytopes originated from well-known lifted linear program relaxations of the Markov Random Field (MRF) energy minimization problem and the Quadratic Assignment Problem (QAP). We derive a closed-form projection on one-sided local polytopes which can be seen as a high-dimensional, generalized version of the row/column-stochastic polytopes. We then use these projections to devise a provably convergent algorithm to solve regularized linear program relaxations of MRF and QAP. Furthermore, as the regularization is decreased both the solution and the optimal energy value converge to that of the respective linear program. The resulting algorithm is considerably more scalable than standard linear solvers and is able to solve significantly larger linear programs.

## Submission history

From: Yam Kushinsky Mr. [view email]**[v1]**Sun, 23 Jul 2017 12:01:27 GMT (1597kb,D)

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In applied mathematics, the maximum **generalized assignment problem** is a problem in combinatorial optimization. This problem is a generalization of the assignment problem in which both tasks and agents have a size. Moreover, the size of each task might vary from one agent to the other.

This problem in its most general form is as follows:

There are a number of agents and a number of tasks. Any agent can be assigned to perform any task, incurring some cost and profit that may vary depending on the agent-task assignment. Moreover, each agent has a budget and the sum of the costs of tasks assigned to it cannot exceed this budget. It is required to find an assignment in which all agents do not exceed their budget and total profit of the assignment is maximized.

## In special cases[edit]

In the special case in which all the agents' budgets and all tasks' costs are equal to 1, this problem reduces to the assignment problem. When the costs and profits of all agents-task assignment are equal, this problem reduces to the multiple knapsack problem. If there is a single agent, then, this problem reduces to the knapsack problem.

## Explanation of definition[edit]

In the following, we have *n* kinds of items, through and *m* kinds of bins through . Each bin is associated with a budget . For a bin , each item has a profit and a weight . A solution is an assignment from items to bins. A feasible solution is a solution in which for each bin the total weight of assigned items is at most . The solution's profit is the sum of profits for each item-bin assignment. The goal is to find a maximum profit feasible solution.

Mathematically the generalized assignment problem can be formulated as an integer program:

## Complexity[edit]

The generalized assignment problem is NP-hard,^{[1]} and it is even APX-hard to approximate it. Recently it was shown that an extension of it is hard to approximate for every .^{[citation needed]}

## Greedy approximation algorithm[edit]

Using any -approximation algorithm ALG for the knapsack problem, it is possible to construct a ()-approximation for the generalized assignment problem in a greedy manner using a residual profit concept. The algorithm constructs a schedule in iterations, where during iteration a tentative selection of items to bin is selected. The selection for bin might change as items might be reselected in a later iteration for other bins. The residual profit of an item for bin is if is not selected for any other bin or – if is selected for bin .

Formally: We use a vector to indicate the tentative schedule during the algorithm. Specifically, means the item is scheduled on bin and means that item is not scheduled. The residual profit in iteration is denoted by , where if item is not scheduled (i.e. ) and if item is scheduled on bin (i.e. ).

Formally:

- Set
- For do:
- Call ALG to find a solution to bin using the residual profit function . Denote the selected items by .
- Update using , i.e., for all .

## See also[edit]

## References[edit]

- Reuven Cohen, Liran Katzir, and Danny Raz, "An Efficient Approximation for the Generalized Assignment Problem", Information Processing Letters, Vol. 100, Issue 4, pp. 162–166, November 2006.
- Lisa Fleischer, Michel X. Goemans, Vahab S. Mirrokni, and Maxim Sviridenko, "Tight Approximation Algorithms for Maximum General Assignment Problems", SODA 2006, pp. 611–620.
- Hans Kellerer, Ulrich Pferschy, David Pisinger,
*Knapsack Problems*, 2005. Springer Verlag ISBN 3-540-40286-1

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