On $k$-max-optimization

Abstract

We generalize bottleneck objectives in combinatorial optimization by minimizing the $k$th largest cost coefficient in a feasible solution. A bisection algorithm is presented which is based on iteratively solving an associated sum objective problem with binary cost coefficients. This algorithm is applicable to general combinatorial optimization problems.

Introduction

We consider combinatorial optimization problems where the kth largest cost coefficient of a feasible solution should be minimized. More precisely, let $E=\{e_1,\dots ,e_n\}$ denote a ground set of elements and let $F\subset 2^E$ denote a subset of the power set of $E$ which we refer to as the feasible set. We call any $S\in F$ a feasible solution. Our considerations are restricted to problems where $\left|S\right|\ge m$ for some $m\in \mathbb{N}$ and all $S\in F$. To simplify notation we may assume without loss of generality that $\left|S\right|=m$ for all $S\in F$. Let $c:E\to \mathbb{Z}$ be a cost function on the elements of the ground set. We may assume that the elements of $E$ are renumbered such that $c\left(e_1\right)\le \cdots \le c\left(e_n\right)$.

Typically, the objective of a combinatorial optimization problem is to minimize either the sum of costs ${\sum }_{e\in S}c\left(e\right)$ or the largest cost coefficient ${max}_{e\in S}c\left(e\right)$ over all feasible solutions $S$. We refer to these problems as combinatorial sum (CSP) and bottleneck (CBP) optimization problems, respectively.

In this paper we introduce a new type of objective function for combinatorial problems which can be seen as a generalization of the bottleneck objective. The task is to find a feasible solution $S\in F$ minimizing the kth largest cost coefficient in $S$.

Let $S=\{e_{i_1},\dots ,e_{i_m}\}$ be a feasible solution with $1\le i_1<\cdots <i_m\le n$. We define an operator $k\text{-}max$ which yields the kth largest among the elements of $S$, i.e., 

$$k\text{-}max\left(S\right)=k\text{-}\underset{e\in S}{max}c\left(e\right)=c\left(e_{i_{m-k+1}}\right)\text{.}$$

The problem of minimizing the kth largest cost coefficient can now be concisely formulated as

$$\left(k\text{MAX}\right)\underset{S\in F}{min}k\text{-}\underset{e\in S}{max}c\left(e\right)\text{.}$$

The special case $k=1$ is the well-known bottleneck or min-max problem.

To the best of our knowledge, problems of type ($k\mathrm{MAX}$) have not been studied in the literature. However, the notation of the kth largest element of a feasible set $S$ can be found in [1]. There, the minimization of the maximum deviation of the cost coefficients of a feasible solution $S\in F$ to its $k$th largest coefficient is discussed for general combinatorial optimization problems.

Although ($k\mathrm{MAX}$) has not been considered, there exist several other interesting generalizations of CBP. Some of them are mentioned in the following since they slightly resemble ($k\mathrm{MAX}$). In [2] the ground set $E$ is partitioned into $p$ non-empty disjoint subsets. For feasible $S\in F$ the objective is to minimize the sum of costs of those elements which have maximum cost in the subsets $S\cap E_k$ for $k=1,\dots ,p$. The considered problem generalizes CBP and CSP simultaneously, since for $p=1$ the problem simplifies to a CBP, while a CSP has to be solved for $p=\left|E\right|$. A generalized CSP is discussed in [3]. Instead of minimizing the complete sum of all cost coefficients of a feasible solution, only the sum of the $k$ largest cost coefficients is considered. Note that this approach also contains the bottleneck case since for $k=1$ the problem simplifies to a CBP, while for $k\ge max\{\left|S\right|:S\in F\}$ an ordinary CSP has to be solved. In lexicographic CBP (see [4], [5]), the largest cost coefficient has to be minimized in first place. Among all optimal solutions to this CBP, the second largest cost coefficient has to be optimized, then the third largest and so on. Moreover, there exist several studies on CBPs with fixed cardinality (for a survey, we refer to [6]), i.e. $\left|S\right|=m$ for all $S\in F$ is required.

Despite these existing generalizations of CBPs, ($k\mathrm{MAX}$) may enable the modeling of many real world problems which could not have been easily formulated so far. We want to demonstrate briefly its potential in image registration. In general registration problems two given data sets have to be rendered in a joint coordinate system such that corresponding features of these data sets are aligned. Such data sets normally correspond to 2- or 3-dimensional images as it is the case for example in medical image registration. For a given model set $A$ of $n$ distinct points in the plane (which may correspond for example to characteristic points of a given reference image) and an image set $B$ of the same cardinality (one may think of characteristic points in a template image), it is assumed that the points in $B$ correspond to points in $A$ but are afflicted with some data errors. The goal is to find the best possible assignment between the points of the two sets such that some distance measure between each pair of aligned points is as small as possible. Neither minimizing the average deviation of the points (which would be modeled by a CSP) nor the worst case assignment (this corresponds to a CBP) adequately deals with the noise-induced errors in set $B$. Instead, it seems more suitable to disregard those $k-1$ pairs of points which are furthest apart from each other. These $k-1$ assignments are considered outliers due to noise and the task in the optimization problem is to minimize the $k$th largest distance in the assignment of set $B$ to set $A$.