Robust balanced optimization

An instance of a balanced optimization problem with vector costs consists of a ground set X, a cost-vector for every element of X, and a system of feasible subsets over X. The goal is to find a feasible subset that minimizes the so-called imbalance of values in every coordinate of the underlying vector costs. Balanced optimization problems with vector costs are equivalent to the robust optimization version of balanced optimization problems under the min-max criterion. We regard these problems as a family of optimization problems; one particular member of this family is the known balanced assignment problem. We investigate the complexity and approximability of robust balanced optimization problems in a fairly general setting. We identify a large family of problems that admit a 2-approximation in polynomial time, and we show that for many problems in this family this approximation factor 2 is best-possible (unless P = NP). We pay special attention to the balanced assignment problem with vector costs and show that this problem is NP-hard even in the highly restricted case of sum costs. We conclude by performing computational experiments for this problem

The transportation problem with conflicts

The transportation problem is a fundamental problem in operations research, where items need to be transported from supply nodes (each with a given supply) to demand nodes (each with a given demand) in the cheapest possible way. Here, we are interested in a generalization of the transportation problem where, each supply node has a (possibly empty) set of conflicting pairs of demand nodes, and each demand node a (possibly empty) set of conflicting pairs of supply nodes. Each supply node may only send supply to at most one demand node of each conflicting pair. Likewise, each demand node may only receive supply from at most one supply node of each conflicting pair. We call the resulting problem the transportation problem with conflicts (TPC). We show that the complexity of TPC depends upon the structure of the so-called conflict graph that follows from the conflicting pairs. More concrete, we show that for many graph-classes the corresponding TPC remains NP-hard, and for some special cases we derive constant factor approximation algorithms

Balanced Optimization with Vector Costs

\u3cp\u3eAn instance of a balanced optimization problem with vector costs consists of a ground set X, a vector cost for every element of X, and a system of feasible subsets over X. The goal is to find a feasible subset that minimizes the spread (or imbalance) of values in every coordinate of the underlying vector costs. We investigate the complexity and approximability of balanced optimization problems in a fairly general setting. We identify a large family of problems that admit a 2-approximation in polynomial time, and we show that for many problems in this family this approximation factor 2 is best-possible (unless P=NP). Special attention is paid to the balanced assignment problem with vector costs, which is shown to be NP-hard even in the highly restricted case of sum costs.\u3c/p\u3

Partitioning vectors into quadruples: Worst-case analysis of a matching-based algorithm

Consider a problem where 4k given vectors need to be partitioned into k clusters of four vectors each. A cluster of four vectors is called a quad, and the cost of a quad is the sum of the component-wise maxima of the four vectors in the quad. The problem is to partition the given 4k vectors into k quads with minimum total cost. We analyze a straightforward matching-based algorithm and prove that this algorithm is a 23 -approximation algorithm for this problem. We further analyze the performance of this algorithm on a hierarchy of special cases of the problem and prove that, in one particular case, the algorithm is a 54 -approximation algorithm. Our analysis is tight in all cases except one

Round-robin tournaments generated by the circle method have maximum carry-over

The Circle Method is widely used in the field of sport scheduling to generate schedules for round-robin tournaments. The so-called carry-over effect value is a number that can be associated to each round-robin schedule; it represents a degree of balance of a schedule. Here, we prove that, for an even number of teams, the Circle Method generates a schedule with maximum carry-over effect value, answering an open question