KAIST Discrete Math Seminar

We consider a problem of minimizing the number of batches of a fixed capacity processing the orders of various sizes on a finite set of items. This batch consolidation problem is motivated by the batch production system typical in raw material industries in which multiple items can be processed in the same batch in case they share sufficiently close production parameters. We focuse on the special case in which up to 2 items can be processed in a single batch. The problem is NP-hard and can not be approximated within 1.00142 of the optimum under the premise, P ≠ NP as can be shown by a polynomial reduction from the vertex cover problem with bounded degree. However, the problem admits a 3/2 -approximation. The idea is to decompose the orders of items so that a maximum matching in the graph on the vertices of the decomposed orders provides a well-consolidated batch set.

Uncoordinated individuals in human society pursuing their personally optimal strategies do not always achieve the social optimum, the most beneficial state to the society as a whole. Instead, strategies form Nash equilibria, which are, in general, socially suboptimal. Society, therefore, has to pay a price of anarchy for the lack of coordination among its members, which is often difficult to quantify in engineering, economics and policymaking. Here, we report on an assessment of this price of anarchy by analyzing the road networks of Boston, New York, and London as well as complex model networks, where one’s travel time serves as the relevant cost to be minimized. Our simulation shows that uncoordinated drivers possibly spend up to 30% more time than they would in socially optimal traffic, which leaves substantial room for improvement. Counterintuitively, simply blocking certain streets can partially improve the traffic condition to a measurable extent based on our result.

One-sentence summaries: Traffic in transportation networks – often substantially suboptimal due to a lack of coordination among users – can be guided to the social optimum without directly controlling individual behavior by adjusting the underlying network structure appropriately.

Many problems in discrete and computational geometry can be reduced to combinatorial questions concerning systems of segments or triangles in the plane. In spite of what (little) we can say about these planar questions, much less is known in higher dimensions. In this talk we will survey some of these questions and classical results, and present the following theorem: Let d>2 and n>d+1 and P a set of n points in d-dimensional Euclidean space. Then P contains a subset Q of d points such that for any point p in P, the simplex spanned by Q and p does not contain the origin in its interior. This answers a question by R.Strausz, and along the way we strengthen the Colored Helly and Caratheodory theorems, due to L. Lovász and I. Bárány, respectively.

Joint work with János Pach and Helge Tverberg.