A dual approach for dynamic pricing in multi-demand markets

Dynamic pricing schemes were introduced as an alternative to posted-price mechanisms. In contrast to static models, the dynamic setting allows to update the prices between buyer-arrivals based on the remaining sets of items and buyers, and so it is capable of maximizing social welfare without the need for a central coordinator. In this paper, we study the existence of optimal dynamic pricing schemes in combinatorial markets. In particular, we concentrate on multi-demand valuations, a natural extension of unit-demand valuations. The proposed approach is based on computing an optimal dual solution of the maximum social welfare problem with distinguished structural properties. Our contribution is twofold. By relying on an optimal dual solution, we show the existence of optimal dynamic prices in unit-demand markets and in multi-demand markets up to three buyers, thus giving new interpretations of results of Cohen-Addad et al. and Berger et al., respectively. Furthermore, we provide an optimal dynamic pricing scheme for bi-demand valuations with an arbitrary number of buyers. In all cases, our proofs also provide efficient algorithms for determining the optimal dynamic prices.Comment: 17 pages, 8 figure

A network flow approach to a common generalization of Clar and Fries numbers

Clar number and Fries number are two thoroughly investigated parameters of plane graphs emerging from mathematical chemistry to measure stability of organic molecules. We consider first a common generalization of these two concepts for bipartite plane graphs, and then extend it to a framework on general (not necessarily planar) directed graphs. The corresponding optimization problem can be transformed into a maximum weight feasible tension problem which is the linear programming dual of a minimum cost network flow (or circulation) problem. Therefore the approach gives rise to a min-max theorem and to a strongly polynomial algorithm that relies exclusively on standard network flow subroutines. In particular, we give the first network flow based algorithm for an optimal Fries structure and its variants

Envy-free Relaxations for Goods, Chores, and Mixed Items

In fair division problems, we are given a set $S$ of $m$ items and a set $N$ of $n$ agents with individual preferences, and the goal is to find an allocation of items among agents so that each agent finds the allocation fair. There are several established fairness concepts and envy-freeness is one of the most extensively studied ones. However envy-free allocations do not always exist when items are indivisible and this has motivated relaxations of envy-freeness: envy-freeness up to one item (EF1) and envy-freeness up to any item (EFX) are two well-studied relaxations. We consider the problem of finding EF1 and EFX allocations for utility functions that are not necessarily monotone, and propose four possible extensions of different strength to this setting. In particular, we present a polynomial-time algorithm for finding an EF1 allocation for two agents with arbitrary utility functions. An example is given showing that EFX allocations need not exist for two agents with non-monotone, non-additive, identical utility functions. However, when all agents have monotone (not necessarily additive) identical utility functions, we prove that an EFX allocation of chores always exists. As a step toward understanding the general case, we discuss two subclasses of utility functions: Boolean utilities that are $\{0,+1\}$-valued functions, and negative Boolean utilities that are $\{0,-1\}$-valued functions. For the latter, we give a polynomial time algorithm that finds an EFX allocation when the utility functions are identical.Comment: 21 pages, 1 figur

Worst case bin packing for OTN electrical layer networks dimensioning

The OTN (Optical Transport Network) standard, defined by ITU-T Recommendation G.709 and G.872, contains a flexible digital hierarchy of ODU (Optical Data Unit) signals. The ODU hierarchy provides sub-wavelength grooming in OTN networks, which is necessary for efficient utilization of the high bit rates of optical channels. When dimensioning the links of a transport network consisting of ODU switches, the packing of lower order ODU signals into higher order ODU signals needs to be taken into account. These networks are expected to be controlled by GMPLS (Generalized MPLS) , which puts specific constraints on the dimensioning. We assume that there is no explicit label control and that the GMPLS control plane is using first-fit strategy for making reservations on a link . With these assumptions the link dimensioning problem is defined as deciding how many higher order ODU component links are required on an OTN GMPLS bundled link for first-fit packing of a given set of lower order ODU demands, in any order of arrival. The paper provides strict bounds for ODU hierarchy-specific item and bin sizes. Then, it introduces an extended variant of the dimensioning problem, when lower order ODU connections which are not controlled by GMPLS are also present

Scalable and Efficient Multipath Routing: Complexity and Algorithms

A fundamental unsolved challenge in multipath routing is to provide disjoint end-to-end paths, each one satisfying certain operational goals (e.g., shortest possible), without overwhelming the data plane with prohibitive amount of forwarding state. In this paper, we study the problem of finding a pair of shortest disjoint paths that can be represented by only two forwarding table entries per destination. Building on prior work on minimum length redundant trees, we show that the underlying mathematical problem is NP-complete and we present heuristic algorithms that improve the known complexity bounds from cubic to the order of a single shortest path search. Finally, by extensive simulations we find that it is possible to very closely attain the absolute optimal path length with our algorithms (the gap is just 1–5%), eventually opening the door for wide-scale multipath routing deployments

Splitting property via shadow systems

Let M_k^r denote the set of r-element multisets over the set {1,...,k. We show that M_k^k has the so-called splitting property introduced by Ahlswede et al. Our approach gives a new interpretation of Sidorenko's construction and is applicable to give an upper bound on weighted Turán numbers, matching previous bounds. We also show how these results are connected to Tuza's conjecture on minimum triangle covers