A lazy approach to on-line bipartite matching

We present a new approach, called a lazy matching, to the problem of on-line matching on bipartite graphs. Imagine that one side of a graph is given and the vertices of the other side are arriving on-line. Originally, incoming vertex is either irrevocably matched to an another element or stays forever unmatched. A lazy algorithm is allowed to match a new vertex to a group of elements (possibly empty) and afterwords, forced against next vertices, may give up parts of the group. The restriction is that all the time each element is in at most one group. We present an optimal lazy algorithm (deterministic) and prove that its competitive ratio equals $1-\pi/\cosh(\frac{\sqrt{3}}{2}\pi)\approx 0.588$. The lazy approach allows us to break the barrier of $1/2$, which is the best competitive ratio that can be guaranteed by any deterministic algorithm in the classical on-line matching

Deferred on-line bipartite matching

We present a new model for the problem of on-line matching on bipartite graphs. Suppose that one part of a graph is given, but the vertices of the other part are presented in an on-line fashion. In the classical version, each incoming vertex is either irrevocably matched to a vertex from the other part or stays unmatched forever. In our version, an algorithm is allowed to match the new vertex to a group of elements (possibly empty). Later on, the algorithm can decide to remove some vertices from the group and assign them to another (just presented) vertex, with the restriction that each element belongs to at most one group. We present an optimal (deterministic) algorithm for this problem and prove that its competitive ratio equals

An easy subexponential bound for online chain partitioning

Bosek and Krawczyk exhibited an online algorithm for partitioning an online poset of width $w$ into $w^{14\lg w}$ chains. We improve this to $w^{6.5 \lg w + 7}$ with a simpler and shorter proof by combining the work of Bosek & Krawczyk with work of Kierstead & Smith on First-Fit chain partitioning of ladder-free posets. We also provide examples illustrating the limits of our approach.Comment: 23 pages, 11 figure

On the Duality of Semiantichains and Unichain Coverings

We study a min-max relation conjectured by Saks and West: For any two posets $P$ and $Q$ the size of a maximum semiantichain and the size of a minimum unichain covering in the product $P\times Q$ are equal. For positive we state conditions on $P$ and $Q$ that imply the min-max relation. Based on these conditions we identify some new families of posets where the conjecture holds and get easy proofs for several instances where the conjecture had been verified before. However, we also have examples showing that in general the min-max relation is false, i.e., we disprove the Saks-West conjecture.Comment: 10 pages, 3 figure

On the density and the structure of the Peirce-like formulae

International audienceWithin the language of propositional formulae built on implication and a finite number of variables $k$, we analyze the set of formulae which are classical tautologies but not intuitionistic (we call such formulae - Peirce's formulae). We construct the large family of so called simple Peirce's formulae, whose sequence of densities for different $k$ is asymptotically equivalent to the sequence $\frac{1}{ 2 k^2}$. We prove that the densities of the sets of remaining Peirce's formulae are asymptotically bounded from above by $\frac{c}{ k^3}$ for some constant $c \in \mathbb{R}$. The result justifies the statement that in the considered language almost all Peirce's formulae are simple. The result gives a partial answer to the question stated in the recent paper by H. Fournier, D. Gardy, A. Genitrini and M. Zaionc - although we have not proved the existence of the densities for Peirce's formulae, our result gives lower and upper bound for it (if it exists) and both bounds are asymptotically equivalent to $\frac{1}{ 2 k^2}$

First-fit coloring of incomparability graphs

One of the simplest heuristics for obtaining a proper coloring of a graph is the first-fit algorithm. First-fit visits each vertex of the graph in the specified order and assigns to every point the least possible number. Let $\mathcal{G}$ be a class of incomparability graphs with bounded maximum clique size, closed under taking induced subgraphs. We prove that first-fit uses a bounded number of colors on the graphs in $\mathcal{G}$ iff there is an incomparability graph of clique size $2$ not contained in $\mathcal{G}$

On the Duality of Semiantichains and Unichain Coverings

Abstract We study a min-max relation conjectured by Saks and West: For any two posets P and Q the size of a maximum semiantichain and the size of a minimum unichain covering in the product P × Q are equal. For positive we state conditions on P and Q that imply the min-max relation. Based on these conditions we identify some new families of posets where the conjecture holds and get easy proofs for several instances where the conjecture had been verified before. However, we also have examples showing that in general the min-max relation is false, i.e., we disprove the Saks-West conjecture. Mathematics Subject Classifications (2010) 06A07, 05B40, 90C46