972 research outputs found
Lower bounds for on-line graph colorings
We propose two strategies for Presenter in on-line graph coloring games. The
first one constructs bipartite graphs and forces any on-line coloring algorithm
to use colors, where is the number of vertices in the
constructed graph. This is best possible up to an additive constant. The second
strategy constructs graphs that contain neither nor as a subgraph
and forces colors. The best known
on-line coloring algorithm for these graphs uses colors
First-Fit is Linear on Posets Excluding Two Long Incomparable Chains
A poset is (r + s)-free if it does not contain two incomparable chains of
size r and s, respectively. We prove that when r and s are at least 2, the
First-Fit algorithm partitions every (r + s)-free poset P into at most
8(r-1)(s-1)w chains, where w is the width of P. This solves an open problem of
Bosek, Krawczyk, and Szczypka (SIAM J. Discrete Math., 23(4):1992--1999, 2010).Comment: v3: fixed some typo
Stackelberg Network Pricing is Hard to Approximate
In the Stackelberg Network Pricing problem, one has to assign tariffs to a
certain subset of the arcs of a given transportation network. The aim is to
maximize the amount paid by the user of the network, knowing that the user will
take a shortest st-path once the tariffs are fixed. Roch, Savard, and Marcotte
(Networks, Vol. 46(1), 57-67, 2005) proved that this problem is NP-hard, and
gave an O(log m)-approximation algorithm, where m denote the number of arcs to
be priced. In this note, we show that the problem is also APX-hard
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