Online Graph Coloring Against a Randomized Adversary

We consider an online model where an adversary constructs a set of [Formula: see text] instances [Formula: see text] instead of one single instance. The algorithm knows [Formula: see text] and the adversary will choose one instance from [Formula: see text] at random to present to the algorithm. We further focus on adversaries that construct sets of [Formula: see text]-chromatic instances. In this setting, we provide upper and lower bounds on the competitive ratio for the online graph coloring problem as a function of the parameters in this model. Both bounds are linear in [Formula: see text] and matching upper and lower bound are given for a specific set of algorithms that we call “minimalistic online algorithms”.

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Coloring Chains for Compression with Uncertain Priors

The Electronic Journal of Combinatorics ◽

10.37236/6468 ◽

2018 ◽

Vol 25 (4) ◽

Author(s):

Noah Golowich

Keyword(s):

Chromatic Number ◽

Lower Bounds ◽

Graph Coloring ◽

Upper And Lower Bounds ◽

Nice Property ◽

Chromatic Numbers ◽

Compression Scheme ◽

Graph Coloring Problem ◽

Close Relationship ◽

Local Graph

Haramaty and Sudan considered the problem of transmitting a message between two people, Alice and Bob, when Alice's and Bob's priors on the message are allowed to differ by at most a given factor. To find a deterministic compression scheme for this problem, they showed that it is sufficient to obtain an upper bound on the chromatic number of a graph, denoted $U(N,s,k)$ for parameters $N,s,k$, whose vertices are nested sequences of subsets and whose edges are between vertices that have similar sequences of sets. In turn, there is a close relationship between the problem of determining the chromatic number of $U(N,s,k)$ and a local graph coloring problem considered by Erdős et al. We generalize the results of Erdős et al. by finding bounds on the chromatic numbers of graphs $H$ and $G$ when there is a homomorphism $\phi :H\rightarrow G$ that satisfies a nice property. We then use these results to improve upper and lower bounds on $\chi(U(N,s,k))$.

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Fully-Dynamic and Kinetic Conflict-Free Coloring of Intervals with Respect to Points

International Journal of Computational Geometry & Applications ◽

10.1142/s021819591940003x ◽

2019 ◽

Vol 29 (01) ◽

pp. 49-72

Author(s):

Mark de Berg ◽

Tim Leijsen ◽

Aleksandar Markovic ◽

André van Renssen ◽

Marcel Roeloffzen ◽

...

Keyword(s):

Lower Bound ◽

Lower Bounds ◽

Upper And Lower Bounds ◽

Worst Case ◽

Insertions And Deletions ◽

Coloring Problem ◽

Trade Offs ◽

Special Cases ◽

The Cost ◽

Case Number

We introduce the fully-dynamic conflict-free coloring problem for a set [Formula: see text] of intervals in [Formula: see text] with respect to points, where the goal is to maintain a conflict-free coloring for [Formula: see text] under insertions and deletions. A coloring is conflict-free if for each point [Formula: see text] contained in some interval, [Formula: see text] is contained in an interval whose color is not shared with any other interval containing [Formula: see text]. We investigate trade-offs between the number of colors used and the number of intervals that are recolored upon insertion or deletion of an interval. Our results include: a lower bound on the number of recolorings as a function of the number of colors, which implies that with [Formula: see text] recolorings per update the worst-case number of colors is [Formula: see text], and that any strategy using [Formula: see text] colors needs [Formula: see text] recolorings; a coloring strategy that uses [Formula: see text] colors at the cost of [Formula: see text] recolorings, and another strategy that uses [Formula: see text] colors at the cost of [Formula: see text] recolorings; stronger upper and lower bounds for special cases. We also consider the kinetic setting where the intervals move continuously (but there are no insertions or deletions); here we show how to maintain a coloring with only four colors at the cost of three recolorings per event and show this is tight.

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Tight Bounds for Restricted Grid Scheduling

International Journal of Foundations of Computer Science ◽

10.1142/s0129054119500102 ◽

2019 ◽

Vol 30 (03) ◽

pp. 375-405

Author(s):

Joan Boyar ◽

Faith Ellen

Keyword(s):

Lower Bound ◽

Lower Bounds ◽

Competitive Ratio ◽

Upper And Lower Bounds ◽

Grid Scheduling ◽

Total Size

The following problem is considered: Items with integer sizes are given and variable sized bins arrive online. A bin must be used if there is still an item remaining which fits in it when the bin arrives. The goal is to minimize the total size of all the bins used. Previously, a lower bound of [Formula: see text] on the competitive ratio of this problem was achieved using items of size [Formula: see text] and [Formula: see text]. For these item sizes and maximum bin size [Formula: see text], we obtain asymptotically matching upper and lower bounds, which vary depending on the ratio of the number of small items to the number of large items.

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A Lower Bound for the Query Phase of Contraction Hierarchies and Hub Labels and a Provably Optimal Instance-Based Schema

Algorithms ◽

10.3390/a14060164 ◽

2021 ◽

Vol 14 (6) ◽

pp. 164

Author(s):

Tobias Rupp ◽

Stefan Funke

Keyword(s):

Lower Bound ◽

Lower Bounds ◽

Real World ◽

Road Network ◽

Upper And Lower Bounds ◽

Bounded Degree ◽

Shortest Path Routing ◽

Graph Classes ◽

Speed Up ◽

To Come

We prove a Ω(n) lower bound on the query time for contraction hierarchies (CH) as well as hub labels, two popular speed-up techniques for shortest path routing. Our construction is based on a graph family not too far from subgraphs that occur in real-world road networks, in particular, it is planar and has a bounded degree. Additionally, we borrow ideas from our lower bound proof to come up with instance-based lower bounds for concrete road network instances of moderate size, reaching up to 96% of an upper bound given by a constructed CH. For a variant of our instance-based schema applied to some special graph classes, we can even show matching upper and lower bounds.

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