370 research outputs found
On Coloring Resilient Graphs
We introduce a new notion of resilience for constraint satisfaction problems,
with the goal of more precisely determining the boundary between NP-hardness
and the existence of efficient algorithms for resilient instances. In
particular, we study -resiliently -colorable graphs, which are those
-colorable graphs that remain -colorable even after the addition of any
new edges. We prove lower bounds on the NP-hardness of coloring resiliently
colorable graphs, and provide an algorithm that colors sufficiently resilient
graphs. We also analyze the corresponding notion of resilience for -SAT.
This notion of resilience suggests an array of open questions for graph
coloring and other combinatorial problems.Comment: Appearing in MFCS 201
Gr\"obner Bases and Nullstellens\"atze for Graph-Coloring Ideals
We revisit a well-known family of polynomial ideals encoding the problem of
graph--colorability. Our paper describes how the inherent combinatorial
structure of the ideals implies several interesting algebraic properties.
Specifically, we provide lower bounds on the difficulty of computing Gr\"obner
bases and Nullstellensatz certificates for the coloring ideals of general
graphs. For chordal graphs, however, we explicitly describe a Gr\"obner basis
for the coloring ideal, and provide a polynomial-time algorithm.Comment: 16 page
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