Faster Graph Coloring in Polynomial Space
Serge Gaspers, Edward Lee

TL;DR
This paper introduces a faster polynomial-space algorithm for counting independent sets and graph coloring, improving previous methods with novel analysis techniques and extending to general graphs.
Contribution
The paper presents a new polynomial-space algorithm for counting independent sets and graph coloring with improved running times, using innovative analysis methods and structural graph properties.
Findings
Faster algorithms for graphs with maximum degree 3.
An exponential-space algorithm for counting independent sets.
Improved overall running time for graph coloring.
Abstract
We present a polynomial-space algorithm that computes the number independent sets of any input graph in time for graphs with maximum degree 3 and in time for general graphs, where n is the number of vertices. Together with the inclusion-exclusion approach of Bj\"orklund, Husfeldt, and Koivisto [SIAM J. Comput. 2009], this leads to a faster polynomial-space algorithm for the graph coloring problem with running time . As a byproduct, we also obtain an exponential-space time algorithm for counting independent sets. Our main algorithm counts independent sets in graphs with maximum degree 3 and no vertex with three neighbors of degree 3. This polynomial-space algorithm is analyzed using the recently introduced Separate, Measure and Conquer approach [Gaspers & Sorkin, ICALP 2015]. Using Wahlstr\"om's compound measure approach, this…
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