Faster Graph Coloring in Polynomial Space

• Published in: Algorithmica Vol. 85; no. 2; pp. 584 - 609
• Main Authors: Gaspers, Serge, Lee, Edward J.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-02-2023; Springer Nature B.V
• Subjects: Algorithm Analysis and Problem Complexity; Algorithms; Apexes; Computer Science; Computer Systems Organization and Communication Networks; Counting; Data Structures and Information Theory; Graph coloring; Graph theory; Graphs; Mathematics of Computing; Polynomials; Theory of Computation; G.2.2 Graph Theory; Graph coloring; Counting independent sets; Branching algorithms; Exponential time algorithms; F.2.2 Nonnumerical Algorithms and Problems; Analysis of algorithms
• ISSN: 0178-4617; 1432-0541
• DOI: 10.1007/s00453-022-01034-7

We present a polynomial-space algorithm that computes the number of independent sets of any input graph in time
O
(
1
.
1389
n
)
for graphs with maximum degree 3 and in time
O
(
1
.
2356
n
)
for general graphs, where
n
is the number of vertices in the input graph. Together with the inclusion-exclusion approach of Björklund, Husfeldt, and Koivisto [SIAM J. Comput. 2009], this leads to a faster polynomial-space algorithm for the graph coloring problem with running time
O
(
2
.
2356
n
)
as well as an exponential-space
O
(
1
.
2330
n
)
time algorithm for counting independent sets. Our main algorithm counts independent sets in graphs with maximum degree at most 3 and no vertex with three neighbors of degree 3. This polynomial-space algorithm is designed and analyzed using the recently introduced Separate, Measure and Conquer approach [Gaspers & Sorkin, ICALP 2015]. Using Wahlström’s compound measure approach, this improvement in running time for small degree graphs is then bootstrapped to larger degrees, giving the improvement for general graphs. Combining both approaches leads to some inflexibility in choosing vertices to branch on for the small-degree cases, which we counter by structural graph properties.