Abstract
We consider the Partition Into Triangles problem on bounded degree graphs. We show that this problem is polynomial-time solvable on graphs of maximum degree three by giving a linear-time algorithm. We also show that this problem becomes $\mathcal{NP}$ -complete on graphs of maximum degree four. Moreover, we show that there is no subexponential-time algorithm for this problem on graphs of maximum degree four unless the Exponential-Time Hypothesis fails. However, the Partition Into Triangles problem on graphs of maximum degree at most four is in many cases practically solvable as we give an algorithm for this problem that runs in $\mathcal{O}(1.02220^{n})$ time and linear space.
Highlights
In his weblog of February 2009 [17], Richard J
Thereafter, we focus on the relation between PARTITION INTO TRIANGLES on graphs of maximum degree four to and EXACT 3-SATISFIABILITY in Sect
We will show that PARTITION INTO TRIANGLES on graphs of maximum degree four is N P-complete, and that no subexponential-time algorithm for this problem exists unless the Exponential-Time Hypothesis [11, 12] fails
Summary
“For every polynomial-time algorithm you have, there is an exponential algorithm that I would rather run.”. Lipton illustrates this quote beautifully: “His point is simple: if your algorithm runs in n4 time, an algorithm that runs in n2n/10 time (alternatively denoted as n1.07178n time) is faster if for example n = 100.”. This paper, we will present such a very fast exponential-time algorithm for the PARTITION INTO TRIANGLES problem restricted to graphs of maximum degree four. This result is further improved to O(1.02220n) or O(2n/31.58) time by a further case analysis in the Appendix These algorithms could solve reasonable size instance as their running times do not include any large factors hidden in the O-notation. Both algorithms use an interesting and powerful relation between PARTITION INTO TRIANGLES on graphs of maximum degree four and the EXACT 3-SATISFIABILITY problem. In the Appendix, one can find the slightly faster O(1.02220n)-time algorithm
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