For the decision problem log clique, we define the log n-partite graph clique problem as its optimization problem. A log n-partite graph G has log n partitions of vertices with each partition n vertices.
The log n-partite graph clique problem: given a log n-partite graph G, try to find the maximum clique of the graph G.
We can see that the size of the maximum clique of a log n-partite graph is less than or equal to log n.
Note that the log n-partite graph clique problem has found applications in computational biology [76, 20].
By our definition, the parameterized version of the optimization problem log n- partite graph clique is
The log n-partite graph clique-para problem: given a log n-partite graph G and a parameter k, is there a clique of size at least k in G?
It is not difficult to show that the log clique-para problem we defined in Chap- ter III is f ptl-reducible to the log n-partite graph clique-para problem. Given
an instance I1 of log clique-para, I1 = (G = (V, E), k), where V = {v1, v2, ..., vn}
and k ≤ log n, we build an instance I2 of log n-partite graph clique-para,
I2 = (G0 = (V0, E0), k) as follows. G0 has log n copies of the vertices in G. We
denote V0 = {V1, V2, ..., Vlog n}, where each copy Vi has n vertices {vi1, vi2, ..., vin}, for
1 ≤ i ≤ log n. The vertex vix in G0, where 1 ≤ i ≤ log n and 1 ≤ x ≤ n, corresponds
to the vertex vx in G. We build edges between two vertices vix and vjy in G0 if and
with each copy of the vertices in G as a partition, and there are edges between vertices from different partitions.
We show that G has a clique of size k if and only if G0 has a clique of size k. Suppose G has a clique C = {vc1, vc2, ..., vck}, where each ci ∈ {1, 2, ..., n}. Then from the construction of G0, there is a clique C0 = {v1c1, v2c2, ..., vkck} in G
0. On the other
hand, suppose there is a clique C0 of size k in G0, we know that all the k vertices in C0 should be from different partitions and any two of them are not copies of the same vertex of G (since by the construction of G0, there are no edges between copies of the same vertex of G). Then the k vertices in C0 corresponds to k different vertices in G. Furthermore, since there is an edge between any two of the k vertices in C0 (C0 is a clique), there is an edge between any two of the corresponding k vertices in G. That is, G has a clique of size k.
The reduction from log clique-para to log n-partite graph clique-para is an f ptl-reduction.
By the above f ptl-reduction, Theorem III.26 and Theorem III.25, we have
Lemma IV.19 The log n-partite graph clique-para problem has no algorithm of time f (k)no(k) for any function f , unless all SNP problems are solvable in subex-
ponential time.
Therefore, by Lemma IV.19 and Theorem IV.2, we have the following theorem. Theorem IV.20 The log n-partite graph clique problem has no PTAS algo- rithm of time f (1/)no(1/) for any function f , unless all SNP problems are solvable
in subexponential time.
Before ending the section, we point out that for the decision problem log chordless path, we do not have a natural optimization version.
CHAPTER V
STUDY OF EPTAS ALGORITHMS ON PLANAR GRAPHS
So far we can prove lower bound results for NP optimization problems when the parameterized versions of these problems are W[t]-hard, t ≥ 1. In this chapter, we discuss the lower bounds for the parameterized problems that are fixed-parameter tractable.
We prove computational lower bounds on the EPTAS algorithms for some famous planar graph NP-hard optimization problems. Based on the result in [17] (Lemma 6), the parameterized versions of these optimization problems are in FPT.
A. EPTAS Lower Bound Results
Based on the outer-planarity of planar graphs, Baker [7] designed EPTAS algorithms of time O(2O(1/)n) for several famous NP-hard optimization problems on planar
graphs, such as planar vertex cover, planar independent set, and pla- nar dominating set, where > 0 is the given error bound, and n is the number of vertices of the planar graph.
Alber et. al [3] designed parameterized algorithms of time 2O(
√
k)nO(1) for the
parameterized versions of the above NP-hard optimization problems on planar graphs. A lot of research has been done on these problems to try to further improve the time complexity of the parameterized algorithms. Interested readers are referred to [2, 56, 41, 42].
Cai et. al [15] proved the following lower bound result for the parameterized algorithms of these problems.
planar dominating set do not have parameterized algorithms of time 2o(
√
k)nO(1),
unless all SNP problems are solvable in subexponential time. From Theorem V.1 and Theorem IV.2, we have
Theorem V.2 planar vertex cover, planar independent set, and planar dominating set have no EPTAS of running time 2o(
√
1/)nO(1), where > 0 is the
given error bound, unless all SNP problems are solvable in subexponential time. Note that the upper bound of the EPTAS algorithms for the above problems in Baker [7] is 2O(1/)nO(1) (also [62]). We can see that there is a gap between the upper
bound and our lower bound result. To come up with new approaches to improve the upper bound of the EPTAS algorithms in [7] will be interesting research. To study this issue, we concentrate on the planar vertex cover problem in the next section.