Finding and Counting Permutations via CSPs

Permutation patterns and pattern avoidance have been intensively studied in combinatorics and computer science, going back at least to the seminal work of Knuth on stack-sorting (1968). Perhaps the most natural algorithmic question in this area is deciding whether a given permutation of length n contains a given pattern of length k. In this work we give two new algorithms for this well-studied problem, one whose running time is $$n^{k/4 + o(k)}$$ n k / 4 + o ( k ) , and a polynomial-space algorithm whose running time is the better of $$O(1.6181^n)$$ O ( 1 . 6181 n ) and $$O(n^{k/2 + 1})$$ O ( n k / 2 + 1 ) . These results improve the earlier best bounds of $$n^{0.47k + o(k)}$$ n 0.47 k + o ( k ) and $$O(1.79^n)$$ O ( 1 . 79 n ) due to Ahal and Rabinovich (2000) resp. Bruner and Lackner (2012) and are the fastest algorithms for the problem when $$k \in \varOmega (\log {n})$$ k ∈ Ω ( log n ) . We show that both our new algorithms and the previous exponential-time algorithms in the literature can be viewed through the unifying lens of constraint-satisfaction. Our algorithms can also count, within the same running time, the number of occurrences of a pattern. We show that this result is close to optimal: solving the counting problem in time $$f(k) \cdot n^{o(k/\log {k})}$$ f ( k ) · n o ( k / log k ) would contradict the exponential-time hypothesis (ETH). For some special classes of patterns we obtain improved running times. We further prove that 3-increasing (4321-avoiding) and 3-decreasing (1234-avoiding) permutations can, in some sense, embed arbitrary permutations of almost linear length, which indicates that a sub-exponential running time is unlikely with the current techniques, even for patterns from these restricted classes.

Improved Hardness of Maximum Common Subgraph Problems on Labeled Graphs of Bounded Treewidth and Bounded Degree

We consider the maximum common connected edge subgraph problem and the maximum common connected induced subgraph problem for simple graphs with labeled vertices (or labeled edges). The former is to find a connected graph with the maximum number of edges that is isomorphic to a subgraph of each of the two input graphs. The latter is to find a common connected induced subgraph with the maximum number of vertices. We prove that both problems are NP-hard for 3-outerplanar labeled graphs even if the maximum vertex degree is bounded by 4. Since the reductions used in the proofs construct graphs with treewidth at most 4, both problems are NP-hard also for such graphs, which significantly improves the previous hardness results for graphs with treewidth 11. We also present improved exponential-time algorithms for both problems on labeled graphs of bounded treewidth and bounded vertex degree.

Application of Algorithmic Manifold to Exponential Time

In the previous paper, I defined algorithmic manifolds simulating polynomial-time algorithms, and I showed topological properties for P problem and NP problem and that NP problem can be transformed into deterministic Turing machine problem. In this paper, I define algorithmic manifolds simulating exponential-time algorithms and, I show topological properties for EXPTIME problem and NEXPTIME problem. I also discuss the relationship between NEXPTIME and deterministic Turing machines.