Given a hereditary family G of admissible graphs and a function λ(G) that linearly depends on the statistics of order-κ subgraphs in a graph G, we consider the extremal problem of determining λ(n, G), the maximum of λ(G) over all admissible graphs G of order n. We call the problem perfectly B-stable for a graph B if there is a constant C such that every admissible graph G of order n C can be made into a blow-up of B by changing at most C(λ(n, G)−λ(G)) n 2 adjacencies. As special cases, this property describes all almost extremal graphs of order n within o(n 2 ) edges and shows that every extremal graph of order n n 0 is a blow-up of B.We develop general methods for establishing stability-type results from flag algebra computations and apply them to concrete examples. In fact, one of our sufficient conditions for perfect stability is stated in a way that allows automatic verification by a computer. This gives a unifying way to obtain computer-assisted proofs of many new results.
Abstract. In the online checkpointing problem, the task is to continuously maintain a set of k checkpoints that allow to rewind an ongoing computation faster than by a full restart. The only operation allowed is to remove an old checkpoint and to store the current state instead. Our aim are checkpoint placement strategies that minimize rewinding cost, i.e., such that at all times T when requested to rewind to some time t ≤ T the number of computation steps that need to be redone to get to t from a checkpoint before t is as small as possible. In particular, we want that the closest checkpoint earlier than t is not further away from t than p k times the ideal distance T /(k + 1), where p k is a small constant. Improving over earlier work showing 1 + 1/k ≤ p k ≤ 2, we show that p k can be chosen less than 2 uniformly for all k. More precisely, we show the uniform bound p k ≤ 1.7 for all k, and present algorithms with asymptotic performance p k ≤ 1.59 + o(1) valid for all k and p k ≤ ln(4) + o(1) ≤ 1.39 + o(1) valid for k being a power of two. For small values of k, we show how to use a linear programming approach to compute good checkpointing algorithms. This gives performances of less than 1.53 for k ≤ 10. One the more theoretical side, we show the first lower bound that is asymptotically more than one, namely p k ≥ 1.30 − o(1). We also show that optimal algorithms (yielding the infimum performance) exist for all k.
This paper enumerates all juxtaposition classes of the form "Av(abc) next to Av(xy)", where abc is a permutation of length three and xy is a permutation of length two. We use Dyck paths decorated by sequences of points to represent elements from such a juxtaposition class. Context free grammars are then used to enumerate these decorated Dyck paths.
We show that, given a suitable combinatorial specification for a permutation class C, one can obtain a specification for the juxtaposition (on either side) of C with Av(21) or Av (12), and that if the enumeration for C is given by a rational or algebraic generating function, so is the enumeration for the juxtaposition. Furthermore this process can be iterated, thereby providing an effective method to enumerate any 'skinny' k × 1 grid class in which at most one cell is non-monotone, with a guarantee on the nature of the enumeration given the nature of the enumeration of the non-monotone cell.In this paper, we consider the effect on combinatorial specifications of juxtapositions. The juxtaposition of two permutations can be thought of as a special kind of merge, in which only the values of the two permutations can be interleaved arbitrarily, and not the positions. However, our primary motivation in studying juxtapositions is towards a generalisation in another direction, namely the broader study of grid classes.Our main result is as follows. Definitions are given in Section 2, but we note here that context-free specifications give rise to algebraic generating functions, while regular ones give rational generating functions.Theorem 1.1. Let C be a permutation class that admits a bottom-to-top combinatorial specification S , and let M ∈ {Av(21), Av(12)}.(i) If S tracks the rightmost entry, then there exists a combinatorial specification for C | M that tracks the rightmost entry.(ii) Similarly, if S tracks the leftmost entry, then there exists one for M | C that tracks the leftmost entry.(iii) If S tracks both the leftmost and the rightmost entries, then there exists specifications for C | M and M | C that do.(iv) If S is context-free (resp. regular), then the specifications for C | M and M | C are context-free (regular).Since the resulting specifications satisfy the same conditions as the theorem requires of C, the process can be iterated, thereby allowing us to generate combinatorial specifications for classes of the form given in Figure 1.Figure 1: The k × 1 grid classes in Corollary 1.2, in which C possesses a rightmost-and leftmost-entry tracking specification, and M i ∈ {Av(21), Av(12)} for all i = j.Corollary 1.2. If a permutation class C possesses a specification S that tracks the rightmost and leftmost entries, then so does any k × 1 grid class of the form given in Figure 1. In particular, if C possesses an algebraic or rational generating function, then so too does the k × 1 grid class.Permutation classes to which our method applies include any class C that contains only finitely many simple permutations, but it is not limited to this (for example, in Section 4 we present a suitable specification for Av(321)). By its nature, there is a parallel between our bottom-to-top specifications and the insertion encoding of Albert, Linton and Ruškuc [4], and we explore this further in Section 4.Prior to this paper, the most general result for k × 1 grids concerns the case where the class C above is itself also monoton...
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