SIAM Journal on Discrete Mathematics, Volume 34, Issue 4, Page 2318-2337, January 2020.
We consider the problem of realizable interval sequences. An interval sequence is comprised of $n$ integer intervals $[a_i,b_i]$ such that $0\le a_i\leq b_i \le n-1$ and is said to be graphic/realizable if there exists a graph with degree sequence, say, $D=(d_1,\ldots,d_n),$ satisfying the condition $a_i\leq d_i\leq b_i$ for each $i\in[1,n]$. There is a characterization (also implying an $O(n)$ verifying algorithm) known for realizability of interval sequences, which is a generalization of the Erdös--Gallai characterization for graphic sequences. However, given any realizable interval sequence, there is no known algorithm for computing a corresponding graphic certificate in $o(n^2)$ time. In this paper, we provide an $O(n \log n)$ time algorithm for computing a graphic sequence for any realizable interval sequence. In addition, when the interval sequence is nonrealizable, we show how to find a graphic sequence having minimum deviation with respect to the given interval sequence in the same time. Finally, we consider variants of the problem, such as computing the most-regular graphic sequence and computing a minimum extension of a length $p$ nongraphic sequence to a graphic one.

中文翻译：

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有效地实现间隔序列

SIAM离散数学杂志，第34卷，第4期，第2318-2337页，2020年1月。
我们考虑可实现的间隔序列的问题。一个间隔序列由$ n $个整数间隔$ [a_i，b_i] $组成，使得$ 0 \ le a_i \ leq b_i \ le n-1 $，并且如果存在具有度序列的图，则可以说是可图形的/可实现的，例如，对于每个$ i \ in [1，n] $，$ D =（d_1，\ ldots，d_n），$满足条件$ a_i \ leq d_i \ leq b_i $。对于间隔序列的可实现性，有一个表征（也暗示着$ O（n）$验证算法），它是图形序列的Erdös-Gallai表征的概括。但是，给定任何可实现的间隔序列，没有已知的算法可以在$ o（n ^ 2）$时间内计算出相应的图形证书。在本文中，我们提供了一个$ O（n \ log n）$时间算法，用于计算任何可实现的间隔序列的图形序列。此外，当间隔序列不可实现时，我们将说明如何同时找到相对于给定间隔序列具有最小偏差的图形序列。最后，我们考虑问题的变体，例如，计算最规则的图形序列，以及计算长度$ p $非图形序列到图形序列的最小扩展。