The 4/3 additive spanner exponent is tight

Amir Abboud, Gregory Bodwin
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引用次数: 20

Abstract

A spanner is a sparse subgraph that approximately preserves the pairwise distances of the original graph. It is well known that there is a smooth tradeoff between the sparsity of a spanner and the quality of its approximation, so long as distance error is measured multiplicatively. A central open question in the field is to prove or disprove whether such a tradeoff exists also in the regime of additive error. That is, is it true that for all ε>0, there is a constant kε such that every graph has a spanner on O(n1+ε) edges that preserves its pairwise distances up to +kε? Previous lower bounds are consistent with a positive resolution to this question, while previous upper bounds exhibit the beginning of a tradeoff curve: all graphs have +2 spanners on O(n3/2) edges, +4 spanners on Õ(n7/5) edges, and +6 spanners on O(n4/3) edges. However, progress has mysteriously halted at the n4/3 bound, and despite significant effort from the community, the question has remained open for all 0 < ε < 1/3. Our main result is a surprising negative resolution of the open question, even in a highly generalized setting. We show a new information theoretic incompressibility bound: there is no function that compresses graphs into O(n4/3 − ε) bits so that distance information can be recovered within +no(1) error. As a special case of our theorem, we get a tight lower bound on the sparsity of additive spanners: the +6 spanner on O(n4/3) edges cannot be improved in the exponent, even if any subpolynomial amount of additive error is allowed. Our theorem implies new lower bounds for related objects as well; for example, the twenty-year-old +4 emulator on O(n4/3) edges also cannot be improved in the exponent unless the error allowance is polynomial. Central to our construction is a new type of graph product, which we call the Obstacle Product. Intuitively, it takes two graphs G,H and produces a new graph G H whose shortest paths structure looks locally like H but globally like G.
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4/3加性扳手指数紧
扳手是一种稀疏子图,它近似地保留了原始图的成对距离。众所周知,只要用乘法测量距离误差,扳手的稀疏性和它的近似值的质量之间就会有一个平滑的权衡。该领域的一个中心开放问题是证明或反驳这种权衡是否也存在于加性误差的制度中。也就是说,对于所有ε>0,是否存在一个常数kε,使得每个图在O(n1+ε)条边上都有一个扳手,使其成对距离保持到+kε?之前的下界与这个问题的积极解决是一致的,而之前的上界显示了权衡曲线的开始:所有图在O(n3/2)边上有+2个扳手,在Õ(n7/5)边上有+4个扳手,在O(n4/3)边上有+6个扳手。然而,进展在n4/3边界神秘地停止了,尽管社区做出了巨大的努力,这个问题仍然对所有0 < ε < 1/3开放。我们的主要结果是,即使在高度一般化的情况下,对开放问题的否定解决也令人惊讶。我们展示了一个新的信息论不可压缩性界:没有函数可以将图压缩成O(n4/3−ε)位,从而在+no(1)误差内恢复距离信息。作为我们的定理的一个特例,我们得到了加性扳手的稀疏性的一个紧下界:0 (n4/3)条边上的+6扳手不能在指数上得到改进,即使允许任何次多项式的加性误差。我们的定理也为相关物体暗示了新的下界;例如,在O(n4/3)边上使用了20年的+4模拟器也不能在指数上进行改进,除非误差允许是多项式。我们构造的核心是一种新的图积,我们称之为障碍积。直观地说,它取两个图G和H,生成一个新的图G H,它的最短路径结构局部像H,全局像G。
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