多股流和分数多股流的流切间隙
Flow-Cut Gaps for Integer and Fractional Multiflows
摘要：
考虑由需求图H和供应图G组成的路由问题。如果该对服从割条件，则该实例的流割间隙是最小值C，使得如果G的每条边被给定容量C，则存在H的可行多流。即使当G是（串-并联）图K_{2,3}时，流切间隙也可以大于1。在本文中，我们主要感兴趣的是“整数”流切间隙。如果G的每条边都给定容量C，那么使得H存在一个可行的整数值多流的最小值C是多少？我们推测，整数流切间隙是定量相关的分数流切间隙。这加强了著名的猜想，即平面图和无子图中的流割间隙是O（1），以表明整数流割间隙是O（1）。我们给出了几个支持这一猜想的非平凡特殊类图的结果，并进一步探索了理解流切间隙的“原始”方法。我们的成果包括：- 令G通过从边st开始的串-并行操作获得，并且考虑将G中的所有边定向在从s到t的方向上。如果一个需求的端点是通过结果导向图中的有向路径连接的，则该需求是兼容的。如果切割条件对于兼容实例成立并且G+H是欧拉的，则存在H的积分路由。- 串并联图的整数流割间隙为5。我们还给出了一类明确的实例，通过初等计算表明，串并联图中的流切间隙至少为2-o（1）;这简化了Lee和Raghavendra的证明。- k-外平面图的整数流割间隙是c^{O（k）}，其中c是固定常数。- 一个简单的证明，流切间隙是O（\log k^*），其中k^* 是H中节点覆盖的大小;这是以前由Günlük通过一个更复杂的证明证明。
英文摘要：
Consider a routing problem consisting of a demand graph H and a supply graph G. If the pair obeys the cut condition, then the flow-cut gap for this instance is the minimum value C such that there is a feasible multiflow for H if each edge of G is given capacity C. The flow-cut gap can be greater than 1 even when G is the (series-parallel) graph K_{2,3}. In this paper we are primarily interested in the "integer" flow-cut gap. What is the minimum value C such that there is a feasible integer valued multiflow for H if each edge of G is given capacity C? We conjecture that the integer flow-cut gap is quantitatively related to the fractional flow-cut gap. This strengthens the well-known conjecture that the flow-cut gap in planar and minor-free graphs is O(1) to suggest that the integer flow-cut gap is O(1). We give several results on non-trivial special classes of graphs supporting this conjecture and further explore the "primal" method for understanding flow-cut gaps. Our results include: - Let G be obtained by series-parallel operations starting from an edge st, and consider orienting all edges in G in the direction from s to t. A demand is compliant if its endpoints are joined by a directed path in the resulting oriented graph. If the cut condition holds for a compliant instance and G+H is Eulerian, then an integral routing of H exists. - The integer flow-cut gap in series-parallel graphs is 5. We also give an explicit class of instances that shows via elementary calculations that the flow-cut gap in series-parallel graphs is at least 2-o(1); this simplifies the proof by Lee and Raghavendra. - The integer flow-cut gap in k-Outerplanar graphs is c^{O(k)} for some fixed constant c. - A simple proof that the flow-cut gap is O(\log k^*) where k^* is the size of a node-cover in H; this was previously shown by Günlük via a more intricate proof.
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