Given an undirected graph and two disjoint vertex pairs @math and @math , the Shortest two disjoint paths problem (S2DP) asks for the minimum total length of two vertex disjoint paths connecting @math with @math , and @math with @math , respectively. We show that for cubic planar graphs there are NC algorithms, uniform circuits of polynomial size and polylogarithmic depth, that compute the S2DP and moreover also output the number of such minimum length path pairs. Previously, to the best of our knowledge, no deterministic polynomial time algorithm was known for S2DP in cubic planar graphs with arbitrary placement of the terminals. In contrast, the randomized polynomial time algorithm by Bj "orklund and Husfeldt, ICALP 2014, for general graphs is much slower, is serial in nature, and cannot count the solutions. Our results are built on an approach by Hirai and Namba, Algorithmica 2017, for a generalisation of S2DP, and fast algorithms for counting perfect matchings in planar graphs.
Bj "orklund and Husfeldt @cite_5 showed that Shortest two disjoint paths in a general unweighted undirected graph has a polynomial time Monte Carlo algorithm. Colin de Verdi 'ere and Schrijver @cite_16 , and Kobayachi and Sommer @cite_9 showed that for planar graphs, deterministic polynomial-time algorithms for the Shortest two disjoint paths exist if the four terminals lie on the boundary of at most two faces. Our new algorithm works for all cubic planar cases, but is much slower. Still it is significantly faster than the general @math time algorithm by Bj "orklund and Husfeldt @cite_5 .
@cite_5 @cite_9 @cite_16
2004779317 65356316 2787728049 2963972775
For a graph G and a collection of vertex pairs (s"1,t"1),...,(s"k,t"k) , the k disjoint paths problem is to find k vertex-disjoint paths P"1,...,P"k, where P"i is a path from s"i to t"i for each i=1,...,k. In the corresponding optimization problem, the shortest disjoint paths problem, the vertex-disjoint paths P"i have to be chosen such that a given objective function is minimized. We consider two different objectives, namely minimizing the total path length (minimum sum, or short: Min-Sum), and minimizing the length of the longest path (Min-Max), for k=2,3. Min-Sum: We extend recent results by Colin de Verdiere and Schrijver to prove that, for a planar graph and for terminals adjacent to at most two faces, the Min-Sum 2 Disjoint Paths Problem can be solved in polynomial time. We also prove that, for six terminals adjacent to one face in any order, the Min-Sum 3 Disjoint Paths Problem can be solved in polynomial time. Min-Max: The Min-Max 2 Disjoint Paths Problem is known to be NP-hard for general graphs. We present an algorithm that solves the problem for graphs with tree-width 2 in polynomial time. We thus close the gap between easy and hard instances, since the problem is weakly NP-hard for graphs with tree-width 3. Given an undirected graph and two pairs of vertices (s i ,t i ) for i ∈ 1,2 we show that there is a polynomial time Monte Carlo algorithm that finds disjoint paths of smallest total length joining s i and t i for i ∈ 1,2 respectively, or concludes that there most likely are no such paths at all. Our algorithm applies to both the vertex- and edge-disjoint versions of the problem. The well-known @math -disjoint path problem ( @math -DPP) asks for pairwise vertex-disjoint paths between @math specified pairs of vertices @math in a given graph, if they exist. The decision version of the shortest @math -DPP asks for the length of the shortest (in terms of total length) such paths. Similarly the search and counting versions ask for one such and the number of such shortest set of paths, respectively. We restrict attention to the shortest @math -DPP instances on undirected planar graphs where all sources and sinks lie on a single face or on a pair of faces. We provide efficient sequential and parallel algorithms for the search versions of the problem answering one of the main open questions raised by Colin de Verdiere and Schrijver for the general one-face problem. We do so by providing a randomised @math algorithm along with an @math time randomised sequential algorithm. We also obtain deterministic algorithms with similar resource bounds for the counting and search versions. In contrast, previously, only the sequential complexity of decision and search versions of the "well-ordered" case has been studied. For the one-face case, sequential versions of our routines have better running times for constantly many terminals. In addition, the earlier best known sequential algorithms (e.g. ) were randomised while ours are also deterministic. The algorithms are based on a bijection between a shortest @math -tuple of disjoint paths in the given graph and cycle covers in a related digraph. This allows us to non-trivially modify established techniques relating counting cycle covers to the determinant. We further need to do a controlled inclusion-exclusion to produce a polynomial sum of determinants such that all "bad" cycle covers cancel out in the sum allowing us to count "good" cycle covers. We present a deterministic algorithm that computes the edge-connectivity of a graph in near-linear time. This is for a simple undirected unweighted graph G with n vertices and m edges. This is the first o(mn) time deterministic algorithm for the problem. Our algorithm is easily extended to find a concrete minimum edge-cut. In fact, we can construct the classic cactus representation of all minimum cuts in near-linear time. The previous fastest deterministic algorithm by Gabow from STOC '91 took O(m+λ2 n), where λ is the edge connectivity, but λ can be as big as n−1. Karger presented a randomized near-linear time Monte Carlo algorithm for the minimum cut problem at STOC’96, but the returned cut is only minimum with high probability. Our main technical contribution is a near-linear time algorithm that contracts vertex sets of a simple input graph G with minimum degree Δ, producing a multigraph Ḡ with O(m Δ) edges, which preserves all minimum cuts of G with at least two vertices on each side. In our deterministic near-linear time algorithm, we will decompose the problem via low-conductance cuts found using PageRank a la Brin and Page (1998), as analyzed by Andersson, Chung, and Lang at FOCS’06. Normally, such algorithms for low-conductance cuts are randomized Monte Carlo algorithms, because they rely on guessing a good start vertex. However, in our case, we have so much structure that no guessing is needed.