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.
The idea of using fast perfect matching counting in restricted graph classes to solve other combinatorial optimisation problems is not new, a prominent example is the polynomial time algorithm for Maximum Cut in an unweighted graph of bounded genus by Galluccio, Loebl, and Vondr ' ak @cite_18 .
@cite_18
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Copyright © 2017, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. Motivated by the fact that in several cases a matching in a graph is stable if and only if it is produced by a greedy algorithm, we study the problem of computing a maximum weight greedy matching on weighted graphs, termed GREEDY-MATCHING. In wide contrast to the maximum weight matching problem, for which many efficient algorithms are known, we prove that GREEDYMATCHING is strongly NP-hard and APX-complete, and thus it does not admit a PTAS unless P=NP, even on graphs with maximum degree at most 3 and with at most three different integer edge weights. Furthermore we prove that GREEDYMATCHING is strongly NP-hard if the input graph is in addition bipartite. Moreover we consider three natural parameters of the problem, for which we establish a sharp threshold behavior between NP-hardness and computational tractability. On the positive side, we present a randomized approximation algorithm (RGMA) for GREEDYMATCHING on a special class of weighted graphs, called bush graphs. We highlight an unexpected connection between RGMA and the approximation of maximum cardinality matching in unweighted graphs via randomized greedy algorithms. We show that, if the approximation ratio of RGMA is θ, then for every ϵ > 0 the randomized MRG algorithm of ( 1995) gives a (ρ - ϵ)-approximation for the maximum cardinality matching. We conjecture that a tight bound for ρ is 2 3; we prove our conjecture true for four subclasses of bush graphs. Proving a tight bound for the approximation ratio of MRG on unweighted graphs (and thus also proving a tight value for ρ) is a long-standing open problem (Poloczek and Szegedy 2012). This unexpected relation of our RGMA algorithm with the MRG algorithm may provide new insights for solving this problem. There are many classical problems in P whose time complexities have not been improved over the past decades. Recent studies of "Hardness in P" have revealed that, for several of such problems, the current fastest algorithm is the best possible under some complexity assumptions. To bypass this difficulty, (SODA 2017) introduced the concept of fully polynomial FPT algorithms. For a problem with the current best time complexity @math , the goal is to design an algorithm running in @math time for a parameter @math and a constant @math . In this paper, we investigate the complexity of graph problems in P parameterized by tree-depth, a graph parameter related to tree-width. We show that a simple divide-and-conquer method can solve many graph problems, including Weighted Matching, Negative Cycle Detection, Minimum Weight Cycle, Replacement Paths, and 2-hop Cover, in @math time or @math time, where @math is the tree-depth of the input graph. Because any graph of tree-width @math has tree-depth at most @math , our algorithms also run in @math time or @math time. These results match or improve the previous best algorithms parameterized by tree-width. Especially, we solve an open problem of fully polynomial FPT algorithm for Weighted Matching parameterized by tree-width posed by We study the perfect matching reconfiguration problem: Given two perfect matchings of a graph, is there a sequence of flip operations that transforms one into the other? Here, a flip operation exchanges the edges in an alternating cycle of length four. We are interested in the complexity of this decision problem from the viewpoint of graph classes. We first prove that the problem is PSPACE-complete even for split graphs and for bipartite graphs of bounded bandwidth with maximum degree five. We then investigate polynomial-time solvable cases. Specifically, we prove that the problem is solvable in polynomial time for strongly orderable graphs (that include interval graphs and strongly chordal graphs), for outerplanar graphs, and for cographs (also known as @math -free graphs). Furthermore, for each yes-instance from these graph classes, we show that a linear number of flip operations is sufficient and we can exhibit a corresponding sequence of flip operations in polynomial time. We use the principle of inclusion and exclusion, combined with polynomial time segmentation and fast Mobius transform, to solve the generic problem of summing or optimizing over the partitions of n elements into a given number of weighted subsets. This problem subsumes various classical graph partitioning problems, such as graph coloring, domatic partitioning, and MAX k-CUT, as well as machine learning problems like decision graph learning and model-based data clustering. Our algorithm runs in O*(2^n ) time, thus substantially improving on the usual O*(3^n )-time dynamic programming algorithm; the notation O* suppresses factors polynomial in n. This result improves, e.g., Byskov?s recent record for graph coloring from O*(2.4023^n ) to O*(2^n ). We note that twenty five years ago, R. M. Karp used inclusion--exclusion in a similar fashion to reduce the space requirement of the usual dynamic programming algorithms from exponential to polynomial.