Cycle-free partial orders and chordal comparability graphs

An implicit representation of chordal comparability graphs in linear time

Construction of a simple elimination scheme for a chordal comparability graph in linear time

Tree structure for distributive lattices and its applications

Planarity, bipartiteness and (directed) acyclicity are basic graph properties with classic linear time recognition algorithms. However, the problems of testing whether a given (di)graph has k vertices whose deletion makes it planar, bipartite or a directed acyclic graph (DAG) are all fundamental NP-complete problems when k is part of the input. As a result, a significant amount of research has been devoted to understanding whether, for every fixed k, these problems admit a polynomial time algorithm (where the exponent in the polynomial is independent of k) and in particular, whether they admit linear time algorithms. While we now know that for any fixed k, we can test in linear time whether a graph is k vertices away from being planar [FOCS 2009, SODA 2014] or bipartite [SODA 2014, SICOMP 2016], the best known algorithms in the case of directed acyclicity are the algorithm of Garey and Tarjan [IPL 78] which runs in time O(nk–1m) and the algorithm of Chen, Liu, Lu, O'Sullivan and Razgon [JACM 2008] which runs in time O(k!4kk4nm). In other words, it has remained open whether it is possible to recognize in linear time, a graph which is 2 vertices away from being acyclic! In this paper, we settle this question by giving an algorithm that decides whether a given graph is k vertices away from being acyclic, in time O(k!4kk5(n + m)). That is, for every fixed k, our algorithm runs in time O(m + n), thus mirroring the case for planarity and bipartiteness. Our algorithm is designed via a general methodology that shaves off a factor of n from some algorithms that use the powerful technique of iterative compression. The two main features of our methodology are: (i) This is the first generic technique for designing linear time algorithms for directed cut-problems and (ii) it can be used in combination with future improvements in algorithms for the compression version of other well-studied cut-problems such as Multicut and Directed Subset Feedback Vertex Set.

Split orders

We present a linear-time algorithm for bounded-type programs that builds a directed graph whose transitive closure gives exactly the results of the standard (cubic-time) Control-Flow Analysis (CFA) algorithm. Our algorithm can be used to list all functions calls from all call sites in (optimal) quadratic time. More importantly, it can be used to give linear-time algorithms for CFA-consuming applications such as:• effects analysis: find the side-effecting expressions in a program.• k-limited CFA: for each call-site, list the functions if there are only a few of them (≤ k) and otherwise output "many".• called-once analysis: identify all functions called from only one call-site.

The analysis of many processes modeled by directed graphs requires the selection of a subset of vertices which cut all the cycles in the graph. Reducing the size of such a cutset usually leads to a simpler and more efficient analysis, but the problem of finding minimum cutsets in general directed graphs is known to be $NP$-complete. In this paper we show that in reducible graphs (and thus in almost all the “practical” flowcharts of programs), minimum cutsets can be found in linear time. We further show that the linear algorithm can check its own applicability to a given graph, thus eliminating the need of prechecking (in nonlinear time) whether it is reducible or not. An immediate application of this result is in program verification systems based on Floyd’s inductive assertions method.

We consider two-machine scheduling problems with job selection. We analyze first the two-machine open shop problem and provide a best possible linear time algorithm. Then, a best possible linear time algorithm is derived for the job selection problem on two unrelated parallel machines. We also show that an exact approach can be derived for both problems with complexity $$O(p(n) \times \sqrt{2}^n)$$O(p(n)×2n), p being a polynomial function of n.

A faster FPTAS for knapsack problem with cardinality constraint

In this article we consider the Directed Steiner Path Cover problem on directed co-graphs. Given a directed graph G=(V,E) and a set T subseteq V of so-called terminal vertices, the problem is to find a minimum number of vertex-disjoint simple directed paths, which contain all terminal vertices and a minimum number of non-terminal vertices (Steiner vertices). The primary minimization criteria is the number of paths. We show how to compute in linear time a minimum Steiner path cover for directed co-graphs. This leads to a linear time computation of an optimal directed Steiner path on directed co-graphs, if it exists. Since the Steiner path problem generalizes the Hamiltonian path problem, our results imply the first linear time algorithm for the directed Hamiltonian path problem on directed co-graphs. We also give binary integer programs for the (directed) Hamiltonian path problem, for the (directed) Steiner path problem, and for the (directed) Steiner path cover problem. These integer programs can be used to minimize change-over times in pick-and-place machines used by companies in electronic industry.

The authors provide distinct algorithms for computing transitive closure and transitive reduction with sequential time complexities of O( Sigma e/sub i/) and O(n/sup 2/+ Sigma e/sub i/), respectively, and parallel time complexities of O(e/sub i/) And O(n+e/sub i/), respectively. The proposed transitive closure algorithms are favorable for the sparse digraphs. It is still open whether transitive reduction may be computed in O( Sigma e/sub i/) (or O(e/sub i/) in parallel). >

Locating an absolute center on graphs that are almost trees

This paper shows that it is possible to find all isomorphic subtrees of a binary tree in linear time and space.

The single-source shortest paths problem (SSSP) is one of the classic problems in algorithmic graph theory: given a positively weighted graph G with a source vertex s , find the shortest path from s to all other vertices in the graph. Since 1959, all theoretical developments in SSSP for general directed and undirected graphs have been based on Dijkstra's algorithm, visiting the vertices in order of increasing distance from s . Thus, any implementation of Dijkstra's algorithm sorts the vertices according to their distances from s . However, we do not know how to sort in linear time. Here, a deterministic linear time and linear space algorithm is presented for the undirected single source shortest paths problem with positive integer weights. The algorithm avoids the sorting bottleneck by building a hierarchical bucketing structure, identifying vertex pairs that may be visited in any order.

We describe two (1+epsilon)-approximation algorithms for computing the Frechet distance between two homeomorphic piecewise linear surfaces R and S of genus zero and total complexity n, with Frechet distance delta. (1) A 2^{O((n + ( (Area(R)+Area(S))/(epsilon.delta)^2 )^2 )} time algorithm if R and S are composed of fat triangles (triangles with angles larger than a constant). (2) An O(D/(epsilon.delta)^2) n + 2^{O(D^4/(epsilon^4.delta^2))} time algorithm if R and S are polyhedral terrains over [0,1]^2 with slope at most D. Although, the Frechet distance between curves has been studied extensively, very little is known for surfaces. Our results are the first algorithms (both for surfaces and terrains) that are guaranteed to terminate in finite time. Our latter result, in particular, implies a linear time algorithm for terrains of constant maximum slope and constant Frechet distance.