The classical Crossing Lemma by Ajtai et al.~and Leighton from 1982 gave an important lower bound of $c \frac{m^3}{n^2}$ for the number of crossings in any drawing of a given graph of $n$ vertices and $m$ edges. The original value was $c= 1/100$, which then has gradually been improved. Here, the bounds for the density of $k$-planar graphs played a central role. Our new insight is that for $k=2,3$ the $k$-planar graphs have substantially fewer edges if specific local configurations that occur in drawings of $k$-planar graphs of maximum density are forbidden. Therefore, we are able to derive better bounds for the crossing number $cr(G)$ of a given graph $G$. In particular, we achieve a bound of $cr(G) \ge \frac{37}{9}m-\frac{155}{9}(n-2)$ for the range of $5n < m \le 6n$, while our second bound $cr(G) \ge 5m - \frac{203}{9}(n-2)$ is even stronger for larger $m>6n$. For $m > 6.77n$, we finally apply the standard probabilistic proof from the BOOK and obtain an improved constant of $c>1/27.48$ in the Crossing Lemma. Note that the previous constant was $1/29$. Although this improvement is not too impressive, we consider our technique as an important new tool, which might be helpful in various other applications.
A tangle $T$ of height $h$ describes the ordering of $n$ elements called wires on each of $h$ levels. A pair of wires $ab$ is called a swap. Between any two adjacent levels only the ordering of adjacent wires may swap and each wire may participate in only one swap. The set of all pairs that are swapped in this way defines a multiset $S(T)$ of swaps called the swap list. A tangle $T$ realizes a given multiset $S$ of swaps if $S(T) = S$. The problem Tangle Height Minimization asks for the minimum height of a tangle that realizes $S$. In this paper, we study the tangle height minimization problem for simple tangles, whose swap list contains each swap at most once. First, we give algorithms with running time $O(n! \cdot 1.325^n)$ and $O((n+1)!)$ space and with running time $O(1.325^{hn})$ time but $O(n^2)$ space, thereby improving over a previous algorithm that uses $O(n! \cdot 1.618^n)$ time and space. Second, we show that the simple tangle height minimization problem can be recast as a graph orientation problem and propose an integer linear program based on this formulation. In an extensive experimental evaluation that considers several variants of each approach, we demonstrate that our exponential-time algorithms and the ILP are capable of solving instances with $n \leq 60$ in a reasonable period of time. Previous approaches can only handle instances up to $n \approx 20$.
This paper introduces a simple and fast method to combinatorially approximate the feedback arc set problem on weighted tournaments, a well-known classical NP-hard problem. Our results are as follows: • A combinatorial 3-approximation algorithm for the \textit{minimum feedback arc set} problem with the expected running time of $O(|V|\log |V|)$ on tournaments that satisfy the probability constraints, where $V$ is the set of vertices of the tournament. The previously best-known approximation factor was 4. • A combinatorial 5/3-approximation algorithm for the \textit{minimum feedback arc set} problem with the expected running time of $O(|V|\log |V|)$ on tournaments that satisfy both the probability and the triangle inequality constraints. The previously best-known approximation factor was 2. Although these problems currently have a PTAS algorithm which returns a $1+\epsilon$ approximation solution, the running time is doubly exponential with respect to $1/\epsilon$, being merely significant in theory or practice. Moreover, there are LP-based algorithms that solve these problems with better approximation factors than ours, but, in expense of solve an LP with $O(|V|^2)$ variables and $O(|V|^3)$ constraints which requires an excessive expenditure of time, making them inappropriate to use in practice. Therefore, the main superiority of our algorithms is that they dramatically reduce the running time by slightly increasing these approximation factors. Another considerable point is that, despite its complex analyses, we obtain these results by a very simple algorithm that can be implemented in several lines of code. We have also shown that the proposed approach can be applied to other problems. Here, we use the method to approximate the rank aggregation problem and show the superiority of the results compared to the current solutions.
In this paper, we first study the problem of decomposing a directed acyclic graph (DAG), $G=(V, E)$ into vertex-disjoint chains and present a fast and practical chain decomposition technique. Our algorithm produces results that are very close to the optimum. The availability of fast and practical chain decomposition algorithms opens the way for novel solutions to fundamental problems. Building on our chain decomposition method, we present fast and practical techniques for computing the reachability information in a directed graph (i.e., its transitive closure). We focus on applicable solutions that enable constant-time reachability queries. Given any path/chain decomposition of $G$, our technique computes a reachability indexing scheme of a DAG, $G=(V, E)$ in parameterized linear time. The experimental results reveal that our method is very fast in practice, indicating that chain decomposition algorithms can be applied to obtain fast and practical solutions to the transitive closure (TC) problem. Furthermore, we show that we can find a substantially large subset of the transitive edges of $G$ in linear time using any chain decomposition. Our extensive experimental results show the interplay between these concepts in various models of DAGs. Additionally, we show the efficiency of this solution by speeding up Fulkerson's method for finding the minimum chain decomposition that corresponds exactly to the width of the graph.
The phylogenetic inference process can produce, for multiple reasons, conflicting hypotheses of the evolutionary history of a set X of biological entities, i.e., phylogenetic trees with the same set of leaf labels X but with distinct topologies. It is natural to wishto quantify the difference between two such trees T1 and T2. We introduce the problem of computing a maximum relaxed agreement forest (MRAF) and use this as a proxy for the dissimilarity of T1 and T2, which in this article we assume to be unrooted and binary. MRAF asks for a partition of the leaf labels X into a minimum number of blocks S1, . . . , Sk such that the two subtrees induced in T1 and T2 by every Si are isomorphic up to suppression of degree-2 nodes and taking the labels X into account. Unlike the earlier introduced maximum agreement forest (MAF) model, the subtrees induced by the Si are allowed to overlap. We prove that it is NP-hard to compute MRAF, by reducing from the problem of partitioning a permutation into a minimum number of monotonic subsequences (PIMS). We further show that MRAF has a O(log n)-approximation algorithm where n = |X| and permits exact algorithms with single-exponential running time. When one of the trees is a caterpillar, we prove that testing whether a MRAF has size at most k can be answered in polynomial time when k is fixed. We also note that on two caterpillars the approximability of MRAF is related to that of PIMS. Finally, we establish a number of bounds on MRAF, compare its behaviour to MAF both theoretically and experimentally and discuss a number of open problems.
