New Algorithms for Maximum Disjoint Paths Based on Tree-Likeness

We study the classical NP-hard problems of finding maximum-size subsets from given sets of k terminal pairs that can be routed via edge-disjoint paths (MaxEDP) or node-disjoint paths (MaxNDP) in a given graph. The approximability of MaxEDP/NDP is currently not well understood; the best known lower bound is Omega(log^{1/2 - varepsilon} n), assuming NP not subseteq ZPTIME(n^{poly log n}). This constitutes a significant gap to the best known approximation upper bound of O(n^1/2) due to Chekuri et al. (2006) and closing this gap is currently one of the big open problems in approximation algorithms. In their seminal paper, Raghavan and Thompson (Combinatorica, 1987) introduce the technique of randomized rounding for LPs; their technique gives an O(1)-approximation when edges (or nodes) may be used by O(log n/log log n) paths. In this paper, we strengthen the above fundamental results. We provide new bounds formulated in terms of the feedback vertex set number r of a graph, which measures its vertex deletion distance to a forest. In particular, we obtain the following. - For MaxEDP, we give an O(r^0.5 log^1.5 kr)-approximation algorithm. As r<=n, up to logarithmic factors, our result strengthens the best known ratio O(n^0.5) due to Chekuri et al. - Further, we show how to route Omega(opt) pairs with congestion O(log(kr)/log log(kr)), strengthening the bound obtained by the classic approach of Raghavan and Thompson. - For MaxNDP, we give an algorithm that gives the optimal answer in time (k+r)^O(r)n. This is a substantial improvement on the run time of 2^kr^O(r)n, which can be obtained via an algorithm by Scheffler. We complement these positive results by proving that MaxEDP is NP-hard even for r=1, and MaxNDP is W[1]-hard for parameter r. This shows that neither problem is fixed-parameter tractable in r unless FPT = W[1] and that our approximability results are relevant even for very small constant values of r

On Structural Parameterizations of the Edge Disjoint Paths Problem

In this paper we revisit the classical Edge Disjoint Paths (EDP) problem, where one is given an undirected graph G and a set of terminal pairs P and asks whether G contains a set of pairwise edge-disjoint paths connecting every terminal pair in P. Our focus lies on structural parameterizations for the problem that allow for efficient (polynomial-time or fpt) algorithms. As our first result, we answer an open question stated in Fleszar, Mnich, and Spoerhase (2016), by showing that the problem can be solved in polynomial time if the input graph has a feedback vertex set of size one. We also show that EDP parameterized by the treewidth and the maximum degree of the input graph is fixed-parameter tractable. Having developed two novel algorithms for EDP using structural restrictions on the input graph, we then turn our attention towards the augmented graph, i.e., the graph obtained from the input graph after adding one edge between every terminal pair. In constrast to the input graph, where EDP is known to remain NP-hard even for treewidth two, a result by Zhou et al. (2000) shows that EDP can be solved in non-uniform polynomial time if the augmented graph has constant treewidth; we note that the possible improvement of this result to an fpt-algorithm has remained open since then. We show that this is highly unlikely by establishing the W[1]-hardness of the problem parameterized by the treewidth (and even feedback vertex set) of the augmented graph. Finally, we develop an fpt-algorithm for EDP by exploiting a novel structural parameter of the augmented graph

The study of probability model for compound similarity searching

Information Retrieval or IR system main task is to retrieve relevant documents according to the users query. One of IR most popular retrieval model is the Vector Space Model. This model assumes relevance based on similarity, which is defined as the distance between query and document in the concept space. All currently existing chemical compound database systems have adapt the vector space model to calculate the similarity of a database entry to a query compound. However, it assumes that fragments represented by the bits are independent of one another, which is not necessarily true. Hence, the possibility of applying another IR model is explored, which is the Probabilistic Model, for chemical compound searching. This model estimates the probabilities of a chemical structure to have the same bioactivity as a target compound. It is envisioned that by ranking chemical structures in decreasing order of their probability of relevance to the query structure, the effectiveness of a molecular similarity searching system can be increased. Both fragment dependencies and independencies assumption are taken into consideration in achieving improvement towards compound similarity searching system. After conducting a series of simulated similarity searching, it is concluded that PM approaches really did perform better than the existing similarity searching. It gave better result in all evaluation criteria to confirm this statement. In terms of which probability model performs better, the BD model shown improvement over the BIR model

Many-to-one matchings with lower quotas : algorithms and complexity

We study a natural generalization of the maximum weight many-to-one matching problem. We are given an undirected bipartite graph G = (AUP,E) with weights on the edges in E, and with lower and upper quotas on the vertices in P. We seek a maximum weight many-toone matching satisfying two sets of constraints: vertices in A are incident to at most one matching edge, while vertices in P are either unmatched or they are incident to a number of matching edges between their lower and upper quota. This problem, which we call maximum weight many-toone matching with lower and upper quotas (wmlq), has applications to the assignment of students to projects within university courses, where there are constraints on the minimum and maximum numbers of students that must be assigned to each project. In this paper, we provide a comprehensive analysis of the complexity of wmlq from the viewpoints of classic polynomial time algorithms, fixed-parameter tractability, as well as approximability. We draw the line between NP-hard and polynomially tractable instances in terms of degree and quota constraints and provide efficient algorithms to solve the tractable ones. We further show that the problem can be solved in polynomial time for instances with bounded treewidth; however, the corresponding runtime is exponential in the treewidth with the maximum upper quota umax as basis, and we prove that this dependence is necessary unless FPT = W[1]. Finally, we also present an approximation algorithm for the general case with performance guarantee umax+1, which is asymptotically best possible unless P = NP

On Directed Feedback Vertex Set parameterized by treewidth

We study the Directed Feedback Vertex Set problem parameterized by the treewidth of the input graph. We prove that unless the Exponential Time Hypothesis fails, the problem cannot be solved in time $2^{o(t\log t)}\cdot n^{\mathcal{O}(1)}$ on general directed graphs, where $t$ is the treewidth of the underlying undirected graph. This is matched by a dynamic programming algorithm with running time $2^{\mathcal{O}(t\log t)}\cdot n^{\mathcal{O}(1)}$. On the other hand, we show that if the input digraph is planar, then the running time can be improved to $2^{\mathcal{O}(t)}\cdot n^{\mathcal{O}(1)}$.Comment: 20

The Power of Cut-Based Parameters for Computing Edge Disjoint Paths

This paper revisits the classical Edge Disjoint Paths (EDP) problem, where one is given an undirected graph G and a set of terminal pairs P and asks whether G contains a set of pairwise edge-disjoint paths connecting every terminal pair in P. Our aim is to identify structural properties (parameters) of graphs which allow the efficient solution of EDP without restricting the placement of terminals in P in any way. In this setting, EDP is known to remain NP-hard even on extremely restricted graph classes, such as graphs with a vertex cover of size 3. We present three results which use edge-separator based parameters to chart new islands of tractability in the complexity landscape of EDP. Our first and main result utilizes the fairly recent structural parameter treecut width (a parameter with fundamental ties to graph immersions and graph cuts): we obtain a polynomial-time algorithm for EDP on every graph class of bounded treecut width. Our second result shows that EDP parameterized by treecut width is unlikely to be fixed-parameter tractable. Our final, third result is a polynomial kernel for EDP parameterized by the size of a minimum feedback edge set in the graph

Measuring what Matters: A Hybrid Approach to Dynamic Programming with Treewidth

We develop a framework for applying treewidth-based dynamic programming on graphs with "hybrid structure", i.e., with parts that may not have small treewidth but instead possess other structural properties. Informally, this is achieved by defining a refinement of treewidth which only considers parts of the graph that do not belong to a pre-specified tractable graph class. Our approach allows us to not only generalize existing fixed-parameter algorithms exploiting treewidth, but also fixed-parameter algorithms which use the size of a modulator as their parameter. As the flagship application of our framework, we obtain a parameter that combines treewidth and rank-width to obtain fixed-parameter algorithms for Chromatic Number, Hamiltonian Cycle, and Max-Cut