Consider the setting of \emph{randomly weighted graphs}, namely, graphs whose edge weights are chosen independently according to probability distributions with finite support over the non-negative reals. Under this setting, properties of weighted graphs typically become random variables and we are interested in computing their statistical features. Unfortunately, this turns out to be computationally hard for some properties albeit the problem of computing them in the traditional setting of algorithmic graph theory is tractable. For example, there are well known efficient algorithms that compute the \emph{diameter} of a given weighted graph, yet, computing the \emph{expected} diameter of a given randomly weighted graph is \SharpP{}-hard even if the edge weights are identically distributed. In this paper, we define a family of properties of weighted graphs and show that for each property in this family, the problem of computing the \emph{$k^{\text{th}}$ moment} (and in particular, the expected value) of the corresponding random variable in a given randomly weighted graph $G$ admits a \emph{fully polynomial time randomized approximation scheme (FPRAS)} for every fixed $k$. This family includes fundamental properties of weighted graphs such as the diameter of $G$, the \emph{radius} of $G$ (with respect to any designated vertex) and the weight of a \emph{minimum spanning tree} of $G$.
We study the complexity of approximating solution structure of the bijective weighted sentence alignment problem of DeNero and Klein (2008). In particular, we consider the complexity of finding an alignment that has a significant overlap with an optimal alignment. We discuss ways of representing the solution for the general weighted sentence alignment as well as phrases-to-words alignment problem, and show that computing a string which agrees with the optimal sentence partition on more than half (plus an arbitrarily small polynomial fraction) positions for the phrases-to-words alignment is NP-…
Consider a scheduling problem in which jobs need to be processed on a single machine. Each job has a weight and is composed of several operations belonging to different families. The machine needs to perform a setup between the processing of operations of different families. A job is completed when its latest operation completes and the goal is to minimize the total weighted completion time of all jobs. We study this problem from the perspective of approximability and provide constant factor approximations as well as an inapproximability result. Prior to this work, only the NP-hardness of the …
In the weighted (minimum) {\sf Label $s$-$t$ Cut} problem, we are given a (directed or undirected) graph $G=(V,E)$, a label set $L = \{\ell_1, \ell_2, \dots, \ell_q \}$ with positive label weights $\{w_\ell\}$, a source $s \in V$ and a sink $t \in V$. Each edge edge $e$ of $G$ has a label $\ell(e)$ from $L$. Different edges may have the same label. The problem asks to find a minimum weight label subset $L'$ such that the removal of all edges with labels in $L'$ disconnects $s$ and $t$. The unweighted {\sf Label $s$-$t$ Cut} problem (i.e., every label has a unit weight) can be approximated with…
We determine the power of the weighted sum scalarization with respect to the computation of approximations for general multiobjective minimization and maximization problems. Additionally, we introduce a new multi-factor notion of approximation that is specifically tailored to the multiobjective case and its inherent trade-offs between different objectives. For minimization problems, we provide an efficient algorithm that computes an approximation of a multiobjective problem by using an exact or approximate algorithm for its weighted sum scalarization. In case that an exact algorithm for the we…
A mixed dominating set of a graph $G = (V, E)$ is a mixed set $D$ of vertices and edges, such that for every edge or vertex, if it is not in $D$, then it is adjacent or incident to at least one vertex or edge in $D$. The mixed domination problem is to find a mixed dominating set with a minimum cardinality. It has applications in system control and some other scenarios and it is $NP$-hard to compute an optimal solution. This paper studies approximation algorithms and hardness of the weighted mixed dominating set problem. The weighted version is a generalization of the unweighted version, where …
We study the problem of finding a minimum weight connected subgraph spanning at least $k$ vertices on planar, node-weighted graphs. We give a $(4+\eps)$-approximation algorithm for this problem. We achieve this by utilizing the recent LMP primal-dual $3$-approximation for the node-weighted prize-collecting Steiner tree problem by Byrka et al (SWAT'16) and adopting an approach by Chudak et al. (Math.\ Prog.\ '04) regarding Lagrangian relaxation for the edge-weighted variant. In particular, we improve the procedure of picking additional vertices (tree merging procedure) given by Sadeghian (2013)…
We study the problem of how well a tree metric is able to preserve the sum of pairwise distances of an arbitrary metric. This problem is closely related to low-stretch metric embeddings and is interesting by its own flavor from the line of research proposed in the literature. As the structure of a tree imposes great constraints on the pairwise distances, any embedding of a metric into a tree metric is known to have maximum pairwise stretch of $Ω(\log n)$. We show, however, from the perspective of average performance, there exist tree metrics which preserve the sum of pairwise distances of the …
We study the {\em min-cost chain-constrained spanning-tree} (abbreviated \mcst) problem: find a min-cost spanning tree in a graph subject to degree constraints on a nested family of node sets. We devise the {\em first} polytime algorithm that finds a spanning tree that (i) violates the degree constraints by at most a constant factor {\em and} (ii) whose cost is within a constant factor of the optimum. Previously, only an algorithm for {\em unweighted} \cst was known \cite{olver}, which satisfied (i) but did not yield any cost bounds. This also yields the first result that obtains an $O(1)$-fac…
Motivated by comparative genomics, Chen et al. [9] introduced the Maximum Duo-preservation String Mapping (MDSM) problem in which we are given two strings $s_1$ and $s_2$ from the same alphabet and the goal is to find a mapping $π$ between them so as to maximize the number of duos preserved. A duo is any two consecutive characters in a string and it is preserved in the mapping if its two consecutive characters in $s_1$ are mapped to same two consecutive characters in $s_2$. The MDSM problem is known to be NP-hard and there are approximation algorithms for this problem [3, 5, 13], but all of th…
In this note, we describe a $α_{GW} + \tildeΩ(1/d^2)$-factor approximation algorithm for Max-Cut on weighted graphs of degree $\leq d$. Here, $α_{GW}\approx 0.878$ is the worst-case approximation ratio of the Goemans-Williamson rounding for Max-Cut. This improves on previous results for unweighted graphs by Feige, Karpinski, and Langberg and Florén. Our guarantee is obtained by a tighter analysis of the solution obtained by applying a natural local improvement procedure to the Goemans-Williamson rounding of the basic SDP strengthened with triangle inequalities.
In some applications of matching, the structural or hierarchical properties of the two graphs being aligned must be maintained. The hierarchical properties are induced by the direction of the edges in the two directed graphs. These structural relationships defined by the hierarchy in the graphs act as a constraint on the alignment. In this paper, we formalize the above problem as the weighted alignment between two directed acyclic graphs. We prove that this problem is NP-complete, show several upper bounds for approximating the solution, and finally introduce polynomial time algorithms for sub…
In this letter, we introduce a novel message-passing algorithm for a class of problems which can be mathematically understood as estimating volume-related properties of random polytopes. Unlike the usual approach consisting in approximating the real-valued cavity marginal distributions by a few parameters, we propose a weighted message-passing algorithm to deal with the entire function. Various alternatives of how to implement our approach are discussed and numerical results for random polytopes are compared with results using the Hit-and-Run algorithm.
Among the most fundamental graph parameters is the Diameter, the largest distance between any pair of vertices. Computing the Diameter of a graph with $m$ edges requires $m^{2-o(1)}$ time under the Strong Exponential Time Hypothesis (SETH), which can be prohibitive for very large graphs, so efficient approximation algorithms for Diameter are desired. There is a folklore algorithm that gives a $2$-approximation for Diameter in $\tilde{O}(m)$ time. Additionally, a line of work concludes with a $3/2$-approximation algorithm for Diameter in weighted directed graphs that runs in $\tilde{O}(m^{3/2})…
A subset $S$ of nodes in a graph $G$ is a $k$-connected $m$-dominating set ($(k,m)$-cds) if the subgraph $G[S]$ induced by $S$ is $k$-connected and every $v \in V \setminus S$ has at least $m$ neighbors in $S$. In the $k$-Connected $m$-Dominating Set ($(k,m)$-CDS) problem the goal is to find a minimum weight $(k,m)$-cds in a node-weighted graph. For $m \geq k$ we obtain the following approximation ratios. For general graphs our ratio $O(k \ln n)$ improves the previous best ratio $O(k^2 \ln n)$ and matches the best known ratio for unit weights. For unit disc graphs we improve the ratio $O(k \ln…
