We revisit Min-Mean-Cycle, the classical problem of finding a cycle in a weighted directed graph with minimum mean weight. Despite an extensive algorithmic literature, previous work falls short of a near-linear runtime in the number of edges $m$. We propose an approximation algorithm that, for graphs with polylogarithmic diameter, achieves a near-linear runtime. In particular, this is the first algorithm whose runtime scales in the number of vertices $n$ as $\tilde{O}(n^2)$ for the complete graph. Moreover, unconditionally on the diameter, the algorithm uses only $O(n)$ memory beyond reading the input, making it "memory-optimal". Our approach is based on solving a linear programming relaxation using entropic regularization, which reduces the problem to Matrix Balancing -- á la the popular reduction of Optimal Transport to Matrix Scaling. The algorithm is practical and simple to implement.
We present a $(1+ε)$-approximation algorithm running in $O(f(ε)\cdot n \log^4 n)$ time for finding the diameter of an undirected planar graph with non-negative edge lengths.
Finding the diameter of a dataset in multidimensional Euclidean space is a well-established problem, with well-known algorithms. However, most of the algorithms found in the literature do not scale well with large values of data dimension, so the time complexity grows exponentially in most cases, which makes these algorithms impractical. Therefore, we implemented 4 simple greedy algorithms to be used for approximating the diameter of a multidimensional dataset; these are based on minimum/maximum l2 norms, hill climbing search, Tabu search and Beam search approaches, respectively. The time comp…
The diameter of a graph is one if its most important parameters, being used in many real-word applications. In particular, the diameter dictates how fast information can spread throughout data and communication networks. Thus, it is a natural question to ask how much can we sparsify a graph and still guarantee that its diameter remains preserved within an approximation $t$. This property is captured by the notion of extremal-distance spanners. Given a graph $G=(V,E)$, a subgraph $H=(V,E_H)$ is defined to be a $t$-diameter spanner if the diameter of $H$ is at most $t$ times the diameter of $G$.…
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})…
We design approximation algorithms for Unique Games when the constraint graph admits good low diameter graph decomposition. For the ${\sf Max2Lin}_k$ problem in $K_r$-minor free graphs, when there is an assignment satisfying $1-\varepsilon$ fraction of constraints, we present an algorithm that produces an assignment satisfying $1-O(r\varepsilon)$ fraction of constraints, with the approximation ratio independent of the alphabet size. A corollary is an improved approximation algorithm for the ${\sf MaxCut}$ problem for $K_r$-minor free graphs. For general Unique Games in $K_r$-minor free graphs,…
In this paper we consider the fundamental problem of approximating the diameter $D$ of directed or undirected graphs. In a seminal paper, Aingworth, Chekuri, Indyk and Motwani [SIAM J. Comput. 1999] presented an algorithm that computes in $\Ot(m\sqrt n + n^2)$ time an estimate $\hat{D}$ for the diameter of an $n$-node, $m$-edge graph, such that $\lfloor 2/3 D \rfloor \leq \hat{D} \leq D$. In this paper we present an algorithm that produces the same estimate in $\Ot(m\sqrt n)$ expected running time. We then provide strong evidence that a better approximation may be hard to obtain if we insist o…
We prove several tight results on the fine-grained complexity of approximating the diameter of a graph. First, we prove that, for any $\varepsilon>0$, assuming the Strong Exponential Time Hypothesis (SETH), there are no near-linear time $2-\varepsilon$-approximation algorithms for the Diameter of a sparse directed graph, even in unweighted graphs. This result shows that a simple near-linear time 2-approximation algorithm for Diameter is optimal under SETH, answering a question from a survey of Rubinstein and Vassilevska-Williams (SIGACT '19) for the case of directed graphs. In the same survey,…
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 comput…
We show, assuming the Strong Exponential Time Hypothesis, that for every $\varepsilon > 0$, approximating directed Diameter on $m$-arc graphs within ratio $7/4 - \varepsilon$ requires $m^{4/3 - o(1)}$ time. Our construction uses nonnegative edge weights but even holds for sparse digraphs, i.e., for which the number of vertices $n$ and the number of arcs $m$ satisfy $m = n \log^{O(1)} n$. This is the first result that conditionally rules out a near-linear time $5/3$-approximation for Diameter.
Betweenness is a well-known centrality measure that ranks the nodes of a network according to their participation in shortest paths. Since an exact computation is prohibitive in large networks, several approximation algorithms have been proposed. Besides that, recent years have seen the publication of dynamic algorithms for efficient recomputation of betweenness in networks that change over time. In this paper we propose the first betweenness centrality approximation algorithms with a provable guarantee on the maximum approximation error for dynamic networks. Several new intermediate algorithm…
We consider a large family of problems in which an ordering (or, more precisely, a chain of subsets) of a finite set must be chosen to minimize some weighted sum of costs. This family includes variations of Min Sum Set Cover (MSSC), several scheduling and search problems, and problems in Boolean function evaluation. We define a new problem, called the Min Sum Ordering Problem (MSOP) which generalizes all these problems using a cost and a weight function defined on subsets of a finite set. Assuming a polynomial time $α$-approximation algorithm for the problem of finding a subset whose ratio of …
This article investigates the approximation quality achievable for biobjective minimization problems with respect to the Pareto cone by solutions that are (approximately) optimal with respect to larger ordering cones. When simultaneously considering $α$-approximations for all closed convex ordering cones of a fixed inner angle $γ\in [\frac π2, π]$, an approximation guarantee between $α$ and $2 α$ is achieved, which depends continuously on $γ$. The analysis is best-possible for any inner angle and it generalizes and unifies the known results that the set of supported solutions is a 2-approximat…
Zwick's $(1+\varepsilon)$-approximation algorithm for the All Pairs Shortest Path (APSP) problem runs in time $\widetilde{O}(\frac{n^ω}{\varepsilon} \log{W})$, where $ω\le 2.373$ is the exponent of matrix multiplication and $W$ denotes the largest weight. This can be used to approximate several graph characteristics including the diameter, radius, median, minimum-weight triangle, and minimum-weight cycle in the same time bound. Since Zwick's algorithm uses the scaling technique, it has a factor $\log W$ in the running time. In this paper, we study whether APSP and related problems admit approx…
This paper is about minimum cost constrained selection of inputs and outputs for generic arbitrary pole placement. The input-output set is constrained in the sense that the set of states that each input can influence and the set of states that each output can sense is pre-specified. Our goal is to optimally select an input-output set that the system has no structurally fixed modes. Polynomial algorithms do not exist for solving this problem unless P=NP. To this end, we propose an approximation algorithm by splitting the problem in to three sub-problems: a) minimum cost accessibility problem, b…
We consider the stochastic geometry model where the location of each node is a random point in a given metric space, or the existence of each node is uncertain. We study the problems of computing the expected lengths of several combinatorial or geometric optimization problems over stochastic points, including closest pair, minimum spanning tree, $k$-clustering, minimum perfect matching, and minimum cycle cover. We also consider the problem of estimating the probability that the length of closest pair, or the diameter, is at most, or at least, a given threshold. Most of the above problems are k…
We prove a strong inapproximability result for the Balanced Minimum Evolution Problem. Our proof also implies that the problem remains NP-hard even when restricted to metric instances. Furthermore, we give a MST-based 2-approximation algorithm for the problem for such instances.
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…
Given a directed graph $G$ with non negative cost on the arcs, a directed tree cover of $G$ is a rooted directed tree such that either head or tail (or both of them) of every arc in $G$ is touched by $T$. The minimum directed tree cover problem (DTCP) is to find a directed tree cover of minimum cost. The problem is known to be $NP$-hard. In this paper, we show that the weighted Set Cover Problem (SCP) is a special case of DTCP. Hence, one can expect at best to approximate DTCP with the same ratio as for SCP. We show that this expectation can be satisfied in some way by designing a purely combi…
We study the problem of augmenting a weighted graph by inserting edges of bounded total cost while minimizing the diameter of the augmented graph. Our main result is an FPT 4-approximation algorithm for the problem.
We present improved algorithms for short cycle decomposition of a graph. Short cycle decompositions were introduced in the recent work of Chu et al, and were used to make progress on several questions in graph sparsification. For all constants $δ\in (0,1]$, we give an $O(mn^δ)$ time algorithm that, given a graph $G,$ partitions its edges into cycles of length $O(\log n)^\frac{1}δ$, with $O(n)$ extra edges not in any cycle. This gives the first subquadratic, in fact almost linear time, algorithm achieving polylogarithmic cycle lengths. We also give an $m \cdot \exp(O(\sqrt{\log n}))$ time algor…
We present a deterministic $(1+o(1))$-approximation $(n^{1/2+o(1)}+D^{1+o(1)})$-time algorithm for solving the single-source shortest paths problem on distributed weighted networks (the CONGEST model); here $n$ is the number of nodes in the network and $D$ is its (hop) diameter. This is the first non-trivial deterministic algorithm for this problem. It also improves (i) the running time of the randomized $(1+o(1))$-approximation $\tilde O(n^{1/2}D^{1/4}+D)$-time algorithm of Nanongkai [STOC 2014] by a factor of as large as $n^{1/8}$, and (ii) the $O(ε^{-1}\log ε^{-1})$-approximation factor of …
Consider the following distance query for an $n$-node graph $G$ undergoing edge insertions and deletions: given two sets of nodes $I$ and $J$, return the distances between every pair of nodes in $I\times J$. This query is rather general and captures several versions of the dynamic shortest paths problem. In this paper, we develop an efficient $(1+ε)$-approximation algorithm for this query using fast matrix multiplication. Our algorithm leads to answers for some open problems for Single-Source and All-Pairs Shortest Paths (SSSP and APSP), as well as for Diameter, Radius, and Eccentricities. Bel…
It has been observed that many complex real-world networks have certain properties, such as a high clustering coefficient, a low diameter, and a power-law degree distribution. A network with a power-law degree distribution is known as scale-free network. In order to study these networks, various random graph models have been proposed, e.g. Preferential Attachment, Chung-Lu, or Hyperbolic. We look at the interplay between the power-law degree distribution and the run time of optimization techniques for well known combinatorial problems. We observe that on scale-free networks, simple evolutionar…
Horn functions form a subclass of Boolean functions and appear in many different areas of computer science and mathematics as a general tool to describe implications and dependencies. Finding minimum sized representations for such functions with respect to most commonly used measures is a computationally hard problem that remains hard even for the important subclass of key Horn functions. In this paper we provide logarithmic factor approximation algorithms for key Horn functions with respect to all measures studied in the literature for which the problem is known to be hard.
We consider directed graphs where each edge is labeled with an integer weight and study the fundamental algorithmic question of computing the value of a cycle with minimum mean weight. Our contributions are twofold: (1) First we show that the algorithmic question is reducible in O(n^2) time to the problem of a logarithmic number of min-plus matrix multiplications of n-by-n matrices, where n is the number of vertices of the graph. (2) Second, when the weights are nonnegative, we present the first (1 + ε)-approximation algorithm for the problem and the running time of our algorithm is \tilde(O)(…
Given a graph with edge costs, the {\em power} of a node is themaximum cost of an edge incident to it, and the power of a graph is the sum of the powers of its nodes. Motivated by applications in wireless networks, we consider the following fundamental problem in wireless network design. Given a graph $G=(V,E)$ with edge costs and degree bounds $\{r(v):v \in V\}$, the {\sf Minimum-Power Edge-Multi-Cover} ({\sf MPEMC}) problem is to find a minimum-power subgraph $J$ of $G$ such that the degree of every node $v$ in $J$ is at least $r(v)$. We give two approximation algorithms for {\sf MPEMC}, wit…
We present a 6-approximation algorithm for the minimum-cost $k$-node connected spanning subgraph problem, assuming that the number of nodes is at least $k^3(k-1)+k$. We apply a combinatorial preprocessing, based on the Frank-Tardos algorithm for $k$-outconnectivity, to transform any input into an instance such that the iterative rounding method gives a 2-approximation guarantee. This is the first constant-factor approximation algorithm even in the asymptotic setting of the problem, that is, the restriction to instances where the number of nodes is lower bounded by a function of $k$.
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 consider the minimum vertex cover problem in hypergraphs in which every hyperedge has size k (also known as minimum hitting set problem, or minimum set cover with element frequency k). Simple algorithms exist that provide k-approximations, and this is believed to be the best possible approximation achievable in polynomial time. We show how to exploit density and regularity properties of the input hypergraph to break this barrier. In particular, we provide a randomized polynomial-time algorithm with approximation factor k/(1 +(k-1)d/(k Delta)), where d and Delta are the average and maximum d…
Conditional lower bounds for dynamic graph problems has received a great deal of attention in recent years. While many results are now known for the fully-dynamic case and such bounds often imply worst-case bounds for the partially dynamic setting, it seems much more difficult to prove amortized bounds for incremental and decremental algorithms. In this paper we consider partially dynamic versions of three classic problems in graph theory. Based on popular conjectures we show that: -- No algorithm with amortized update time $O(n^{1-\varepsilon})$ exists for incremental or decremental maximum c…
Approximating Subset Sum is a classic and fundamental problem in computer science and mathematical optimization. The state-of-the-art approximation scheme for Subset Sum computes a $(1-\varepsilon)$-approximation in time $\tilde{O}(\min\{n/\varepsilon, n+1/\varepsilon^2\})$ [Gens, Levner'78, Kellerer et al.'97]. In particular, a $(1-1/n)$-approximation can be computed in time $O(n^2)$. We establish a connection to Min-Plus-Convolution, a problem that is of particular interest in fine-grained complexity theory and can be solved naively in time $O(n^2)$. Our main result is that computing a $(1-1…
The neighbourhood function N(t) of a graph G gives, for each t, the number of pairs of nodes <x, y> such that y is reachable from x in less that t hops. The neighbourhood function provides a wealth of information about the graph (e.g., it easily allows one to compute its diameter), but it is very expensive to compute it exactly. Recently, the ANF algorithm (approximate neighbourhood function) has been proposed with the purpose of approximating NG(t) on large graphs. We describe a breakthrough improvement over ANF in terms of speed and scalability. Our algorithm, called HyperANF, uses the new H…
We propose to study unweighted graphs of constant distance VC-dimension as a broad generalization of many graph classes for which we can compute the diameter in truly subquadratic-time. In particular for any fixed $H$, the class of $H$-minor free graphs has distance VC-dimension at most $|V(H)|-1$. Our first main result is that on graphs of distance VC-dimension at most $d$, for any fixed $k$ we can either compute the diameter or conclude that it is larger than $k$ in time $\tilde{\cal O}(k\cdot mn^{1-\varepsilon_d})$, where $\varepsilon_d \in (0;1)$ only depends on $d$. Then as a byproduct of…
We consider a network design problem called the generalized terminal backup problem. Whereas earlier work investigated the edge-connectivity constraints only, we consider both edge- and node-connectivity constraints for this problem. A major contribution of this paper is the development of a strongly polynomial-time 4/3-approximation algorithm for the problem. Specifically, we show that a linear programming relaxation of the problem is half-integral, and that the half-integral optimal solution can be rounded to a 4/3-approximate solution. We also prove that the linear programming relaxation of…
We give a nearly optimal sublinear-time algorithm for approximating the size of a minimum vertex cover in a graph G. The algorithm may query the degree deg(v) of any vertex v of its choice, and for each 1 <= i <= deg(v), it may ask for the i-th neighbor of v. Letting VC_opt(G) denote the minimum size of vertex cover in G, the algorithm outputs, with high constant success probability, an estimate VC_estimate(G) such that VC_opt(G) <= VC_estimate(G) <= 2 * VC_opt(G) + epsilon*n, where epsilon is a given additive approximation parameter. We refer to such an estimate as a (2,epsilon)-estimate. The…
In multistage perfect matching problems we are given a sequence of graphs on the same vertex set and asked to find a sequence of perfect matchings, corresponding to the sequence of graphs, such that consecutive matchings are as similar as possible. More precisely, we aim to maximize the intersections, or minimize the unions between consecutive matchings. We show that these problems are NP-hard even in very restricted scenarios. We propose new approximation algorithms and present methods to transfer results between different problem variants without loosing approximation guarantees.
We present approximation algorithms for almost all variants of the multi-criteria traveling salesman problem (TSP). First, we devise randomized approximation algorithms for multi-criteria maximum traveling salesman problems (Max-TSP). For multi-criteria Max-STSP, where the edge weights have to be symmetric, we devise an algorithm with an approximation ratio of 2/3 - eps. For multi-criteria Max-ATSP, where the edge weights may be asymmetric, we present an algorithm with a ratio of 1/2 - eps. Our algorithms work for any fixed number k of objectives. Furthermore, we present a deterministic algori…
We consider the problem of adding a fixed number of new edges to an undirected graph in order to minimize the diameter of the augmented graph, and under the constraint that the number of edges added for each vertex is bounded by an integer. The problem is motivated by network-design applications, where we want to minimize the worst case communication in the network without excessively increasing the degree of any single vertex, so as to avoid additional overload. We present three algorithms for this task, each with their own merits. The special case of a matching augmentation, when every verte…
Given a metric space $(F \cup C, d)$, we consider star covers of $C$ with balanced loads. A star is a pair $(f, C_f)$ where $f \in F$ and $C_f \subseteq C$, and the load of a star is $\sum_{c \in C_f} d(f, c)$. In minimum load $k$-star cover problem $(\mathrm{MLkSC})$, one tries to cover the set of clients $C$ using $k$ stars that minimize the maximum load of a star, and in minimum size star cover $(\mathrm{MSSC})$ one aims to find the minimum number of stars of load at most $T$ needed to cover $C$, where $T$ is a given parameter. We obtain new bicriteria approximations for the two problems us…
We study the minimum diameter spanning tree problem under the reload cost model (DIAMETER-TREE for short) introduced by Wirth and Steffan (2001). In this problem, given an undirected edge-colored graph $G$, reload costs on a path arise at a node where the path uses consecutive edges of different colors. The objective is to find a spanning tree of $G$ of minimum diameter with respect to the reload costs. We initiate a systematic study of the parameterized complexity of the DIAMETER-TREE problem by considering the following parameters: the cost of a solution, and the treewidth and the maximum de…
Graph connectivity and network design problems are among the most fundamental problems in combinatorial optimization. The minimum spanning tree problem, the two edge-connected spanning subgraph problem (2-ECSS) and the tree augmentation problem (TAP) are all examples of fundamental well-studied network design tasks that postulate different initial states of the network and different assumptions on the reliability of network components. In this paper we motivate and study \emph{Flexible Graph Connectivity} (FGC), a problem that mixes together both the modeling power and the complexities of all …
We present a new algorithm, which solves the problem of distributively finding a minimum diameter spanning tree of any (non-negatively) real-weighted graph $G = (V,E,ω)$. As an intermediate step, we use a new, fast, linear-time all-pairs shortest paths distributed algorithm to find an absolute center of $G$. The resulting distributed algorithm is asynchronous, it works for named asynchronous arbitrary networks and achieves $\mathcal{O}(|V|)$ time complexity and $\mathcal{O}\left(|V|\,|E|\right)$
The diameter of an undirected or a directed graph is defined to be the maximum shortest path distance over all pairs of vertices in the graph. Given an undirected graph $G$, we examine the problem of assigning directions to each edge of $G$ such that the diameter of the resulting oriented graph is minimized. The minimum diameter over all strongly connected orientations is called the oriented diameter of $G$. The problem of determining the oriented diameter of a graph is known to be NP-hard, but the time-complexity question is open for planar graphs. In this paper we compute the exact value of …
