pinecone/core-2020-05-10-deduplication huggingface dataset:
Polynomial algorithms for partitioning a tree into single-center subtrees to minimize flat service costs

This paper deals with the following graph partitioning problem. Consider a connected graph with n nodes, p of which are centers, while the remaining ones are units. For each unit-center pair there is a fixed service cost and the goal is to find a partition into connected components such that each component contains only one center and the total service cost is minimum. This problem is known to be NP-hard on general graphs, and here we show that it remains such even if the service cost is monotone and the graph is bipartite. However, in this paper we derive some polynomial time algorithms for trees. For this class of graphs we provide several reformulations of the problem as integer linear programs proving the integrality of the corresponding polyhedra. As a consequence, the tree partitioning problem can be solved in polynomial time either by linear programming or by suitable convex nondifferentiable optimization algorithms. Moreover, we develop a dynamic programming algorithm, whose recursion is based on sequences of minimum weight closure problems, which solves the problem on trees in O(np) time
Transversals of subtree hypergraphs and the source location problem in digraphs.

A hypergraph H = (V,E) is a subtree hypergraph if there is a tree T on V such that each hyperedge of E induces a subtree of T. Since the number of edges of a subtree hypergraph can be exponential in n = |V|, one can not always expect to be able to find a minimum size transversal in time polynomial in n. In this paper, we show that if it is possible to decide if a set of vertices W ⊆ V is a transversal in time S(n) (where n = |V|), then it is possible to find a minimum size transversal in O(n3S(n)). This result provides a polynomial algorithm for the Source Location Problem: a set of (k,l)-sources for a digraph D = (V,A) is a subset S of V such that for any v ∈ V there are k arc-disjoint paths that each join a vertex of S to v and l arc-disjoint paths that each join v to S. The Source Location Problem is to find a minimum size set of (k,l)-sources. We show that this is a case of finding a transversal of a subtree hypergraph, and that in this case S(n) is polynomial
Minimal selectors and fault tolerant networks

International audienceIn this paper we study a combinatorial optimization problem issued from on-board networks in satellites. In this kind of networks the entering signals (inputs) should be routed to amplifiers (outputs). The connections are made via expensive switches with four links available. The paths connecting inputs to outputs should be link-disjoint. More formally, we call it $\plk-$network an undirected graph with $p+\lambda$ inputs, $p+k$ outputs and internal vertices of degree four. A $\plk-$network is valid if it is tolerant to a restricted number of faults in the network, i.e. if for any choice of at most $k$ faulty inputs and $\lambda$ faulty outputs, there exist $p$ edge-disjoint paths from the remaining inputs to the remaining outputs. In the special case $\lambda=0$, a $\plk-$network is already known as a selector. Our optimization problem consists of determining $N\plk$, the minimum number of nodes in a valid $\plk-$network. For this, we present validity certificates and a gluing lemma from which derive lower bounds for $N\plk$. We also provide constructions, and hence upper bounds, based on expanders. The problem is very sensitive to the order of $\lambda$ and $k$. For instance, when $\lambda$ and $k$ are small compared to $p$, the question reduces to avoid certain forbidden local configurations. For larger values of $\lambda$ and $k$, the problem is to find graphs with a good expansion property for small sets. This leads us to introduce a new parameter called $\alpha$-robustness. We use $\alpha$-robustness to generalize our constructions to higher order values of $k$ and $\lambda$. In many cases, we provide asymptotically tight bounds for $N\plk$
Stackelberg Network Pricing is Hard to Approximate

In the Stackelberg Network Pricing problem, one has to assign tariffs to a
certain subset of the arcs of a given transportation network. The aim is to
maximize the amount paid by the user of the network, knowing that the user will
take a shortest st-path once the tariffs are fixed. Roch, Savard, and Marcotte
(Networks, Vol. 46(1), 57-67, 2005) proved that this problem is NP-hard, and
gave an O(log m)-approximation algorithm, where m denote the number of arcs to
be priced. In this note, we show that the problem is also APX-hard
Traffic grooming in bidirectional WDM ring networks

International audienceWe study the minimization of ADMs (Add-Drop Multiplexers) in optical WDM bidirectional rings considering symmetric shortest path routing and all-to-all unitary requests. We precisely formulate the problem in terms of graph decompositions, and state a general lower bound for all the values of the grooming factor C and N, the size of the ring. We first study exhaustively the cases C = 1, C = 2, and C = 3, providing improved lower bounds, optimal constructions for several infinite families, as well as asymptotically optimal constructions and approximations. We then study the case C > 3, focusing specifically on the case C = k(k + 1)/2 for some k ≥ 1. We give optimal decompositions for several congruence classes of N using the existence of some combinatorial designs. We conclude with a comparison of the cost functions in unidirectional and bidirectional WDM rings
The Complexity of Welfare Maximization in Congestion Games

We investigate issues of complexity related to welfare maximization in congestion games. In particular, we provide a full classification of complexity results for the problem of finding a minimum cost solution to a congestion game, under the model of Rosenthal. We consider both network and general congestion games, and we examine several variants of the problem concerning the structure of the game and the properties of its associated cost functions. Many of these problem variants turn out to be NP-hard, and some are hard to approximate to within any finite factor, unless P = NP. We also identify several versions of the problem that are solvable in polynomial time.United States. Dept. of Energy (Grant Number: DE-AC52-07NA27344)Lawrence Livermore National Laboratory (Grant Number: LLNL-JRNL-410585)United States. Office of Naval Research (Grant Number: N000141110056
A note on a generalization of eigenvector centrality for bipartite
  graphs and applications

Eigenvector centrality is a linear algebra based graph invariant used in
various rating systems such as webpage ratings for search engines. A
generalization of the eigenvector centrality invariant is defined which is
motivated by the need to design rating systems for bipartite graph models of
time-sensitive and other processes. The linear algebra connection and some
applications are described.Comment: Published in Networks: Peteris Daugulis: A note on a generalization
  of eigenvector centrality for bipartite graphs and applications. Networks
  59(2): 261-264 (2012
Deception tactics for network interdiction: a multiobjective approach

The article of record as published may be found at:http://dx.doi:10.1002/net.20458This article develops defender-attacker network interdiction models with deception.  Here, deception refers to a preemptive and intelligent use of concealed interdiction assets and decoys by the defender, in addition to transparent assets commonly employed in modeling defender-attacker problems.  These models can help security planners to locate a limited number of checkpoints and sensors of various types to, for example, detect the smuggling of illegal products.  The problem is complex, in part, because the objective function of the defender and the attacker are different, and because the latter (which represents the attacker's behavior) is difficult to predict by the defender.  First, we use duality theory and a generalized network flow model to devise an equivalent mixed-integer programming formulation, and develops its Benders decomposition.  We extend this formulation with a multiobjective approach to account for several behaviors simultaneously.  The computational effort to solve these models is considerable, as exemplified by our testing on a variety of cases for a medium-sized, notional network.U.S. Center for Army Analysi
A model of self-avoiding random walks for searching complex networks

Random walks have been proven useful in several applications in networks. Some variants of the basic random walk have been devised pursuing a suitable trade-off between better performance and limited cost. A self-avoiding random walk (SAW) is one that tries not to revisit nodes, therefore covering the network faster than a random walk. Suggested as a network search mechanism, the performance of the SAW has been analyzed using essentially empirical studies. A strict analytical approach is hard since, unlike the random walk, the SAW is not a Markovian stochastic process. We propose an analytical model to estimate the average search length of a SAW when used to locate a resource in a network. The model considers single or multiple instances of the resource sought and the possible availability of one-hop replication in the network (nodes know about resources held by their neighbors). The model characterizes networks by their size and degree distribution, without assuming a particular topology. It is, therefore, a mean-field model, whose applicability to real networks is validated by simulation. Experiments with sets of randomly built regular networks, Erdős–Rényi networks, and scale-free networks of several sizes and degree averages, with and without one-hop replication, show that model predictions are very close to simulation results, and allow us to draw conclusions about the applicability of SAWs to network search
The orienteering problem with variable profits

This article introduces, models, and solves a generalization of the orienteering problem, called the the orienteering problem with variable profits (OPVP). The OPVP is defined on a complete undirected graph G = (V,E), with a depot at vertex 0. Every vertex i∈V \{0} has a profit pi to be collected, and an associated collection parameter αi∈[0, 1]. The vehicle may make a number of “passes,” collecting 100αi percent of the remaining profit at each pass. In an alternative model, the vehicle may spend a continuous amount of time at every vertex, collecting a percentage of the profit given by a function of the time spent. The objective is to determine a maximal profit tour for the vehicle, starting and ending at the depot, and not exceeding a travel time limit
Scheduling unit processing time arc shutdown jobs to maximize network
  flow over time: complexity results

We study the problem of scheduling maintenance on arcs of a capacitated
network so as to maximize the total flow from a source node to a sink node over
a set of time periods. Maintenance on an arc shuts down the arc for the
duration of the period in which its maintenance is scheduled, making its
capacity zero for that period. A set of arcs is designated to have maintenance
during the planning period, which will require each to be shut down for exactly
one time period. In general this problem is known to be NP-hard. Here we
identify a number of characteristics that are relevant for the complexity of
instance classes. In particular, we discuss instances with restrictions on the
set of arcs that have maintenance to be scheduled; series parallel networks;
capacities that are balanced, in the sense that the total capacity of arcs
entering a (non-terminal) node equals the total capacity of arcs leaving the
node; and identical capacities on all arcs
Revisiting deadlock prevention: a probabilistic approach

We revisit the deadlock-prevention problem by focusing on priority digraphs
instead of the traditional wait-for digraphs. This has allowed us to formulate
deadlock prevention in terms of prohibiting the occurrence of directed cycles
even in the most general of wait models (the so-called AND-OR model, in which
prohibiting wait-for directed cycles is generally overly restrictive). For a
particular case in which the priority digraphs are somewhat simplified, we
introduce a Las Vegas probabilistic mechanism for resource granting and analyze
its key aspects in detail
Bounds on the radius and status of graphs

Two classical concepts of centrality in a graph are the median and the
center. The connected notions of the status and the radius of a graph seem to
be in no relation. In this paper, however, we show a clear connection of both
concepts, as they obtain their minimum and maximum values at the same type of
tree graphs. Trees with fixed maximum degree and extremum radius and status,
resp., are characterized. The bounds on radius and status can be transferred to
general connected graphs via spanning trees.
  A new method of proof allows not only to regain results of Lin et al. on
graphs with extremum status, but it allows also to prove analogous results on
graphs with extremum radius
Online traveling salesman problems with rejection options

In this article, we consider online versions of the traveling salesman problem on metric spaces for which requests to visit points are not mandatory. Associated with each request is a penalty (if rejected). Requests are revealed over time (at their release dates) to a server who must decide which requests to accept and serve in order to minimize a linear combination of the time to serve all accepted requests and the total penalties of all rejected requests. In the basic online version of the problem, a request can be accepted any time after its release date. In the real-time online version, a request must be accepted or rejected at the time of its release date. For the basic version, we provide a best possible 2-competitive online algorithm for the problem on a general metric space. For the real-time version, we first consider special metric spaces: on the nonnegative real line, we provide a best possible 2.5-competitive polynomial time online algorithm; on the real line, we prove a Ω(√ln n) lower bound of 2.64 on any competitive ratios and give a 3-competitive online algorithm. We then consider the case of a general metric space and prove a inline image lower bound on the competitive ratio of any online algorithms. Finally, among the restricted class of online algorithms with prior knowledge about the total number of requests n, we propose an asymptotically best possible O(√ln n)-competitive algorithm.United States. Office of Naval Research (Grant N00014-09-1-0326)United States. Office of Naval Research (Grant N00014-12-1-0033)United States. Air Force Office of Scientific Research (Grant FA9550-10-1-0437
Centroidal bases in graphs

We introduce the notion of a centroidal locating set of a graph $G$, that is,
a set $L$ of vertices such that all vertices in $G$ are uniquely determined by
their relative distances to the vertices of $L$. A centroidal locating set of
$G$ of minimum size is called a centroidal basis, and its size is the
centroidal dimension $CD(G)$. This notion, which is related to previous
concepts, gives a new way of identifying the vertices of a graph. The
centroidal dimension of a graph $G$ is lower- and upper-bounded by the metric
dimension and twice the location-domination number of $G$, respectively. The
latter two parameters are standard and well-studied notions in the field of
graph identification.
  We show that for any graph $G$ with $n$ vertices and maximum degree at
least~2, $(1+o(1))\frac{\ln n}{\ln\ln n}\leq CD(G) \leq n-1$. We discuss the
tightness of these bounds and in particular, we characterize the set of graphs
reaching the upper bound. We then show that for graphs in which every pair of
vertices is connected via a bounded number of paths,
$CD(G)=\Omega\left(\sqrt{|E(G)|}\right)$, the bound being tight for paths and
cycles. We finally investigate the computational complexity of determining
$CD(G)$ for an input graph $G$, showing that the problem is hard and cannot
even be approximated efficiently up to a factor of $o(\log n)$. We also give an
$O\left(\sqrt{n\ln n}\right)$-approximation algorithm
Centroidal bases in graphs

We introduce the notion of a centroidal locating set of a graph G, that is, a set L of vertices such that all vertices in G are uniquely determined by their relative distances to the vertices of L. A centroidal locating set of G of minimum size is called a centroidal basis, and its size is the centroidal dimension CD (G). This notion, which is related to previous concepts, gives a new way of identifying the vertices of a graph. The centroidal dimension of a graph G is lower- and upper-bounded by the metric dimension and twice the location-domination number of G, respectively. The latter two parameters are standard and well-studied notions in the field of graph identification. We show that for any graph G with n vertices and maximum degree at least 2, (1 + o(1)) In n/InInn <= CD <= (G) n - 1. We discuss the tightness of these bounds and in particular, we characterize the set of graphs reaching the upper bound. We then show that for graphs in which every pair of vertices is connected via a bounded number of paths, CD(G) = Omega(root vertical bar E(G)vertical bar) , the bound being tight for paths and cycles. We finally investigate the computational complexity of determining CD(G) for an input graph G, showing that the problem is hard and cannot even be approximated efficiently up to a factor of o(log n). We also give an O (root nInn)-approximation algorithm.Peer ReviewedPostprint (author’s final draft
The School Bus Routing and Scheduling Problem with Transfers

In this article, we study the school bus routing and scheduling problem with transfers arising in the field of nonperiodic public transportation systems. It deals with the transportation of pupils from home to their school in the morning taking the possibility that pupils may change buses into account. Allowing transfers has several consequences. On the one hand, it allows more flexibility in the bus network structure and can, therefore, help to reduce operating costs. On the other hand, transfers have an impact on the service level: the perceived service quality is lower due to the existence of transfers; however, at the same time, user ride times may be reduced and, thus, transfers may also have a positive impact on service quality. The main objective is the minimization of the total operating costs. We develop a heuristic solution framework to solve this problem and compare it with two solution concepts that do not consider transfers. The impact of transfers on the service level in terms of time loss (or user ride time) and the number of transfers is analyzed. Our results show that allowing transfers reduces total operating costs significantly while average and maximum user ride times are comparable to solutions without transfers. (authors' abstract
An incremental algorithm for uncapacitated facility location problem

We study the incremental facility location problem, wherein we are given an instance of the uncapacitated facility location problem (UFLP) and seek an incremental sequence of opening facilities and an incremental sequence of serving customers along with their fixed assignments to facilities open in the partial sequence. We say that a sequence has a competitive ratio of k, if the cost of serving the first ℓ customers in the sequence is at most k times the optimal solution for serving any ℓ customers for all possible values of ℓ. We provide an incremental framework that computes a sequence with a competitive ratio of at most eight and a worst-case instance that provides a lower bound of three for any incremental sequence. We also present the results of our computational experiments carried out on a set of benchmark instances for the UFLP. The problem has applications in multistage network planning
On the complexity of equal shortest path routing

International audienceIn telecommunication networks packets are carried from a source s to a destination t on a path determined by the underlying routing protocol. Most routing protocols belong to the class of shortest path routing protocols. In such protocols, the network operator assigns a length to each link. A packet going from s to t follows a shortest path according to these lengths. For better protection and efficiency, one wishes to use multiple (shortest) paths between two nodes. Therefore the routing protocol must determine how the traffic from s to t is distributed among the shortest paths. In the protocol called OSPF-ECMP (for Open Shortest Path First-Equal Cost Multiple Path) the traffic incoming at every node is uniformly balanced on all outgoing links that are on shortest paths. In that context, the operator task is to determine the " best " link lengths, toward a goal such as maximizing the network throughput for given link capacities. In this work, we show that the problem of maximizing even a single commodity flow for the OSPF-ECMP protocol cannot be approximated within any constant factor ratio. Besides this main theorem, we derive some positive results which include polynomial-time approximations and an exponential-time exact algorithm
Graph orientation and flows over time

Flows over time are used to model many real-world logistic and routing problems. The networks underlying such problems - streets, tracks, etc. - are inherently undirected and directions are only imposed on them to reduce the danger of colliding vehicles and similar problems. Thus, the question arises, what influence the orientation of the network has on the network flow over time problem that is being solved on the oriented network. In the literature, this is also referred to as the contraflow or lane reversal problem. We introduce and analyze the price of orientation: How much flow is lost in any orientation of the network if the time horizon remains fixed? We prove that there is always an orientation where we can still send one-third of the flow and this bound is tight. For the special case of networks with a single source or sink, this fraction is half, which is again tight. We present more results of similar flavor and also show nonapproximability results for finding the best orientation for single and multicommodity maximum flows over time
A characterization of irreducible infeasible subsystems in flow networks

Infeasible network flow problems with supplies and demands can be characterized via violated cut-inequalities of the classical Gale-Hoffman theorem. Written as a linear program, irreducible infeasible subsystems (IISs) provide a different means of infeasibility characterization. In this article, we answer a question left open in the literature by showing a one-to-one correspondence between IISs and Gale-Hoffman-inequalities in which one side of the cut has to be weakly connected. We also show that a single max-flow computation allows one to compute an IIS. Moreover, we prove that finding an IIS of minimal cardinality in this special case of flow networks is strongly NP-hard
Minimal disconnected cuts in planar graphs.

The problem of finding a disconnected cut in a graph is NP-hard in general but polynomial-time solvable on planar graphs. The problem of finding a minimal disconnected cut is also NP-hard but its computational complexity was not known for planar graphs. We show that it is polynomial-time solvable on 3-connected planar graphs but NP-hard for 2-connected planar graphs. Our technique for the first result is based on a structural characterization of minimal disconnected cuts in 3-connected  inline image-free-minor graphs and on solving a topological minor problem in the dual. In addition we show that the problem of finding a minimal connected cut of size at least 3 is NP-hard for 2-connected apex graphs. Finally, we relax the notion of minimality and prove that the problem of finding a so-called semi-minimal disconnected cut is still polynomial-time solvable on planar graphs