Popularity is an approach in mechanism design to find fair structures in a graph, based on the votes of the nodes. Popular matchings are the relaxation of stable matchings: given a graph with strict preferences on the neighbors of the nodes, a matching is popular if there is no other matching such that the number of nodes preferring is more than those preferring . This paper considers the popularity testing problem, when the task is to decide whether a given matching is popular or not. Previous algorithms applied reductions to maximum weight matchings. We give a new algorithm for testing popularity by reducing the problem to maximum matching testing, thus attaining a linear running time . Linear programming-based characterization of popularity is often applied for proving the popularity of a certain matching. As a consequence of our algorithm we derive a more structured dual witness than previous ones. Based on this result we give a combinatorial characterization of fractional popular matchings, which is a special class of popular matchings.

Survivable routing is crucial in backbone networks to ensure connectivity, even during failures. During network design, groups of network elements prone to potential failure events are identified. These groups are referred to asShared Risk Link Groups(SRLGs). When these SRLGs consist of a set of links intersected by a connected region of the plane, they are termed regional-SRLGs. A recent study has presented a polynomial-time algorithm for finding amaximum number of regional-SRLG-disjoint pathsbetween two given nodes in a planar topology, where the paths are node-disjoint. However, existing algorithms for this problem are not practical due to their runtime and implementation complexities. This paper investigates a more general model in two aspects. First, instead of node-disjointness, we search for non-crossing regional-SRLG-disjoint paths. Second, we show how the algorithm can be extended to solve problems in directed networks. It introduces an efficient and easily implementable algorithmic framework, leveraging an arbitrarily chosen shortest path finding subroutine for graphs with possibly negative weights. Depending on the subroutine chosen, the framework either improves the previous worst-case runtime complexity or can solve the problem with high probability (w.h.p.) in near-linear expected time. The proposed framework enables the first additive approximation for a more generalNP-hard version of the problem, where the objective is to find the maximum number of regional-SRLG-disjoint paths. We validate our findings through extensive simulations.

Survivable routing is crucial in backbone networks to ensure connectivity, even during failures. At network design, groups of network elements prone to potential failure events are identified. These groups are referred to as Shared Risk Link Groups (SRLGs), and if they are a set of links intersected by a connected region of the plane, we call them regional-SRLGs. A recent study has presented a polynomial-time algorithm for finding a maximum number of regional-SRLG-disjoint paths between two given nodes in a planar topology, with the paths being nodedisjoint. However, existing algorithms for this problem are not practical due to their runtime and implementation complexities.This paper investigates a more general model, the maximum number of non-crossing, regional-SRLG-disjoint paths problem. It introduces an efficient and easily implementable algorithmic framework, leveraging an arbitrarily chosen shortest path finding subroutine for graphs with possibly negative weights. Depending on the subroutine chosen, the framework improves the previous worst-case runtime complexity, or can solve the problem w.h.p. in near-linear expected time. The proposed framework enables the first additive approximation for a more general ${\mathcal{N}}{\mathcal{P}}$ -hard version of the problem, where the objective is to find the maximum number of regional-SRLG-disjoint paths. We validate our findings through extensive simulations.

In fair division problems, we are given a set S of m items and a set N of n agents with individual preferences, and the goal is to find an allocation of items among agents so that each agent finds the allocation fair. There are several established fairness concepts and envy-freeness is one of the most extensively studied ones. However envy-free allocations do not always exist when items are indivisible and this has motivated relaxations of envy-freeness: envy-freeness up to one item (EF1) and envy-freeness up to any item (EFX) are two well-studied relaxations. We consider the problem of finding EF1 and EFX allocations for utility functions that are not necessarily monotone, and propose four possible extensions of different strength to this setting. In particular, we present a polynomial time algorithm for finding an EF1 allocation for two agents with arbitrary utility functions. An example is given showing that EFX allocations need not exist for two agents with non-monotone, non-additive, identical utility functions. However, when all agents have monotone (not necessarily additive) identical utility functions, we give a pseudo-polynomial time algorithm that always finds an EFX allocation of chores. As a step toward understanding the general case, we discuss two subclasses of utility functions: Boolean utilities that are {0,+1}-valued functions, and negative Boolean utilities that are {0,−1}-valued functions. For the latter, we give a polynomial time algorithm that finds an EFX allocation when the utility functions are identical.

P4 is a widely used Domain-specific Language for Programmable Data Planes. A critical step in P4 compilation is finding a feasible and efficient mapping of the high-level P4 source code constructs to the physical resources exposed by the underlying hardware, while meeting data and control flow dependencies in the program. In this paper, we take a new look at the algorithmic aspects of this problem, with the motivation to understand the fundamental theoretical limits and obtain better P4 pipeline embeddings, and to speed up practical P4 compilation times for RMT and dRMT target architectures. We report mixed results: we find that P4 compilation is computationally hard even in a severely relaxed formulation, and there is no polynomial-time approximation of arbitrary precision (unless$\mathcal {P}$=$\mathcal {N}$$\mathcal {P}$), while the good news is that, despite its inherent complexity, P4 compilation is approximable in linear time with a small constant bound even for the most complex, nearly real-life models.

Abstract. Dynamic pricing schemes were introduced as an alternative to posted-price mechanisms. In contrast to static models, the dynamic setting allows us to update the prices between buyer-arrivals based on the remaining sets of items and buyers, and so it is capable of maximizing social welfare without the need for a central coordinator. In this paper, we study the existence of optimal dynamic pricing schemes in combinatorial markets. In particular, we concentrate on multidemand valuations, a natural extension of unit-demand valuations. The proposed approach is based on computing an optimal dual solution of the maximum social welfare problem with distinguished structural properties. Our contribution is twofold. By relying on an optimal dual solution, we show the existence of optimal dynamic prices in unit-demand markets and in multidemand markets up to three buyers, thus giving new interpretations of results of Cohen-Addad et al. [ Proceedings of the ACM Conference on Economics and Computation, 2016, pp. 383–400] and Berger, Eden, and Feldman [ Proceedings of the International Conference on Web and Internet Economics, Springer, 2020, pp. 206–219], respectively. Furthermore, we provide an optimal dynamic pricing scheme for bidemand valuations with an arbitrary number of buyers. In all cases, our proofs also provide efficient algorithms for determining the optimal dynamic prices.

The current best practice in survivable routing is to compute link or node disjoint paths in the network topology graph. It can protect single-point failures; however, several failure events may cause the interruption of multiple network elements. The set of network elements subject to potential failure events is called Shared Risk Link Group (SRLG), identified during network planning. Unfortunately, for any given list of SRLGs, finding two paths that can survive a single SRLG failure is NP-Complete. In this paper, we provide a polynomial-time SRLG-disjoint routing algorithm for planar network topologies and a large set of SRLGs. Namely, we focus on regional failures, where the failed network elements must not be far from each other. We use a flexible definition of regional failure, where the only restrictions are that i) the topology is a planar graph, ii) each SRLG forms a set of connected edges in the dual of the planar graph, and iii) for each node$v$, the links incident to$v$are part of an SRLG. The proposed algorithm is based on a max-min theorem. Through extensive simulations, we show that the algorithm scales well with the network size, and one of the paths returned by the algorithm is only 4% longer than the shortest path on average.

The current best practice in survivable routing is to compute link or node disjoint paths in the network topology graph. It can protect single-point failures; however, several failure events may cause the interruption of multiple network elements. The set of network elements subject to potential failure events is called Shared Risk Link Group (SRLG), identified during network planning. Unfortunately, for any given list of SRLGs, finding two paths that can survive a single SRLG failure is NP-Complete. In this paper, we provide a polynomial-time SRLG-disjoint routing algorithm for planar network topologies and a large set of SRLGs. Namely, we focus on regional failures, where the failed network elements must not be far from each other. We use a flexible definition of regional failure, where the only restriction is that the topology is a planar graph, and the SRLGs form a set of connected edges in the dual of the planar graph. The proposed algorithm is based on a max-min theorem. Through extensive simulations, we show that the algorithm scales well with the network size, and one of the paths returned by the algorithm is only 4% longer than the shortest path on average.

Many recent studies shed light on the vulnerability of networks against large-scale natural disasters. The corresponding network failures, called regional failures, are manifested at failing multiple network elements that are physically close to each other. The recovery mechanisms of current backbone networks protect failures listed as Shared Risk Link Groups (SRLGs). We aim to design an algorithm for the routing engines, which can generate a reasonable list of SRLGs based on the limited geometric information available. As a first step towards this direction, in this paper, we propose a limited geographic information failure model for the network topology that enables efficient algorithms to compute the set of links that are expected to be close to each other. More precisely, we work with (1) relative node positions without knowing the real distances, (2) an area in the map defines the route of each physical cable, and (3) a regional failure is a circular disk with k=0,1, ... nodes in its interior. We describe an efficient algorithm for listing SRLGs based on our limited geographic information failure model and show that under realistic assumptions, the obtained list of SRLGs is short, having approximately 1.2 n and 2.2n elements for k=0 and k=1, respectively, where n is the number of nodes of the network.

Generalized diversity coding is a promising proactive recovery scheme against single edge failures for unicast connections in transport networks. At the source node, the user data is split into two parts, and their bitwise XOR is computed as a third redundancy sub-flow. In order to guarantee instantaneous failure recovery without costly node upgrades, the network must ensure that any two of the three sub-flows reach the destination node in case of a single edge failure only by allowing flow duplication or merging identical flows, and avoiding any coding operation in the core network. In this paper, we investigate the corresponding routing problem to calculate capacity-efficient routes for these sub-flows. We propose a polynomial-time algorithm for topologies without capacity constraints on the links and without capability limitations of the nodes. We show that with node limitations the presented algorithm (as well as a minimum cost disjoint path-pair) provides a 4/3-approximation for the routing problem. Furthermore, we formulate an integer linear program to provide a minimum cost solution with arbitrary constraints in general graphs and we propose a polynomial-time algorithm in directed acyclic graphs. Our simulation results suggest that with upgrading only a small set of core network nodes with flow duplication and merging capabilities most of the benefits of generalized diversity coding can be achieved.

Nowadays, a majority of the Internet service providers are either piloting or migrating to software-defined networking (SDN) in their networks. In an SDN architecture a central network controller has a top-down view of the network and can directly configure each of their physical switches. It opens up several fundamental unsolved challenges, such as deploying efficient multipath routing that can provide disjoint end-to-end paths, each one satisfying specific operational goals (e.g., shortest possible), without overwhelming the data plane with a prohibitive amount of forwarding state. In this paper, we study the problem of finding a pair of shortest (node- or edge-) disjoint paths that can be represented by only two forwarding table entries per destination. Building on prior work on minimum length redundant trees, we show that the complexity of the underlying mathematical problem is NP-complete and we present fast heuristic algorithms. By extensive simulations, we find that it is possible to very closely attain the absolute optimal path length with our algorithms (the gap is just 1%-5%), eventually opening the door for wide-scale multipath routing deployments. Finally, we show that even if a primary tree is already given it remains NP-complete to find a minimum length secondary tree concerning this primary tree.

In this article we give network coding algorithms for multi‐layered video streaming. The problem is motivated by video broadcasting in a communication network to users with varying demands. We give a polynomial time algorithm for deciding feasibility for the case of two layers, and show that the problem becomes NP‐hard if the task is to maximize the number of satisfied demands. For the case of three layers we also show NP‐hardness of the problem. Finally, we propose a heuristic for three layers and give experimental comparison with previous approaches. © 2017 Wiley Periodicals, Inc. NETWORKS, Vol. 71(1), 51–59 2018

In this paper, we propose a new proactive recovery scheme against single edge failures for unicast connections in transport networks. The new scheme is a generalization of diversity coding where the source data AB are split into two parts A and B and three data flows A, B, and their exclusive OR (XOR) A⊕B are sent along the network between the source and the destination node of the connection. By ensuring that two data flows out of the three always operate even if a single edge fails, the source data can be instantaneously recovered at the destination node. In contrast with diversity coding, we do not require the three data flows to be routed along three disjoint paths; however, in our scheme, a data flow is allowed to split into two parallel segments and later merge back. Thus, our generalized diversity coding (GDC) scheme can be used in sparse but still two-connected network topologies. Our proof improves an earlier result of network coding, by using purely graph theoretical tool set instead of algebraic argument. In particular, we show that when the source data are divided into two parts, robust intra-session network coding against single edge failures is always possible without any in-network algebraic operation. We present linear-time robust code construction algorithms for this practical special case in minimal coding graphs. We further characterize this question, and show that by increasing the number of edge failures and source data parts, we lose these desired properties.

A fundamental unsolved challenge in multipath routing is to provide disjoint end-to-end paths, each one satisfying certain operational goals (e.g., shortest possible), without overwhelming the data plane with prohibitive amount of forwarding state. In this paper, we study the problem of finding a pair of shortest disjoint paths that can be represented by only two forwarding table entries per destination. Building on prior work on minimum length redundant trees, we show that the underlying mathematical problem is NP-complete and we present heuristic algorithms that improve the known complexity bounds from cubic to the order of a single shortest path search. Finally, by extensive simulations we find that it is possible to very closely attain the absolute optimal path length with our algorithms (the gap is just 1 -- 5%), eventually opening the door for wide-scale multipath routing deployments.

Survivable routing methods have been thoroughly investigated in the past decades in transport networks. However, the proposed approaches suffered either from slow recovery time, poor bandwidth utilization, high computational or operational complexity, and could not really provide an alternative to the widely deployed single edge failure resilient dedicated 1 + 1 protection approach. Diversity coding is a candidate to overcome these difficulties with a relatively simple technique: dividing the connection data into two parts, and adding some redundancy at the source node. However, a missing link to make diversity coding a real alternative to 1+1 in transport networks is finding its minimum cost survivable routing, even in sparse topologies, where previous approaches may fail. In this paper we propose a polynomial-time algorithm with O(|V||E| log |V|) complexity for this routing problem. On the other hand, we show that the same routing problem turns to be NP-hard as soon as we limit the forwarding capabilities of some nodes and the capacities of some links of the network.