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Following the methods used by Derksen-Weyman in \cite{DW11} and Chindris in \cite{Chi08}, we use quiver theory to represent the generalized Littlewood-Richardson coefficients for the branching rule for the diagonal embedding of $\gl(n)$ as the dimension of a weight space of semi-invariants. Using this, we prove their saturation and investigate when they are nonzero. We also show that for certain partitions the associated stretched polynomials satisfy the same conjectures as single Littlewood-Richardson coefficients. We then provide a polytopal description of this multiplicity and show that its positivity may be computed in strongly polynomial time. Finally, we remark that similar results hold for certain other generalized Littlewood-Richardson coefficients.

We consider two single-machine scheduling problems in which two competing projects share one common resource. Each project has multiple interim assessments, and its own jobs are ordered completely. A tardy job incurs a tardiness penalty cost which can be avoided by compressing some jobs, which requires an additional cost. The performance measure of each project is the total tardiness penalty cost plus the total compression cost. The first problem minimizes the weighted sum of the performance measures of two projects, and the second problem minimizes the performance measure of one project with a constraint on that of the other. We show that the first problem is solvable in strongly polynomial time and the second is weakly NP-hard. Furthermore, we analyze how the computational complexity of each problem changes if the number of projects is more than two.

We consider the inverse optimal value problem on minimum spanning tree under unit $$l_{\infty }$$ norm. Given an edge weighted connected undirected network $$G=(V, E, {\varvec{w}})$$ and a spanning trees $$T^0$$ , we aim to modify the weights of the edges such that $$T^0$$ is the minimum spanning tree under the new weight vector whose weight is equal to a given value K and the modification cost under unit $$l_{\infty }$$ norm is minimized. We present a mathematical model of the problem. After analyzing the properties, we propose a sufficient and necessary condition for optimal solutions of the problem. Then we develop a strongly polynomial time algorithm with running time O(|V||E|). Finally, we give an example to demonstrate the algorithm.

International audience; Edmonds (1973) characterized the condition for the existence of a packing of spanning arborescences and also that of spanning branchings in a directed graph. Durand de Gevigney, Nguyen and Szigeti (2013) generalized the spanning arborescence packing problem to a matroid-based arborescence packing problem and gave a necessary and sufficient condition for the existence of a packing and a polynomial-time algorithm.In this paper, a generalization of this latter problem – the polymatroid-based arborescence packing problem – is considered. Two problem settings are formulated: an unsplittable version and a splittable version. The unsplittable version is shown to be strongly NP-complete. Whereas, the splittable version, which generalizes the capacitated version of the spanning arborescence packing problem, can be solved in strongly polynomial time. For convenience, we provide a strongly polynomial-time algorithm for the problem of the polymatroid-based capacitated packing of branchings for the splittable version.

J. De Loera & T. McAllister and K. D. Mulmuley & H. Narayanan & M. Sohoni independently proved that determining the vanishing of Littlewood–Richardson coefficients has strongly polynomial time computational complexity. Viewing these as Schubert calculus numbers, we prove the generalization to the Littlewood–Richardson polynomials that control equivariant cohomology of Grassmannians. We construct a polytope using the edge-labeled tableau rule of H. Thomas, A. Yong. Our proof then combines a saturation theorem of D. Anderson, E. Richmond, A. Yong, a reading order independence property, and E. Tardos’ algorithm for combinatorial linear programming.

Motivated by applications in evacuation planning, we consider a problem of optimizing flow with arc reversals in which the transit time depends on the orientation of the arc. In the considered problems, the transit time on an arc may change when it is reversed, contrary to the problems considered in the existing literature. Extending the existing idea of auxiliary network construction to allow asymmetric transit time on arcs, we present strongly polynomial time algorithms for solving single-source-single-sink maximum dynamic contraflow problem and quickest contraflow problem. The results are substantiated by a computational experiment in a Kathmandu road network. An algorithm to solve the corresponding earliest arrival contraflow problem with a pseudo-polynomial-time complexity is also presented. The partial contraflow approach for the corresponding problems has also been discussed.

© 2020 Wiley Periodicals LLC. In 2013, Orlin proved that the max flow problem could be solved in O(nm) time. His algorithm ran in O(nm + m1.94) time, which was the fastest for graphs with fewer than n1.06 arcs. If the graph was not sufficiently sparse, the fastest running time was an algorithm due to King, Rao, and Tarjan. We describe a new variant of the excess scaling algorithm for the max flow problem whose running time strictly dominates the running time of the algorithm by King et al. For graphs in which m = O(nlog n), the running time of our algorithm dominates that of King et al. by a factor of O(loglog n). Moreover, our algorithm achieves this improved performance without reliance on dynamic trees.

Let G be an n-node graph without two disjoint odd cycles. The algorithm of Artmann, Weismantel and Zenklusen (STOC’17) for bimodular integer programs can be used to find a maximum weight stable set in G in strongly polynomial time. Building on structural results characterizing sufficiently connected graphs without two disjoint odd cycles, we construct a size-\(O(n^2)\) extended formulation for the stable set polytope of G.

Applying game-theoretical tools for measuring the reliability of a network has become very common. The basic idea is very natural: analyzing an appropriately defined attacker–defender game might give rise to a relevant security metric. In this paper we consider a very natural set of games: the Defender chooses a path P between two given nodes and the Attacker chooses a network element a (that is, an edge or a node). In all cases, the Attacker has to pay a given cost of attack c(a); if, however, a is on P then he also gains a given profit of d(a). We determine the value of various versions of this game and show that the thus arising reliability metrics provide a generalization of weighted connectivity of graphs. We also prove that the values of the games and optimum mixed strategies for both players can be computed in strongly polynomial time.

This paper presents an $$O(n(m+n\log n)\log n)$$ time algorithm to solve the minimum cost tension problem, where n and m denote the number of nodes and number of arcs, respectively. The algorithm is inspired by Orlin (Oper Res 41:338–350, 1993) and improves upon the previous best strongly polynomial time of $$O(\max \{m^3n, m^2\log n(m+n\log n)\})$$ due to Ghiyasvand (J Comb Optim 34:203–217, 2017).

Let G = (V, E) be a simple graph and $\mathbf {w}\in \mathbb {Z}^{E}_{>0}$ assign each edge e ∈ E a positive integer weight w(e). A subset of E that intersects every triangle of G is called a triangle cover of (G, w), and its weight is the total weight of its edges. A collection of triangles in G (repetition allowed) is called a triangle packing of (G, w) if each edge e ∈ E appears in at most w(e) members of the collection. Let τ t (G, w) and ν t (G, w) denote the minimum weight of a triangle cover and the maximum cardinality of a triangle packing of (G, w), respectively. Generalizing Tuza’s conjecture for unit weight, Chapuy et al. conjectured that τ t (G, w)/ν t (G, w) ≤ 2 holds for every simple graph G and every $\mathbf {w}\in \mathbb {Z}^{E}_{>0}$ . In this paper, using a hypergraph approach, we design polynomial-time combinatorial algorithms for finding triangle covers of small weights. These algorithms imply new sufficient conditions for the conjecture of Chapuy et al. More precisely, given (G, w), suppose that all edges of G are covered by the set ${\mathscr {T}}_{G}$ consisting of edge sets of triangles in G. Let $|E|_{w}={\sum }_{e\in E}w(e)$ and $|{\mathscr {T}}_{G}|_{w}={\sum }_{\{e,f,g\}\in {\mathscr {T}}_{G}}w(e)w(f)w(g)$ denote the weighted numbers of edges and triangles in (G, w), respectively. We show that a triangle cover of (G, w) of weight at most 2ν t (G, w) can be found in strongly polynomial time if one of the following conditions is satisfied: (i) $\nu _{t}(G,\mathbf {w})/|{\mathscr {T}}_{G}|_{w}\ge \frac {1}{3}$ , (ii) $\nu _{t}(G,\mathbf {w})/|E|_{w}\ge \frac {1}{4}$ , (iii) $|E|_{w}/|{\mathscr {T}}_{G}|_{w}\ge 2$ .

We prove that primal-dual log-barrier interior point methods are not strongly polynomial, by constructing a family of linear programs with $3r+1$ inequalities in dimension $2r$ for which the number of iterations performed is in $\Omega(2^r)$. The total curvature of the central path of these linear programs is also exponential in $r$, disproving a continuous analogue of the Hirsch conjecture proposed by Deza, Terlaky and Zinchenko. Our method is to tropicalize the central path in linear programming. The tropical central path is the piecewise-linear limit of the central paths of parameterized families of classical linear programs viewed through logarithmic glasses. This allows us to provide combinatorial lower bounds for the number of iterations and the total curvature, in a general setting., Comment: This paper supersedes arXiv:1405.4161. 31 pages, 5 figures, 1 table

The quickest path problem consists of finding a path in a directed network to transmit a given amount $$\sigma $$ of items from a source node to a sink node with minimal transmission time, where the transmission time depends on the traversal times $$\tau $$ and the capacities u of the arcs. We suppose that there exist more than one quickest path and finds a quickest path with a special property. In this paper, first, some algorithms to find a quickest path with minimum capacity, minimum lead time, and min-max arc lead time are presented, which runs in $$O(r(m+n \log n))$$ and $$ O((r(m+n \log n))\log r') $$ time, where r and $$ r' $$ are the number of distinct capacity and lead time values and m and n are the number of arcs and nodes in the given network. Then, we suppose that values $$\sigma , \tau $$ and u change to $$\sigma ', \tau '$$ , and $$u'$$ . The purpose is to find a quickest path such that it has the minimum transmission time value among all quickest paths, by changing $$\sigma $$ to $$\sigma '$$ , $$\tau $$ to $$\tau '$$ , or u to $$u'$$ . We show that some of these problems are solved in strongly polynomial time and the others remain as open problems.

This paper addresses the solution of the capacitated minimum cost flow problem on a network containingn nodes and m arcs. Satisfying necessary and sufficient optimality conditions can be done on the residual network although it can be quite time consuming as testified by the minimum mean cycle-canceling algorithm (MMCC). We introduce a contracted network which exploits these conditions on a much smaller network. Since the construction of this contracted network is very flexible, we study its properties depending on the construction choice. A generic contraction algorithm is then produced around the contracted network. Interestingly enough, it turns out it encapsulates both the MMCC and primal network simplex algorithms as extreme cases. By guiding the solution using a particular expansion scheme, we are able to recuperate theoretical results from MMCC. As such, we obtain a strongly polynomial Contraction-Expansion algorithm which runs in O(m3n2) time. There is thus no improvement of the runtime complexity, yet the expansion scheme sticks to very practical observations of MMCCs behavior, namely that of phases and jumps on the optimality parameter. The solution time is ultimately significantly reduced, even more so as the size of the instance increases.

Let $G$ be an $n$-node graph without two disjoint odd cycles. The algorithm of Artmann, Weismantel and Zenklusen (STOC'17) for bimodular integer programs can be used to find a maximum weight stable set in $G$ in strongly polynomial time. Building on structural results characterizing sufficiently connected graphs without two disjoint odd cycles, we construct a size-$O(n^2)$ extended formulation for the stable set polytope of $G$., 19 pages, 3 figures

In this paper, a double-pivot simplex method is proposed. Two upper bounds of iteration numbers are derived. Applying one of the bounds to some special linear programming (LP) problems, such as LP with a totally unimodular matrix and Markov Decision Problem (MDP) with a fixed discount rate, indicates that the double-pivot simplex method solves these problems in a strongly polynomial time. A variant of Klee-Minty cube is used to show that the estimated bounds of the iteration numbers are very tight. Numerical test on three variants of Klee-Minty cubes is performed for the problems with sizes as big as $200$ constraints and $400$ variables. Dantzig's simplex method cannot handle Klee-Minty cube problem with $200$ constraints because it needs about $2^{200} \approx 10^{60}$ iterations. But the proposed algorithm performs extremely good for all three variants., 22 pages, 4 figures

Fair division is a fundamental problem in various multi-agent settings, where the goal is to divide a set of resources among agents in a fair manner. We study the case where m indivisible items need to be divided among n agents with additive valuations using the popular fairness notion of maximin share (MMS). An MMS allocation provides each agent a bundle worth at least her maximin share. While it is known that such an allocation need not exist, a series of work provided approximation algorithms for a 2/3-MMS allocation in which each agent receives a bundle worth at least 2/3 times her maximin share. More recently, Ghodsi et al. [EC'2018] showed the existence of a 3/4-MMS allocation and a PTAS to find a (3/4-\epsilon)-MMS allocation for an \epsilon > 0. Most of the previous works utilize intricate algorithms and require agents' approximate MMS values, which are computationally expensive to obtain. In this paper, we develop a new approach that gives a simple algorithm for showing the existence of a 3/4-MMS allocation. Furthermore, our approach is powerful enough to be easily extended in two directions: First, we get a strongly polynomial-time algorithm to find a 3/4-MMS allocation, where we do not need to approximate the MMS values at all. Second, we show that there always exists a (3/4 + 1/(12n))-MMS allocation, improving the best previous factor. This improves the approximation guarantee, most notably for small n. We note that 3/4 was the best factor known for n> 4., Comment: Fixed typos and added more details. A two-page abstract appeared in ACM EC 2020

The shortest path tree problem is one of the most studied problems in network optimization. Given a directed weighted graph, the aim is to find a shortest path from a given origin node to any other node of the graph. When any change occurs (i.e., the origin node is changed, some nodes/arcs are added/removed to/from the graph, the cost of a subset of arcs is increased/decreased), in order to determine a (still) optimal solution, two different strategies can be followed: a re-optimization algorithm is applied starting from the current optimal solution or a new optimal solution is built from scratch. Generally speaking, the Re-optimization Shortest Path Tree Problem (R-SPTP) consists in solving a sequence of shortest path problems, where the kth problem differs only slightly from the $$(k-1){th}$$ one, by exploiting the useful information available after each shortest path tree computation. In this paper, we propose an exact algorithm for the R-SPTP, in the case of origin node change. The proposed strategy is based on a dual approach, which adopts a strongly polynomial auction algorithm to extend the solution under construction. The approach is evaluated on a large set of test problems. The computational results underline that it is very promising and outperforms or at least is not worse than the solution approaches reported in the literature.

We identify a class of Linear Complementarity Problems (LCPs) that are solvable in strongly polynomial time by Lemke's Algorithm (Scheme 1) or by the Parametric Principal Pivoting Method (PPPM). This algorithmic feature for the class of problems under consideration here is attributable to the proper selection of the covering vector in Scheme 1 or the parametric direction vector in the PPPM which leads to solutions of limited and monotonically increasing support size; such solutions are sparse. These and other LCPs may very well have multiple solutions, many of which are unattainable by either algorithm and thus are said to be elusive. The initial conditions imposed on the new matrix class identified in Sect. 2 are subsequently relaxed in later sections.

In this paper, a new problem on a directed network is presented. Let D be a feasible network such that all arc capacities are equal to U. Given a t > 0, the network D with arc capacities U - t is called the t-network. The goal of the problem is to compute the largest t such that the t-network is feasible. First, we present a weakly polynomial time algorithm to solve this problem, which runs in O(log(nU)) maximum flow computations, where n is the number of nodes. Then, an O(m2n) time approach is presented, where m is the number of arcs. Both the weakly and strongly polynomial algorithms are inspired by McCormick and Ervolina (1994).

Locating source of diffusion in networks is crucial for controlling and preventing epidemic risks. It has been studied under various probabilistic models. In this paper, we study source location from a deterministic point of view by modeling it as the minimum cost doubly resolving set (DRS) problem, which is a strengthening of the well-known metric dimension problem.Let G be an undirected graph on n vertices, where each vertex has a nonnegative cost. A vertex subset S of G is a doubly resolving set (DRS) of G if for every pair of vertices u,v in G, there exist x,yS such that the difference of distances (in terms of number of edges) between u and x,y is not equal to the difference of distances between v and x,y. The minimum cost DRS problem consists of finding a DRS in G with minimum total cost. We establish (lnn) approximability of the minimum DRS problem on general graphs for both weighted and unweighted versions. This provides the first explicit lower and upper bounds on approximation for the minimum (cost) DRS, which are nearly tight. Moreover, we design the first known strongly polynomial time exact algorithms for the minimum cost DRS problem on general wheels and trees with additional constant k0 edges.

This paper considers transient total-cost MDPs with transition rates whose values may be greater than one, and average-cost MDPs satisfying the condition that the expected time to hit a certain state from any initial state and under any stationary policy is bounded above by a constant. Linear programming formulations for such MDPs are provided that are solvable in strongly polynomial time.

Rectilinear Steiner arborescences under linear delay constraints play an important role for buffering. We present exact algorithms for either minimizing the total length subject to delay constraints, or minimizing the total length plus the (weighted) absolute total negative slack. Our main theoretical contribution is the first minimum cost flow formulation for embedding Steiner arborescences at minimum length subject to delay constraints, resulting in the first strongly polynomial time algorithm for this subproblem. We use the minimum cost flow formulation to quickly compute lower bounds in a branch-&-bound algorithm for optimum Steiner arborescences. We demonstrate the benefit of our new algorithm experimentally.

Partitioning a sequence of length n into k coherent segments ( Seg ) is one of the classic optimization problems. As long as the optimization criterion is additive, Seg can be solved exactly in O ( n 2 k ) time using a classic dynamic program. Due to the quadratic term, computing the exact segmentation may be too expensive for long sequences, which has led to development of approximate solutions. We consider an existing estimation scheme that computes ( 1 + ϵ ) approximation in polylogarithmic time. We augment this algorithm, making it strongly polynomial. We do this by first solving a slightly different segmentation problem ( MaxSeg ), where the quality of the segmentation is the maximum penalty of an individual segment. By using this solution to initialize the estimation scheme, we are able to obtain a strongly polynomial algorithm. In addition, we consider a cumulative version of Seg , where we are asked to discover the optimal segmentation for each prefix of the input sequence. We propose a strongly polynomial algorithm that yields ( 1 + ϵ ) approximation in O ( n k 2 / ϵ ) time. Finally, we consider a cumulative version of MaxSeg , and show that we can solve the problem in O ( n k log ⁡ k ) time.

We present a new class of polynomial-time algorithms for submodular function minimization (SFM), as well as a unified framework to obtain strongly polynomial SFM algorithms. Our algorithms are based on simple iterative methods for the minimum-norm problem, such as the conditional gradient and Fujishige-Wolfe algorithms. We exhibit two techniques to turn simple iterative methods into polynomial-time algorithms. Firstly, we adapt the geometric rescaling technique, which has recently gained attention in linear programming, to SFM and obtain a weakly polynomial bound $O(({n}^4\cdot \textrm{EO} + {n}^5)\log ({n} L))$. Secondly, we exhibit a general combinatorial black-box approach to turn $\varepsilon L$-approximate SFM oracles into strongly polynomial exact SFM algorithms. This framework can be applied to a wide range of combinatorial and continuous algorithms, including pseudo-polynomial ones. In particular, we can obtain strongly polynomial algorithms by a repeated application of the conditional gradient or of the Fujishige-Wolfe algorithm. Combined with the geometric rescaling technique, the black-box approach provides an $O(({n}^5\cdot \textrm{EO} +{n}^6)\log^2{n})$ algorithm. Finally, we show that one of the techniques we develop in the paper can also be combined with the cutting-plane method of Lee, Sidford, and Wong \cite{LSW}, yielding a simplified variant of their $O(n^3 \log^2 n \cdot \textrm{EO} + n^4\log^{O(1)} n)$ algorithm.

In this paper we consider the single machine scheduling problem with one non-availability interval to minimize the maximum lateness where jobs have positive tails. Two cases are considered. In the first one, the non-availability interval is due to the machine maintenance. In the second case, the non-availibility interval is related to the operator who is organizing the execution of jobs on the machine. The contribution of this paper consists in an improved FPTAS for the maintenance non-availability interval case and its extension to the operator non-availability interval case. The two FPTASs are strongly polynomial and outperform the recent ones by Kacem, Kellerer and Seifaddini presented in [12].

We consider the problem of optimally compressing and caching data across a communication network. Given the data generated at edge nodes and a routing path, our goal is to determine the optimal data compression ratios and caching decisions across the network in order to minimize average latency, which can be shown to be equivalent to maximizing the compression and caching gain under an energy consumption constraint. We show that this problem is NP-hard in general and the hardness is caused by the caching decision subproblem, while the compression sub-problem is polynomial-time solvable. We then propose an approximation algorithm that achieves a $(1-1/e)$-approximation solution to the optimum in strongly polynomial time. We show that our proposed algorithm achieve the near-optimal performance in synthetic-based evaluations. In this paper, we consider a tree-structured network as an illustrative example, but our results easily extend to general network topology at the expense of more complicated notations.

Suppose that we are given a graph whose each vertex is either a supply vertex or a demand vertex and is assigned a nonnegative integer supply or demand value. We consider partitioning G into connected components by removing edges from G so that each connected component has exactly one supply vertex and there exists a flow in each connected component satisfying the supply/demand constraints. The problem that determines the existence of such a partition is called the partition problem. Ito et al. (2005) showed that the partition problem is \(\mathcal {NP}\)-complete in general and it can be solved in linear time if the given graph is a tree. When the graph does not have such a partition, we scale the demand values uniformly by scale factor r so that the obtained graph has a desired partition. The maximum supply rate problem is the problem that finds the maximum value of such r. Whereas the maximum supply rate problem is \(\mathcal {NP}\)-hard in general in the same way as the partition problem, Morishita and Nishizeki (2015) gave a weakly polynomial-time algorithm for the problem on trees.

We consider a version of parallel machine scheduling with rejection. An instance of the problem is given by m identical parallel machines and a set of n independent jobs, with each job having a processing time and a penalty. A job may be accepted to be processed or be rejected at its penalty. The objective of the problem is to partition the set of jobs into two subsets, the subset of accepted and the subset of rejected jobs, and to schedule the set of accepted jobs such that the makespan is minimized under the constraint that the total penalty of the rejected jobs is no more than a given bound. In this paper, we present a 2-approximation algorithm within strongly polynomial time for the problem. We also present a polynomial time approximation scheme whose running time is O ( n m O ( 1 � 2 ) + mn 2 ) for the problem. Moreover, for the case where the number of machines is a fixed constant m, our results lead to a fully polynomial time approximation scheme for the problem. Our result is fairly good in the sense that in a reasonable size of jobs, our FPTAS improves previous best running time from O ( n m + 2 / � m ) to O ( 1 / � 2 m + 3 + mn 2 ) .

We prove that the simplex method with the highest gain/most-negative-reduced cost pivoting rule converges in strongly polynomial time for deterministic Markov decision processes (MDPs) regardless of the discount factor. For a deterministic MDP with n states and m actions, we prove the simplex method runs in O(n^3m^2log^2 n) iterations if the discount factor is uniform and O(n^5m^3log^2 n) iterations if each action has a distinct discount factor. Previously the simplex method was known to run in polynomial time only for discounted MDPs where the discount was bounded away from 1 [Ye11]. Unlike in the discounted case, the algorithm does not greedily converge to the optimum, and we require a more complex measure of progress. We identify a set of layers in which the values of primal variables must lie and show that the simplex method always makes progress optimizing one layer, and when the upper layer is updated the algorithm makes a substantial amount of progress. In the case of nonuniform discounts, we define a polynomial number of "milestone" policies and we prove that, while the objective function may not improve substantially overall, the value of at least one dual variable is always making progress towards some milestone, and the algorithm will reach the next milestone in a polynomial number of steps., Comment: Minor typo fixes and improvements over version 1. Appeared in SODA 2013

We deal with a recently proposed method of Chubanov [A polynomial projection algorithm for linear feasibility problems. Math. Program. 153 (2015), pp. 687–713] for solving linear homogeneous systems with positive variables. Some improvements of Chubanov's method and its analysis are presented. We propose a new and simple cut criterion and show that the cuts defined by the new criterion are at least as sharp as in [1]. The new cut criterion reduces the iteration bound for his Basic Procedure by a factor 5, without changing the order of its strongly polynomial complexity. Our Modified Main Algorithm is in essence the same as Chubanov's Main Algorithm, except that it uses our Modified Basic Procedure as a subroutine. It is shown that it has (Formula presented.) time complexity, just as in [1]. Some promising computational results are presented, in comparison with the optimization package Gurobi.

In this article, we study compact Mixed-Integer Programming (MIP) models for the Resource-Constrained Project Scheduling Problem (RCPSP). Compared to the classical time-indexed formulation, the size of compact models is strongly polynomial in the number of jobs. In addition to two compact models from the literature, we propose a new compact model. We can show that all three compact models are equivalent by successive linear transformations. For their LP-relaxations, however, we state a full inclusion hierarchy where our new model dominates the previous models in terms of polyhedral strength. Moreover, we reveal a polyhedral relationship to the common time-indexed model. Furthermore, a general class of valid cutting planes for the compact models is introduced and finally all models are evaluated by computational experiments.

Braess's paradox states that removing a part of a network may improve the players' latency at equilibrium. In this work, we study the approximability of the best subnetwork problem for the class of random $${\mathcal {G}}_{n,p}$$Gn,p instances proven prone to Braess's paradox by Valiant and Roughgarden RSA '10 (Random Struct Algorithms 37(4):495---515, 2010), Chung and Young WINE '10 (LNCS 6484:194---208, 2010) and Chung et al. RSA '12 (Random Struct Algorithms 41(4):451---468, 2012). Our main contribution is a polynomial-time approximation-preserving reduction of the best subnetwork problem for such instances to the corresponding problem in a simplified network where all neighbors of source s and destination t are directly connected by 0 latency edges. Building on this, we consider two cases, either when the total rate r is sufficiently low, or, when r is sufficiently high. In the first case of low$$r= O(n_{+})$$r=O(n+), here $$n_{+}$$n+ is the maximum degree of $$\{s, t\}$${s,t}, we obtain an approximation scheme that for any constant $$\varepsilon > 0$$ź>0 and with high probability, computes a subnetwork and an $$\varepsilon $$ź-Nash flow with maximum latency at most $$(1+\varepsilon )L^*+ \varepsilon $$(1+ź)Lź+ź, where $$L^*$$Lź is the equilibrium latency of the best subnetwork. Our approximation scheme runs in polynomial time if the random network has average degree $$O(\mathrm {poly}(\ln n))$$O(poly(lnn)) and the traffic rate is $$O(\mathrm {poly}(\ln \ln n))$$O(poly(lnlnn)), and in quasipolynomial time for average degrees up to o(n) and traffic rates of $$O(\mathrm {poly}(\ln n))$$O(poly(lnn)). Finally, in the second case of high$$r= {\varOmega }(n_{+})$$r=Ω(n+), we compute in strongly polynomial time a subnetwork and an $$\varepsilon $$ź-Nash flow with maximum latency at most $$(1+2\varepsilon + o(1))L^*$$(1+2ź+o(1))Lź.

We present a new strongly polynomial algorithm for generalized flow maximization that is significantly simpler and faster than the previous strongly polynomial algorithm [34]. For the uncapacitated problem formulation, the complexity bound O ( mn ( m + n log n )log ( n 2 / m )) improves on the previous estimate by almost a factor O ( n 2 ). Even for small numerical parameter values, our running time bound is comparable to the best weakly polynomial algorithms. The key new technical idea is relaxing the primal feasibility conditions. This allows us to work almost exclusively with integral flows, in contrast to all previous algorithms for the problem.

We present a market-based approach to the Air Traffic Flow Management (ATFM) problem. The goods in our market are delays and buyers are airline companies; the latter pay money to the Federal Aviation Administration (FAA) to buy away the desired amount of delay on a per flight basis. We give a notion of equilibrium for this market and an LP whose every optimal solution gives an equilibrium allocation of flights to landing slots as well as equilibrium prices for the landing slots. Via a reduction to matching, we show that this equilibrium can be computed combinatorially in strongly polynomial time. Moreover, there is a special set of equilibrium prices, which can be computed easily, that is identical to the VCG solution, and therefore the market is incentive compatible in dominant strategy.

A strongly polynomial-time algorithm is proposed for the strict homogeneous linear-inequality feasibility problem in the positive orthant, that is, to obtain \(x\in \mathbb {R}^n\), such that \(Ax > 0\), \(x> 0\), for an \(m\times n\) matrix \(A\), \(m\ge n\). This algorithm requires \(O(p)\) iterations and \(O(m^2(n+p))\) arithmetical operations to ensure that the distance between the solution and the iteration is \(10^{-p}\). No matrix inversion is needed. An extension to the non-homogeneous linear feasibility problem is presented.

The hypergraph k -cut problem is the problem of finding a minimum capacity set of hyperedges whose removal divides a given hypergraph into at least k connected components. We present an algorithm for this problem, that runs in strongly polynomial time if both k and the maximum size of the hyperedges are constants. Our algorithm extends the algorithm proposed by Thorup (2008) for computing minimum k -cuts of graphs from greedy packings of spanning trees.