Matroid intersection is a classical optimization problem where, given two matroids over the same ground set, the goal is to find the largest common independent set. In this paper, we show that there exists a certain “sparsifer”: a subset of elements, of size O(|Sopt| · 1/ɛ), where Sopt denotes the optimal solution, that is guaranteed to contain a 3/2 + ɛ approximation, while guaranteeing certain robustness properties. We call such a small subset a Density Constrained Subset (DCS), which is inspired by the Edge-Degree Constrained, Subgraph, (EDCS) [Bernstein and Stein, 2015], originally designed for the maximum cardinality matching problem in a graph. Our proof is constructive and hinges on a greedy decomposition of matroids, which we call the density-based decomposition. We show that this sparsifier has certain robustness properties that can be used in one-way communication and random-order streaming models.

Abstract We devise the first constant-factor approximation algorithm for finding an integral multi-commodity flow of maximum total value for instances where the supply graph together with the demand edges can be embedded on an orientable surface of bounded genus. This extends recent results for planar instances. Our techniques include an uncrossing algorithm, which is significantly more difficult than in the planar case, a partition of the cycles in the support of an LP solution into free homotopy classes, and a new rounding procedure for freely homotopic non-separating cycles.

Abstract. We consider the problem of optimizing a coverage function under an [Formula: see text]-matchoid of rank [Formula: see text]. We design fixed-parameter algorithms as well as streaming algorithms to compute an exact solution. Unlike previous work that presumes linear representativity of matroids, we consider the general oracle model. For the special case where the coverage function is linear, we give a deterministic fixed-parameter algorithm parameterized by [Formula: see text] and [Formula: see text]. This result, combined with the lower bounds of Lovasz [ Algebraic Methods in Graph Theory, Vol. II (Colloquium Szeged 1978 ), North-Holland, Amsterdam, 1981, pp. 495–517] and Jensen and Korte [ SIAM J. Comput., 11 (1982), pp. 184–190], demonstrates a separation between the [Formula: see text]-matchoid and the matroid [Formula: see text]-parity problems in the setting of fixed-parameter tractability. For a general coverage function, we give both deterministic and randomized fixed-parameter algorithms, parameterized by [Formula: see text] and [Formula: see text], where [Formula: see text] is the number of points covered in an optimal solution. The resulting algorithms can be directly translated into streaming algorithms. For unweighted coverage functions, we show that we can find an exact solution even when the function is given in the form of a value oracle (and so we do not have access to an explicit representation of the set system). Our result can be implemented in the streaming setting and stores a number of elements depending only on [Formula: see text] and [Formula: see text] but is completely independent of the total size [Formula: see text] of the ground set. This shows that it is possible to circumvent the recent space lower bound of Feldman et al. [ Proceedings of STOC, 2020, pp. 1363–1374] by parameterizing the solution value. This result, combined with existing lower bounds, also provides a new separation between the space and time complexity of maximizing an arbitrary submodular function and a coverage function in the value oracle model.

We consider the maximum weight $b$-matching problem in the random-order semi-streaming model. Assuming all weights are small integers drawn from $[1,W]$, we present a $2 - \frac{1}{2W} + \varepsilon$ approximation algorithm, using a memory of $O(\max(|M_G|, n) \cdot poly(\log(m),W,1/\varepsilon))$, where $|M_G|$ denotes the cardinality of the optimal matching. Our result generalizes that of Bernstein [Bernstein, 2015], which achieves a $3/2 + \varepsilon$ approximation for the maximum cardinality simple matching. When $W$ is small, our result also improves upon that of Gamlath et al. [Gamlath et al., 2019], which obtains a $2 - δ$ approximation (for some small constant $δ\sim 10^{-17}$) for the maximum weight simple matching. In particular, for the weighted $b$-matching problem, ours is the first result beating the approximation ratio of $2$. Our technique hinges on a generalized weighted version of edge-degree constrained subgraphs, originally developed by Bernstein and Stein [Bernstein and Stein, 2015]. Such a subgraph has bounded vertex degree (hence uses only a small number of edges), and can be easily computed. The fact that it contains a $2 - \frac{1}{2W} + \varepsilon$ approximation of the maximum weight matching is proved using the classical Kőnig-Egerváry's duality theorem.

In the k-edge-connected spanning subgraph (kECSS) problem, our goal is to compute a minimum-cost sub-network that is resilient against up to k link failures: Given an n-node m-edge graph with a cost function on the edges, our goal is to compute a minimum-cost k-edge-connected spanning subgraph. This NP-hard problem generalizes the minimum spanning tree problem and is the "uniform case" of a much broader class of survival network design problems (SNDP). A factor of two has remained the best approximation ratio for polynomial-time algorithms for the whole class of SNDP, even for a special case of 2ECSS. The fastest 2-approximation algorithm is however rather slow, taking O(mn k) time [Khuller, Vishkin, STOC'92]. A faster time complexity of O(n²) can be obtained, but with a higher approximation guarantee of (2k-1) [Gabow, Goemans, Williamson, IPCO'93]. Our main contribution is an algorithm that (1+ε)-approximates the optimal fractional solution in Õ(m/ε²) time (independent of k), which can be turned into a (2+ε) approximation algorithm that runs in time Õ(m/(ε²) + {k²n^{1.5}}/ε²) for (integral) kECSS; this improves the running time of the aforementioned results while keeping the approximation ratio arbitrarily close to a factor of two.

Maximizing a monotone submodular function under various constraints is a classical and intensively studied problem. However, in the single-pass streaming model, where the elements arrive one by one and an algorithm can store only a small fraction of input elements, there is large gap in our knowledge, even though several approximation algorithms have been proposed in the literature. In this work, we present the first lower bound on the approximation ratios for cardinality and matroid constraints that beat $1-\frac{1}{e}$ in the single-pass streaming model. Let $n$ be the number of elements in the stream. Then, we prove that any (randomized) streaming algorithm for a cardinality constraint with approximation ratio $2-\sqrt{2}+\varepsilon$ requires $\Omega(\frac{n}{K^2})$ space for any $\varepsilon>0$, where $K$ is the size limit of the output set. We also prove that any (randomized) streaming algorithm for a (partition) matroid constraint with approximation ratio $\frac{K}{2K-1}+\varepsilon$ requires $\Omega(\frac{n}{K^2})$ space for any $\varepsilon>0$, where $K$ is the rank of the given matroid. In addition, we give streaming algorithms that assume access to the objective function via a weak oracle that can only be used to evaluate function values on feasible sets. Specifically, we show weak-oracle streaming algorithms for cardinality and matroid constraints with approximation ratios $\frac{K}{2K-1}$ and $\frac{1}{2}$, respectively, whose space complexity is exponential in $K$ but is independent of $n$. The former one exactly matches the known inapproximability result for a cardinality constraint in the weak oracle model. The latter one almost matches our lower bound of $\frac{K}{2K-1}$ for a matroid constraint, which almost settles the approximation ratio for a matroid constraint that can be obtained by a streaming algorithm whose space complexity is independent of $n$.

We consider the problem of maximizing a submodular function under the b-matching constraint, in the semi-streaming model. Our main results can be summarized as follows. - When the function is linear, i.e. for the maximum weight b-matching problem, we obtain a 2+ε approximation. This improves the previous best bound of 3+ε [Roie Levin and David Wajc, 2021]. - When the function is a non-negative monotone submodular function, we obtain a 3 + 2 √2 ≈ 5.828 approximation. This matches the currently best ratio [Roie Levin and David Wajc, 2021]. - When the function is a non-negative non-monotone submodular function, we obtain a 4 + 2 √3 ≈ 7.464 approximation. This ratio is also achieved in [Roie Levin and David Wajc, 2021], but only under the simple matching constraint, while we can deal with the more general b-matching constraint. We also consider a generalized problem, where a k-uniform hypergraph is given with an extra matroid constraint imposed on the edges, with the same goal of finding a b-matching that maximizes a submodular function. We extend our technique to this case to obtain an algorithm with an approximation of 8/3k+O(1). Our algorithms build on the ideas of the recent works of Levin and Wajc [Roie Levin and David Wajc, 2021] and of Garg, Jordan, and Svensson [Paritosh Garg et al., 2021]. Our main technical innovation is to introduce a data structure and associate it with each vertex and the matroid, to record the extra information of the stored edges. After the streaming phase, these data structures guide the greedy algorithm to make better choices.

We devise a constant-factor approximation algorithm for the maximization version of the edge-disjoint paths problem if the supply graph together with the demand edges forms a planar graph. By planar duality, this is equivalent to packing cuts in a planar graph such that each cut contains exactly one demand edge. We also show that the natural linear programming relaxations have constant integrality gap, yielding an approximate max-multiflow min-multicut theorem.

We give improved multi-pass streaming algorithms for the problem of maximizing a monotone or arbitrary non-negative submodular function subject to a general p-matchoid constraint in the model in which elements of the ground set arrive one at a time in a stream. The family of constraints we consider generalizes both the intersection of p arbitrary matroid constraints and p-uniform hypergraph matching. For monotone submodular functions, our algorithm attains a guarantee of p+1+ε using O(p/ε)-passes and requires storing only O(k) elements, where k is the maximum size of feasible solution. This immediately gives an O(1/ε)-pass (2+ε)-approximation for monotone submodular maximization in a matroid and (3+ε)-approximation for monotone submodular matching. Our algorithm is oblivious to the choice ε and can be stopped after any number of passes, delivering the appropriate guarantee. We extend our techniques to obtain the first multi-pass streaming algorithms for general, non-negative submodular functions subject to a p-matchoid constraint. We show that a randomized O(p/ε)-pass algorithm storing O(p³klog(k)/ε³) elements gives a (p+1+γ+O(ε))-approximation, where γ is the guarantee of the best-known offline algorithm for the same problem.

Given a set D of n unit disks in the plane and an integer k <= n, the maximum area connected subset problem asks for a set D' subseteq D of size k that maximizes the area of the union of disks, under the constraint that this union is connected. This problem is motivated by wireless router deployment and is a special case of maximizing a submodular function under a connectivity constraint. We prove that the problem is NP-hard and analyze a greedy algorithm, proving that it is a 1/2-approximation. We then give a polynomial-time approximation scheme (PTAS) for this problem with resource augmentation, i.e., allowing an additional set of epsilon k disks that are not drawn from the input. Additionally, for two special cases of the problem we design a PTAS without resource augmentation.

We study classical deadline-based preemptive scheduling of jobs in a computing environment equipped with both dynamic speed scaling and sleep state capabilities: Each job is specified by a release time, a deadline and a processing volume, and has to be scheduled on a single, speed-scalable processor that is supplied with a sleep state. In the sleep state, the processor consumes no energy, but a constant wake-up cost is required to transition back to the active state. In contrast to speed scaling alone, the addition of a sleep state makes it sometimes beneficial to accelerate the processing of jobs in order to transition the processor to the sleep state for longer amounts of time and incur further energy savings. The goal is to output a feasible schedule that minimizes the energy consumption. Since the introduction of the problem by Irani et al. (ACM Trans Algorithms 3(4), 2007), its exact computational complexity has been repeatedly posed as an open question (see e.g. Albers and Antoniadis in ACM Trans Algorithms 10(2):9, 2014; Baptiste et al. in ACM Trans Algorithms 8(3):26, 2012; Irani and Pruhs in SIGACT News 36(2):63–76, 2005). The currently best known upper and lower bounds are a 4 / 3-approximation algorithm and NP-hardness due to Albers and Antoniadis (2014) and Kumar and Shannigrahi (CoRR, 2013. arXiv:1304.7373 ), respectively. We close the aforementioned gap between the upper and lower bound on the computational complexity of speed scaling with sleep state by presenting a fully polynomial-time approximation scheme for the problem. The scheme is based on a transformation to a non-preemptive variant of the problem, and a discretization that exploits a carefully defined lexicographical ordering among schedules.

In this paper, we consider the problem of maximizing a monotone submodular function subject to a knapsack constraint in the streaming setting. In particular, the elements arrive sequentially and at any point of time, the algorithm has access only to a small fraction of the data stored in primary memory. For this problem, we propose a (0.363-epsilon)-approximation algorithm, requiring only a single pass through the data; moreover, we propose a (0.4-epsilon)-approximation algorithm requiring a constant number of passes through the data. The required memory space of both algorithms depends only on the size of the knapsack capacity and epsilon.

We study computing all-pairs shortest paths (APSP) on distributed networks (the CONGEST model). The goal is for every node in the (weighted) network to know the distance from every other node using communication. The problem admits (1+o(1))-approximation Õ(n)-time algorithms [2], [3], which are matched with Ω(n)-time lower bounds [3], [4], [5]1. No ω(n) lower bound or o(m) upper bound were known for exact computation. In this paper, we present an Õ(n5/4)-time randomized (Las Vegas) algorithm for exact weighted APSP; this provides the first improvement over the naive O(m)-time algorithm when the network is not so sparse. Our result also holds for the case where edge weights are asymmetric (a.k.a. the directed case where communication is bidirectional). Our techniques also yield an Õ(n3/4k1/2+ n)-time algorithm for the k-source shortest paths problem where we want every node to know distances from k sources; this improves Elkin's recent bound [6] when k = ω̃(n1/4). We achieve the above results by developing distributed algorithms on top of the classic scaling technique, which we believe is used for the first time for distributed shortest paths computation. One new algorithm which might be of an independent interest is for the reversed r-sink shortest paths problem, where we want every of r sinks to know its distances from all other nodes, given that every node already knows its distance to every sink. We show an Õ(n√r)-time algorithm for this problem. Another new algorithm is called short range extension, where we show that in Õ(n√h) time the knowledge about distances can be “extended” for additional h hops. For this, we use weight rounding to introduce small additive errors which can be later fixed.

Our input instance is a bipartite graph G = (A ∪ B, E) where A is a set of applicants, B is a set of jobs, and each vertex u ∊ A ∪ B has a preference list ranking its neighbors in a strict order of preference. For any two matchings M and T in G, let φ (M, T) be the number of vertices that prefer M to T. A matching M is popular if φ(Μ, T) ≥ φ(Τ,M) for all matchings T in G. There is a utility function w : E → ℚ and we consider the problem of matching applicants to jobs in a popular and utility-optimal manner. A popular mixed matching could have a much higher utility than all popular matchings, where a mixed matching is a probability distribution over matchings, i.e., a mixed matching Π = {(M0, p0),…, (Mk, pk)} for some matchings M0,…,Mk and for all i. The function φ(·,) easily extends to mixed matchings; a mixed matching Π is popular if φ(Π,Λ) > φ(Λ, Π) for all mixed matchings Λ in G. Motivated by the fact that a popular mixed matching could have a much higher utility than all popular matchings, we study the popular fractional matching polytope Pg. Our main result is that this polytope is half-integral and in the special case where a stable matching in G is a perfect matching, this polytope is integral. This implies that there is always a max-utility popular mixed matching Π such that where M0 and M1 are matchings in G. As Π can be computed in polynomial time, an immediate consequence of our result is that in order to implement a max-utility popular mixed matching in G, we need just a single random bit. We analyze PG whose description may have exponentially many constraints via an extended formulation with a linear number of constraints. The linear program that gives rise to this formulation has an unusual property: self-duality. In other words, this linear program is identical to its dual program. This is a rare case where an LP of a natural problem has such a property. The self-duality of this LP plays a crucial role in our proof of half-integrality of PG. We also show that our result carries over to the roommates problem, where the graph G need not be bipartite. The polytope of popular fractional matchings is still half-integral here and so we can compute a max-utility popular half-integral matching in G in polynomial time. To complement this result, we also show that the problem of computing a max-utility popular (integral) matching in a roommates instance is NP-hard.

We are given a bipartite graph $G = (A \cup B, E)$ where each vertex has a preference list ranking its neighbors: In particular, every $a \in A$ ranks its neighbors in a strict order of preference, whereas the preference list of any $b \in B$ may contain ties. A matching $M$ is popular if there is no matching $M'$ such that the number of vertices that prefer $M'$ to $M$ exceeds the number of vertices that prefer $M$ to $M'$. We show that the problem of deciding whether $G$ admits a popular matching or not is $\mathsf{NP}$-hard. This is the case even when every $b \in B$ either has a strict preference list or puts all its neighbors into a single tie. In contrast, we show that the problem becomes polynomially solvable in the case when each $b \in B$ puts all its neighbors into a single tie. That is, all neighbors of $b$ are tied in $b$'s list and $b$ desires to be matched to any of them. Our main result is an $O(n^2)$ algorithm (where $n = |A \cup B|$) for the popular matching problem in this model. Note that this model is quite different from the model where vertices in $B$ have no preferences and do not care whether they are matched or not.

Makespan minimization in restricted assignment (R|p_{ij} in {p_j, infinity}|C_{max}) is a classical problem in the field of machine scheduling. In a landmark paper, [Lenstra, Shmoys, and Tardos, Math. Progr. 1990] gave a 2-approximation algorithm and proved that the problem cannot be approximated within 1.5 unless P=NP. The upper and lower bounds of the problem have been essentially unimproved in the intervening 25 years, despite several remarkable successful attempts in some special cases of the problem recently. In this paper, we consider a special case called graph-balancing with light hyper edges, where heavy jobs can be assigned to at most two machines while light jobs can be assigned to any number of machines. For this case, we present algorithms with approximation ratios strictly better than 2. Specifically, - Two job sizes: Suppose that light jobs have weight w and heavy jobs have weight W, and w < W. We give a 1.5-approximation algorithm (note that the current 1.5 lower bound is established in an even more restrictive setting). Indeed, depending on the specific values of w and W, sometimes our algorithm guarantees sub-1.5 approximation ratios. - Arbitrary job sizes: Suppose that W is the largest given weight, heavy jobs have weights in the range of (beta W, W], where 4/7 <= beta < 1, and light jobs have weights in the range of (0,beta W]. We present a (5/3+beta/3)-approximation algorithm. Our algorithms are purely combinatorial, without the need of solving a linear program as required in most other known approaches.

The problem of finding a maximum cardinality stable matching in the presence of ties and unacceptable partners, called MAX SMTI, is a well-studied NP-hard problem. The MAX SMTI is NP-hard even for highly restricted instances where (i) ties appear only in women's preference lists and (ii) each tie appears at the end of each woman's preference list. The current best lower bounds on the approximation ratio for this variant are 1.1052 unless P=NP and 1.25 under the unique games conjecture, while the current best upper bound is 1.4616. In this paper, we improve the upper bound to 1.25, which matches the lower bound under the unique games conjecture. Note that this is the first special case of the MAX SMTI where the tight approximation bound is obtained. The improved ratio is achieved via a new analysis technique, which avoids the complicated case-by-case analysis used in earlier studies. As a by-product of our analysis, we show that the integrality gap of natural IP and LP formulations for this variant is 1.25. We also show that the unrestricted MAX SMTI cannot be approximated with less than 1.5 unless the approximation ratio of a certain special case of the minimum maximal matching problem can be improved.

We study classical deadline-based preemptive scheduling of jobs in a computing environment equipped with both dynamic speed scaling and sleep state capabilities: Each job is specified by a release time, a deadline and a processing volume, and has to be scheduled on a single, speed-scalable processor that is supplied with a sleep state. In the sleep state, the processor consumes no energy, but a constant wake-up cost is required to transition back to the active state. In contrast to speed scaling alone, the addition of a sleep state makes it sometimes beneficial to accelerate the processing of jobs in order to transition the processor to the sleep state for longer amounts of time and incur further energy savings. The goal is to output a feasible schedule that minimizes the energy consumption. Since the introduction of the problem by Irani et al. [17], its exact computational complexity has been repeatedly posed as an open question (see e.g. [2,9,16]). The currently best known upper and lower bounds are a 4/3-approximation algorithm and NP-hardness due to [2] and [2,18], respectively. We close the aforementioned gap between the upper and lower bound on the computational complexity of speed scaling with sleep state by presenting a fully polynomial-time approximation scheme for the problem. The scheme is based on a transformation to a non-preemptive variant of the problem, and a discretization that exploits a carefully defined lexicographical ordering among schedules.

Let G = (A union B, E) be a bipartite graph, where every vertex ranks its neighbors in an order of preference (with ties allowed) and let r be the worst rank used. A matching M is fair in G if it has maximum cardinality, subject to this, M matches the minimum number of vertices to rank r neighbors, subject to that, M matches the minimum number of vertices to rank (r-1) neighbors, and so on. We show an efficient combinatorial algorithm based on LP duality to compute a fair matching in G. We also show a scaling based algorithm for the fair b-matching problem. Our two algorithms can be extended to solve other profile-based matching problems. In designing our combinatorial algorithm, we show how to solve a generalized version of the minimum weighted vertex cover problem in bipartite graphs, using a single-source shortest paths computation---this can be of independent interest.

Our input is a graph $G = (V, E)$ where each vertex ranks its neighbors in a strict order of preference. The problem is to compute a matching in $G$ that captures the preferences of the vertices in a popular way. Matching $M$ is more popular than matching $M'$ if the number of vertices that prefer $M$ to $M'$ is more than those that prefer $M'$ to $M$. The unpopularity factor of $M$ measures by what factor any matching can be more popular than $M$. We show that $G$ always admits a matching whose unpopularity factor is $O(\log|V|)$, and such a matching can be computed in linear time. In our problem the optimal matching would be a least unpopularity factor matching---we show that computing such a matching is NP-hard. In fact, for any $\epsilon > 0$, it is NP-hard to compute a matching whose unpopularity factor is at most $4/3 - \epsilon$ of the optimal.

Let G = (V, E) be a graph with positive integral edge weights. Our problem is to find a matching of maximum weight in G. We present a simple iterative algorithm for this problem that uses a maximum cardinality matching algorithm as a subroutine. Using the current fastest maximum cardinality matching algorithms, we solve the maximum weight matching problem in O(W√nm logn(n2/m)) time, or in O(W nω) time with high probability, where n = |V|, m = |E|, W is the largest edge weight, and ω < 2.376 is the exponent of matrix multiplication. In relatively dense graphs, our algorithm performs better than all existing algorithms with W = o(log1.5 n). Our technique hinges on exploiting Edmonds’ matching polytope and its dual.

We introduce a bidding language for expressing negative value externalities in position auctions for online advertising. The unit-bidder constraints (UBC) language allows a bidder to condition a bid on its allocated slot and on the slots allocated to other bidders. We introduce a natural extension of the Generalized Second Price (GSP) auction, the expressive GSP (eGSP) auction, that induces truthful revelation of constraints for a rich subclass of unit-bidder types, namely downward-monotonic UBC. We establish the existence of envy-free Nash equilibrium in eGSP under a further restriction to a subclass of exclusion constraints, for which the standard GSP has no pure strategy Nash equilibrium. The equilibrium results are obtained by reduction to equilibrium analysis for reserve price GSP (Even-Dar et al. 2008). In considering the winner determination problem, which is NP-hard, we bound the approximation ratio for social welfare in eGSP and provide parameterized complexity results.

We present an algorithm to solve the group mutual exclusion, problem in the cache-coherent (CC) model. For the same problem in the distributed shared memory (DSM) model, Danek and Hadzilacos presented algorithms of O(n) remote memory references (RMR) and proved a matching lower bound, where n is the number of processes. We show that in the CC model, using registers and LL/SC variables, our algorithm achieves O(min(log n,k)) RMR, where k is the point contention, which is so far the best. Moreover, given a recent result of Attiya, Hendler and Woelfel showing that exclusion problems have a Ω(log n) RME lower bound using registers, comparison primitives and LL/SC variables, our algorithm thus achieves the best theoretical bound.

We introduce the classified stable matching problem, a problem motivated by academic hiring. Suppose that a number of institutes are hiring faculty members from a pool of applicants. Both institutes and applicants have preferences over the other side. An institute classifies the applicants based on their research areas (or any other criterion), and, for each class, it sets a lower bound and an upper bound on the number of applicants it would hire in that class. The objective is to find a stable matching from which no group of participants has reason to deviate. Moreover, the matching should respect the upper/lower bounds of the classes. In the first part of the paper, we study classified stable matching problems whose classifications belong to a fixed set of “order types.” We show that if the set consists entirely of downward forests, there is a polynomial-time algorithm; otherwise, it is NP-complete to decide the existence of a stable matching.

We consider the following version of the stable matching problem. Suppose that men have preferences for women, women have preferences for dogs, and dogs have preferences for men. The goal is to organize them into family units so that no three of them have incentive to desert their assigned family members to join in a new family. This problem is called circular stable matching, allegedly originated by Knuth. We also investigate a generalized version of this problem, in which every participant has preference among all others. The goal is similarly to partition them into oriented triples so that no three persons have incentive to deviate from the assignment. This problem is motivated by recent innovations in kidney exchange, and we call it the 3-way kidney transplant problem. We report complexity, structural and counting results on these two problems.

We introduce and study the {\em donation center location} problem, which has an additional application in network testing and may also be of independent interest as a general graph-theoreticproblem.Given a set of agents and a set of centers, where agents have preferences over centers and centers have capacities, the goal is to open a subset of centers and to assign a maximum-sized subset of agents to their most-preferred open centers, while respecting the capacity constraints. We prove that in general, the problem is hard to approximate within $n^{1/2-\epsilon}$ for any $\epsilon>0$. In view of this, we investigate two special cases. In one, every agent has a bounded number of centers on her preference list, and in the other, all preferences are induced by a line-metric. We present constant-factor approximation algorithms for the former and exact polynomial-time algorithms for the latter. Of particular interest among our techniques are an analysis of the greedy algorithm for a variant of the maximum coverage problem called\emph{frugal coverage}, the use of maximum matching subroutine with subsequent modification, analyzed using a counting argument, and a reduction to the independent set problem on \emph{terminal intersection graphs}, which we show to be a subclass of trapezoid graphs.

In routing games with infinitesimal players, it follows from well-known convexity arguments that equilibria exist and are unique (up to induced delays, and under weak assumptions on delay functions). In routing games with players that control large amounts of flow, uniqueness has been demonstrated only in limited cases: in 2-terminal, nearly-parallel graphs; when all players control exactly the same amount of flow; when latency functions are polynomials of degree at most three. In this work, we answer an open question posed by Cominetti, Correa, and Stier-Moses (ICALP 2006) and show that there may be multiple equilibria in atomic player routing games. We demonstrate this multiplicity via two specific examples. In addition, we show our examples are topologically minimal by giving a complete characterization of the class of network topologies for which unique equilibria exist. Our proofs and examples are based on a novel characterization of these topologies in terms of sets of circulations.