Scarf’s algorithm - a pivoting procedure that finds a dominating extreme point in a down-monotone polytope - can be used to show the existence of a fractional stable matching in hypergraphs. The problem of finding a fractional stable matching in hypergraphs, however, is PPAD-complete. In this work, we study the behavior of Scarf’s algorithm on arborescence hypergraphs, the family of hypergraphs in which hyperedges correspond to the paths of an arborescence. For arborescence hypergraphs, we prove that Scarf’s algorithm can be implemented to find an integral stable matching in polynomial time. En route to our result, we uncover novel structural properties of bases and pivots for the more general family of network hypergraphs. Our work provides the first proof of polynomial-time convergence of Scarf’s algorithm on hypergraphic stable matching problems, giving hope to the possibility of polynomial-time convergence of Scarf’s algorithm for other families of polytopes.

Many centralized mechanisms for two-sided matching markets that enjoy strong theoretical properties assume that the planner solicits full information on the preferences of each participating agent. In particular, they expect that participants compile and communicate their complete preference lists over agents from the other side of the market. However, real-world markets are often very large and agents cannot always be expected to even produce a ranking of all options on the other side. It is therefore important to understand the impact of incomplete or truncated lists on the quality of the resultant matching.

The assignment game, introduced by Shapley and Shubik [1971], is a classic model for two-sided matching markets between buyers and sellers. In the original assignment game, it is assumed that payments lead to transferable utility and that buyers have unit-demand valuations for the items being sold. There has since been substantial work studying various extensions of the assignment game. The first main area of extension is to imperfectly transferable utility, which is when frictions, taxes, or fees impede the transfer of money between agents. The second is with more complex valuation functions, in particular gross substitutes valuations, which describe substitutable goods. Multiple efficient algorithms have been proposed for computing a competitive equilibrium, the standard solution concept in assignment games, in each of these two settings. However, these lines of work have been mostly independent, with no algorithmic results combining the two.

In the online minimum cost matching problem, there are n servers and, at each of n time steps, a request arrives and must be irrevocably matched to a server that has not yet been matched, with the goal of minimizing the sum of the distances between the matched pairs. Online minimum cost matching is a central problem in applications such as ride-hailing platforms and food delivery services. Despite achieving a worst-case competitive ratio that is exponential in n even on the line, the simple greedy algorithm, which matches each request to its nearest available server, performs well in practice and has a number of attractive features such as strategyproofness. A major question is thus to explain greedy's strong empirical performance. In this paper, we aim to understand the performance of greedy on the line over instances that are at least partially random.

Discovery program (DISC) is a policy used by the New York City Department of Education (NYC DOE) to increase the number of admissions of students from low socio-economic background to specialized high schools. This policy has been instrumental in increasing the number of disadvantaged students attending these schools, by reserving a percentage of seats to disadvantaged students that complete a three-week summer program (with a very high success rate [Hu, 2018]). However, assuming that students care more about the school they are assigned to rather than the type of seat they occupy (school-over-seat hypothesis), our empirical analysis using NYC DOE data from 12 recent academic years (2005--06 to 2016--17) shows that DISC creates about 950 in-group blocking pairs each year amongst disadvantaged students, impacting about 650 disadvantaged students every year. Moreover, we find that this program does not respect improvements as it benefits lower-performing disadvantaged students more than top-performing disadvantaged students by matching some of the former to more preferred schools, thus unintentionally creating an incentive to under-perform. These experimental results are confirmed by our theoretical analysis.

An instance of the marriage problem is given by a graph $G = (A \cup B,E)$, together with, for each vertex of $G$, a strict preference order over its neighbors. A matching $M$ of $G$ is popular in the marriage instance if $M$ does not lose a head-to-head election against any matching where vertices are voters. Every stable matching is a min-size popular matching; another subclass of popular matchings that always exists and can be easily computed is the set of dominant matchings. A popular matching $M$ is dominant if $M$ wins the head-to-head election against any larger matching. Thus, every dominant matching is a max-size popular matching, and it is known that the set of dominant matchings is the linear image of the set of stable matchings in an auxiliary graph. Results from the literature seem to suggest that stable and dominant matchings behave, from a complexity theory point of view, in a very similar manner within the class of popular matchings. The goal of this paper is to show that there are instead differences in the tractability of stable and dominant matchings and to investigate further their importance for popular matchings. First, we show that it is easy to check if all popular matchings are also stable; however, it is co-NP hard to check if all popular matchings are also dominant. Second, we show how some new and recent hardness results on popular matching problems can be deduced from the NP-hardness of certain problems on stable matchings, also studied in this paper, thus showing that stable matchings can be employed to show not only positive results on popular matchings (as is known) but also most negative ones. Problems for which we show new hardness results include finding a min-size (resp., max-size) popular matching that is not stable (resp., dominant). A known result for which we give a new and simple proof is the NP-hardness of finding a popular matching when $G$ is nonbipartite.

Integer programming is a general optimization framework with a wide variety of applications, e.g., in scheduling, production planning, and graph optimization. As Integer Programs (IPs) model many provably hard to solve problems, modern IP solvers rely on heuristics. These heuristics are often human-designed, and tuned over time using experience and data. The goal of this work is to show that the performance of those solvers can be greatly enhanced using reinforcement learning (RL). In particular, we investigate a specific methodology for solving IPs, known as the Cutting Plane Method. This method is employed as a subroutine by all modern IP solvers. We present a deep RL formulation, network architecture, and algorithms for intelligent adaptive selection of cutting planes (aka cuts). Across a wide range of IP tasks, we show that our trained RL agent significantly outperforms human-designed heuristics, and effectively generalizes to larger instances and across IP problem classes. The trained agent is also demonstrated to benefit the popular downstream application of cutting plane methods in Branch-and-Cut algorithm, which is the backbone of state-of-the-art commercial IP solvers.

Let G = (A ∪ B, E) be an instance of the stable marriage problem where every vertex ranks its neighbors in a strict order of preference. A matching M in G is popular if M does not lose a head-to-head election against any matching. Popular matchings are a well-studied generalization of stable matchings, introduced with the goal of enlarging the set of admissible solutions, while maintaining a certain level of fairness. Every stable matching is a min-size popular matching. Unfortunately, when there are edge costs, it is NP-hard to find a popular matching of minimum cost – even worse, the min-cost popular matching problem is hard to approximate up to any factor. Let opt be the cost of a min-cost popular matching. Our goal is to efficiently compute a matching of cost at most opt by paying the price of mildly relaxing popularity. Our main positive result is a bi-criteria algorithm that finds in polynomial time a near-popular or “quasi-popular” matching of cost at most opt. Key to the algorithm are a number of results for certain polytopes related to matchings. In particular, we give a polynomial-size extended formulation for an integral polytope sandwiched between the popular and quasi-popular matching polytopes. We complement these results by showing that it is NP-hard to find a quasi-popular matching of minimum cost, and that both the popular and quasi-popular matching polytopes have near-exponential extension complexity.

We consider popular matching problems in both bipartite and non-bipartite graphs with strict preference lists. It is known that every stable matching is a min-size popular matching. A subclass of max-size popular matchings called dominant matchings has been well-studied in bipartite graphs: they always exist and there is a simple linear time algorithm to find one. We show that it is NP-complete to decide if a bipartite graph admits a popular matching that is neither stable nor dominant. This gives rise to the anomaly that though it is easy to find min-size and max-size popular matchings in bipartite graphs, it is NP-complete to decide if there exists any popular matching whose size is sandwiched between the two extremes. We also show a number of related hardness results, such as (tight) 1/2-inapproximability of the maximum cost popular matching problem when costs are nonnegative. In non-bipartite graphs, we show a strong negative result: it is NP-hard to decide whether a popular matching exists or not, and the same result holds if we replace popular with dominant. On the positive side, we show the following results in any graph: we identify a subclass of dominant matchings called strongly dominant matchings and show a linear time algorithm to decide if a strongly dominant matching exists or not; we show an efficient algorithm to compute a popular matching of minimum cost in a graph with edge costs and bounded treewidth, or decide there is no popular matching.

$2$-level polytopes naturally appear in several areas of pure and applied mathematics, including combinatorial optimization, polyhedral combinatorics, communication complexity, and statistics. In this paper, we present a study of some $2$-level polytopes arising in combinatorial settings. Our first contribution is proving that $f_0(P)f_{d-1}(P)\leq d2^{d+1}$ for a large collection of families of such polytopes $P$. Here $f_0(P)$ (resp., $f_{d-1}(P)$) is the number of vertices (resp., facets) of $P$, and $d$ is its dimension. Whether this holds for all 2-level polytopes was asked in [A. Bohn, Y. Faenza, S. Fiorini, V. Fisikopoulos, M. Macchia, and K. Pashkovich, in Algorithms--ESA 2015, Springer, Berlin, 2015, pp. 191--202], and experimental results from [S. Fiorini, V. Fisikopoulos, and M. Macchia, in Combinatorial Optimization, Springer, Cham, 2016, pp. 285--296] showed it true for $d\leq 7$. The key to most of our proofs is a deeper understanding of the relations among those polytopes and their underlying combinatorial structure. This leads to a number of results that we believe to be of independent interest: a trade-off formula for the number of cliques and stable sets in a graph, a description of stable matching polytopes as affine projections of certain order polytopes, and a linear-size description of the base polytope of matroids that are 2-level in terms of cuts of an associated tree.

We show that the problem of finding the simplex of largest volume in the convex hull of n points in ℚd can be approximated with a factor of O(log d) d/2 in polynomial time. This improves upon the previously best known approximation guarantee of d(d–1)/2 by Khachiyan. On the other hand, we show that there exists a constant c > 1 such that this problem cannot be approximated with a factor of cd, unless P = NP. Our hardness result holds even if n = O(d), in which case there exists a d-approximation algorithm that relies on recent sampling techniques, where is again a constant. We show that similar results hold for the problem of finding the largest absolute value of a subdeterminant of a d × n matrix.

We introduce the reverse Chvátal--Gomory rank $r^*(P)$ of an integral polyhedron $P$, defined as the supremum of the Chvátal--Gomory ranks of all rational polyhedra whose integer hull is $P$. A well-known example in dimension two shows that there exist integral polytopes $P$ with $r^*(P)=+\infty$. We provide a geometric characterization of polyhedra with this property in every dimension, and investigate upper bounds on $r^*(P)$ when this value is finite.

The classic stable set problem asks to find a maximum cardinality set of pairwise non-adjacent vertices in an undirected graph G. This problem is NP-hard to approximate with factor n^{1-epsilon} for any constant epsilon>0 [Hastad/Acta Mathematica/1996; Zuckerman/STOC/2006], where n is the number of vertices, and therefore there is no hope for good approximations in the general case. We study the stable set problem when restricted to graphs with bounded odd cycle packing number ocp(G), possibly by a function of n. This is the largest number of vertex-disjoint odd cycles in G. Equivalently, it is the logarithm of the largest absolute value of a sub-determinant of the edge-node incidence matrix A_G of G. Hence, if A_G is totally unimodular, then ocp(G)=0. Therefore, ocp(G) is a natural distance measure of A_G to the set of totally unimodular matrices on a scale from 1 to n/3. When ocp(G)=0, the graph is bipartite and it is well known that stable set can be solved in polynomial time. Our results imply that the odd cycle packing number indeed strongly influences the approximability of stable set. More precisely, we obtain a polynomial-time approximation scheme for graphs with ocp(G)=o(n/log(n)), and an alpha-approximation algorithm for any graph where alpha smoothly increases from a constant to n as ocp(G) grows from O(n/log(n)) to n/3. On the hardness side, we show that, assuming the exponential-time hypothesis, stable set cannot be solved in polynomial time if ocp(G)=Omega(log^{1+epsilon}(n)) for some epsilon>0. Finally, we generalize a theorem by Györi et al. [Györi et al./Discrete Mathematics/1997] and show that graphs without odd cycles of small weight can be made bipartite by removing a small number of vertices. This allows us to extend some of our above results to the weighted stable set problem.

We propose an algorithm for solving the maximum weighted stable set problem on claw-free graphs that runs in O (| V |(| E | + | V | log| V |))-time, drastically improving the previous best known complexity bound. This algorithm is based on a novel decomposition theorem for claw-free graphs, which is also introduced in the present article. Despite being weaker than the structural results for claw-free graphs given by Chudnovsky and Seymour [2005, 2008a, 2008b] our decomposition theorem is, on the other hand, algorithmic, that is, it is coupled with an O (| V || E |)-time algorithm that actually produces the decomposition.

Given an integral polyhedron $P\subseteq\mathbb{R}^n$ and a rational polyhedron $Q\subseteq\mathbb{R}^n$ containing the same integer points as $P$, we investigate how many iterations of the Chvátal--Gomory closure operator have to be performed on $Q$ to obtain a polyhedron contained in the affine hull of $P$. We show that if $P$ contains an integer point in its relative interior, then such a number of iterations can be bounded by a function depending only on $n$. On the other hand, we prove that if $P$ is not full-dimensional and does not contain any integer point in its relative interior, then no finite bound on the number of iterations exists.

In this paper, we provide the first linear programming formulations for the stable set problem in claw-free graphs, together with polynomial time separation routines for those formulations (they are not compact). We then exploit one of those extended formulations and propose a new polytime algorithm for solving the separation problem for the stable set polytope of claw-free graphs. This routine combines a separation algorithm for the matching polytope due to Padberg and Rao and the solution of (moderate size) compact linear programs. Hence, it does not rely on the ellipsoid method and seems to be appropriate to be inserted in branch and cut frameworks for solving real world problems.

We propose an algorithm for solving the maximum weighted stable set problem on claw-free graphs that runs in O(n3)-time, drastically improving the previous best known complexity bound. This algorithm is based on a novel decomposition theorem for claw-free graphs, which is also introduced in the present paper. Despite being weaker than the well-known structure result for claw-free graphs given by Chudnovsky and Seymour [5], our decomposition theorem is, on the other hand, algorithmic, i.e. it is coupled with an O(n3)-time procedure that actually produces the decomposition. We also believe that our algorithmic decomposition result is interesting on its own and might be also useful to solve other kind of problems on claw-free graphs.