A Simply Exponential Upper Bound on the Maximum Number of Stable Matchings

Stable matching is a classical combinatorial problem that has been the subject of intense theoretical and empirical study since its introduction in 1962 in a seminal paper by Gale and Shapley. In this paper, we provide a new upper bound on $f(n)$, the maximum number of stable matchings that a stable matching instance with $n$ men and $n$ women can have. It has been a long-standing open problem to understand the asymptotic behavior of $f(n)$ as $n\to\infty$, first posed by Donald Knuth in the 1970s. Until now the best lower bound was approximately $2.28^n$, and the best upper bound was $2^{n\log n- O(n)}$. In this paper, we show that for all $n$, $f(n) \leq c^n$ for some universal constant $c$. This matches the lower bound up to the base of the exponent. Our proof is based on a reduction to counting the number of downsets of a family of posets that we call "mixing". The latter might be of independent interest

Locally Stable Marriage with Strict Preferences

We study stable matching problems with locality of information and control. In our model, each agent is a node in a fixed network and strives to be matched to another agent. An agent has a complete preference list over all other agents it can be matched with. Agents can match arbitrarily, and they learn about possible partners dynamically based on their current neighborhood. We consider convergence of dynamics to locally stable matchings -- states that are stable with respect to their imposed information structure in the network. In the two-sided case of stable marriage in which existence is guaranteed, we show that the existence of a path to stability becomes NP-hard to decide. This holds even when the network exists only among one partition of agents. In contrast, if one partition has no network and agents remember a previous match every round, a path to stability is guaranteed and random dynamics converge with probability 1. We characterize this positive result in various ways. For instance, it holds for random memory and for cache memory with the most recent partner, but not for cache memory with the best partner. Also, it is crucial which partition of the agents has memory. Finally, we present results for centralized computation of locally stable matchings, i.e., computing maximum locally stable matchings in the two-sided case and deciding existence in the roommates case.Comment: Conference version in ICALP 2013; to appear in SIAM J. Disc Mat

Counting Houses of Pareto Optimal Matchings in the House Allocation Problem

Let $A,B$ with $|A| = m$ and $|B| = n\ge m$ be two sets. We assume that every element $a\in A$ has a reference list over all elements from $B$. We call an injective mapping $\tau$ from $A$ to $B$ a matching. A blocking coalition of $\tau$ is a subset $A'$ of $A$ such that there exists a matching $\tau'$ that differs from $\tau$ only on elements of $A'$, and every element of $A'$ improves in $\tau'$, compared to $\tau$ according to its preference list. If there exists no blocking coalition, we call the matching $\tau$ an exchange stable matching (ESM). An element $b\in B$ is reachable if there exists an exchange stable matching using $b$. The set of all reachable elements is denoted by $E^*$. We show $$|E^*| \leq \sum_{i = 1,\ldots, m}{\left\lfloor\frac{m}{i}\right\rfloor} = \Theta(m\log m).$$ This is asymptotically tight. A set $E\subseteq B$ is reachable (respectively exactly reachable) if there exists an exchange stable matching $\tau$ whose image contains $E$ as a subset (respectively equals $E$). We give bounds for the number of exactly reachable sets. We find that our results hold in the more general setting of multi-matchings, when each element $a$ of $A$ is matched with $\ell_a$ elements of $B$ instead of just one. Further, we give complexity results and algorithms for corresponding algorithmic questions. Finally, we characterize unavoidable elements, i.e., elements of $B$ that are used by all ESM's. This yields efficient algorithms to determine all unavoidable elements.Comment: 24 pages 2 Figures revise

Lifting Linear Extension Complexity Bounds to the Mixed-Integer Setting

Mixed-integer mathematical programs are among the most commonly used models for a wide set of problems in Operations Research and related fields. However, there is still very little known about what can be expressed by small mixed-integer programs. In particular, prior to this work, it was open whether some classical problems, like the minimum odd-cut problem, can be expressed by a compact mixed-integer program with few (even constantly many) integer variables. This is in stark contrast to linear formulations, where recent breakthroughs in the field of extended formulations have shown that many polytopes associated to classical combinatorial optimization problems do not even admit approximate extended formulations of sub-exponential size. We provide a general framework for lifting inapproximability results of extended formulations to the setting of mixed-integer extended formulations, and obtain almost tight lower bounds on the number of integer variables needed to describe a variety of classical combinatorial optimization problems. Among the implications we obtain, we show that any mixed-integer extended formulation of sub-exponential size for the matching polytope, cut polytope, traveling salesman polytope or dominant of the odd-cut polytope, needs $\Omega(n/\log n)$ many integer variables, where $n$ is the number of vertices of the underlying graph. Conversely, the above-mentioned polyhedra admit polynomial-size mixed-integer formulations with only $O(n)$ or $O(n \log n)$ (for the traveling salesman polytope) many integer variables. Our results build upon a new decomposition technique that, for any convex set $C$, allows for approximating any mixed-integer description of $C$ by the intersection of $C$ with the union of a small number of affine subspaces.Comment: A conference version of this paper will be presented at SODA 201

The matching polytope does not admit fully-polynomial size relaxation schemes

The groundbreaking work of Rothvo{\ss} [arxiv:1311.2369] established that every linear program expressing the matching polytope has an exponential number of inequalities (formally, the matching polytope has exponential extension complexity). We generalize this result by deriving strong bounds on the polyhedral inapproximability of the matching polytope: for fixed $0 < \varepsilon < 1$, every polyhedral $(1 + \varepsilon / n)$-approximation requires an exponential number of inequalities, where $n$ is the number of vertices. This is sharp given the well-known $\rho$-approximation of size $O(\binom{n}{\rho/(\rho-1)})$ provided by the odd-sets of size up to $\rho/(\rho-1)$. Thus matching is the first problem in $P$, whose natural linear encoding does not admit a fully polynomial-size relaxation scheme (the polyhedral equivalent of an FPTAS), which provides a sharp separation from the polynomial-size relaxation scheme obtained e.g., via constant-sized odd-sets mentioned above. Our approach reuses ideas from Rothvo{\ss} [arxiv:1311.2369], however the main lower bounding technique is different. While the original proof is based on the hyperplane separation bound (also called the rectangle corruption bound), we employ the information-theoretic notion of common information as introduced in Braun and Pokutta [http://eccc.hpi-web.de/report/2013/056/], which allows to analyze perturbations of slack matrices. It turns out that the high extension complexity for the matching polytope stem from the same source of hardness as for the correlation polytope: a direct sum structure.Comment: 21 pages, 3 figure