An important aspect of multi-agent systems concerns the formation of coalitions that are stable or optimal in some well-defined way The notion of popularity has recently received a lot of attention in this context A partition is popular if there is no other partition in which more agents are better off than worse off In 2019, a long-standing open problem concerning popularity was solved by proving that computing popular partitions in roommate games is NP-hard, even when preferences are strict We show that this result breaks down when allowing for randomization: mixed popular partitions can be found efficiently via linear programming and a separation oracle Mixed popular partitions are particularly attractive because they are guaranteed to exist in any coalition formation game Our result implies that one can efficiently verify whether a given partition in a roommate game is popular and that strongly popular partitions can be found in polynomial time (resolving an open problem) By contrast, we prove that both problems become computationally intractable when moving from coalitions of size~2 to coalitions of size~3, even when preferences are strict and globally ranked Moreover, we give elaborate proofs showing the NP-hardness of finding popular, strongly popular, and mixed popular partitions in additively separable hedonic games and finding popular partitions in fractional hedonic games

Finding and Recognizing Popular Coalition Structures

We study popular matchings in the many-to-many matching problem, which is given by a graph $$G = (V,E)$$
 and, for every agent $$u\in V$$
 , a capacity $$\textsf {cap}(u) \ge 1$$
 and a preference list strictly ranking her neighbors. A matching in G is popular if it weakly beats every matching in a majority vote when agents cast votes for one matching versus the other according to their preferences. First, we show that when $$G = (A\cup B,E)$$
 is bipartite, e.g., when matching students to courses, every pairwise stable matching is popular. In particular, popular matchings are guaranteed to exist. Our main contribution is to show that a max-size popular matching in G can be computed in linear time by the 2-level Gale–Shapley algorithm, which is an extension of the classical Gale–Shapley algorithm. We prove its correctness via linear programming. Second, we consider the problem of computing a max-size popular matching in $$G = (V,E)$$
 (not necessarily bipartite) when every agent has capacity 1, e.g., when matching students to dorm rooms. We show that even when G admits a stable matching, this problem is $$\mathsf {NP}$$
 -hard, which is in contrast to the tractability result in bipartite graphs.

Two Problems in Max-Size Popular Matchings

Let G be a digraph where every node has preferences over its incoming edges. The preferences of a node extend naturally to preferences over branchings, i.e., directed forests; a branching B is popular if B does not lose a head-to-head election (where nodes cast votes) against any branching. Such popular branchings have a natural application in liquid democracy. The popular branching problem is to decide if G admits a popular branching or not. We give a characterization of popular branchings in terms of dual certificates and use this characterization to design an efficient combinatorial algorithm for the popular branching problem. When preferences are weak rankings, we use our characterization to formulate the popular branching polytope in the original space and also show that our algorithm can be modified to compute a branching with least unpopularity margin. When preferences are strict rankings, we show that “approximately popular” branchings always exist.

/pdf/popular-branchings-and-their-dual-certificates-djg5ji9b0c.pdf

Popular branchings and their dual certificates

Chebyshev polynomials of the first and second kind for a set K are monic polynomials with minimal $$L_\infty $$- and $$L_1$$-norm on K, respectively. This article presents numerical procedures based on semidefinite programming to compute these polynomials in case K is a finite union of compact intervals. For Chebyshev polynomials of the first kind, the procedure makes use of a characterization of polynomial non-negativity. It can incorporate additional constraints, e.g. that all the roots of the polynomial lie in K. For Chebyshev polynomials of the second kind, the procedure exploits the method of moments.

/pdf/computation-of-chebyshev-polynomials-for-union-of-intervals-1ypf9ijzfa.pdf

Computation of Chebyshev Polynomials for Union of Intervals

/pdf/popular-branchings-and-their-dual-certificates-5ffkkf37u0.pdf

Popular Branchings and Their Dual Certificates

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.

Popular matchings and limits to tractability

This article details the theoretical grounds for a semidefinite-programming-based method that computes best approximants by splines under some general constraints and relative to several function norms, notably the max-norm. The method has been implemented as a matlab package called Basc, which relies on the two external packages Chebfun and CVX.

/pdf/basc-constrained-approximation-by-semidefinite-programming-3bs6sneo2k.pdf

Basc: constrained approximation by semidefinite programming

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 stable and dominant matchings are the only two tractable subclasses of popular matchings in bipartite graphs; more precisely, we show that it is NP-complete to decide if $G$ admits a popular matching that is neither stable nor dominant. We also show a number of related hardness results, such as (tight) inapproximability of the maximum weight popular matching problem. 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.

Popular Matchings and Limits to Tractability

Let $G = (A \cup B, E)$ be an instance of the stable marriage problem with strict preference lists. A matching $M$ is popular in $G$ 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 exist 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 indeed there are 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 not only to show positive results on popular matching (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 non-bipartite.

Understanding popular matchings via stable matchings.

Let $G = (A \cup B, E)$ be an instance of the stable marriage problem with strict preference lists. A matching $M$ is popular in $G$ if $M$ does not lose a head-to-head election against any matching where vertices are voters. Every stable matching is popular; another subclass of popular matchings that always exist 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. The set of dominant matchings is the linear image of the set of stable matchings in an auxiliary graph. 
In this paper, we investigate the difference between the tractability of stable and dominant matchings, and its consequence for popular matchings. We give the first known complete description of the dominant matching polytope in the original space and show that it has an exponential number of facets (recall that the stable matching polytope has a linear number of facets). This polyhedral asymmetry is reflected by a complexity asymmetry: We show that it is easy to decide if every popular matching in $G$ is also stable, however it is co-NP hard to decide if every popular matching in $G$ is also dominant. 
We show that several hardness results in popular matchings, including the above result and the hardness of finding a popular matching in a non-bipartite graph, can be attributed to the NP-hardness of the following two stable matching problems: 
- does $G$ admit a stable matching that is not dominant? 
- does $G$ admit a stable matching that is also dominant?

Vladlena Powers

Papers

Popular matchings and limits to tractability

Basc: constrained approximation by semidefinite programming

Popular Matchings and Limits to Tractability

Understanding popular matchings via stable matchings.

Popularity, stability, and the dominant matching polytope.