A key issue in cooperative game theory is coalitional stability, usually captured by the notion of the core---the set of outcomes that are resistant to group deviations. However, some coalitional games have empty cores, and any outcome in such a game is unstable. We investigate the possibility of stabilizing a coalitional game by using subsidies. We consider scenarios where an external party that is interested in having the players work together offers a supplemental payment to the grand coalition, or, more generally, a particular coalition structure. This payment is conditional on players not deviating from this coalition structure, and may be divided among the players in any way they wish. We define the cost of stability as the minimum external payment that stabilizes the game. We provide tight bounds on the cost of stability, both for games where the coalitional values are nonnegative (profit-sharing games) and for games where the coalitional values are nonpositive (cost-sharing games), under natural assumptions on the characteristic function, such as superadditivity, anonymity, or both. We also investigate the relationship between the cost of stability and several variants of the least core. Finally, we study the computational complexity of problems related to the cost of stability, with a focus on weighted voting games.

Andries Brouwer maintains a public database of existence results for strongly regular graphs on $$n\le 1300$$ vertices. We have implemented most of the infinite families of graphs listed there in the open-source software Sagemath (The Sage Developers, http://www.sagemath.org ), as well as provided constructions of the “sporadic” cases, to obtain a graph for each set of parameters with known examples. Besides providing a convenient way to verify these existence results from the actual graphs, it also extends the database to higher values of n.

A book with $k$ pages consists of a straight line (the spine) and $k$ half-planes (the pages), such that the boundary of each page is the spine. If a graph is drawn on a book with $k$ pages in such a way that the vertices lie on the spine, and each edge is contained in a page, the result is a k-page book drawing (or simply a $k$-page drawing). The $k$-page crossing number $\nu_k(G)$ of a graph $G$ is the minimum number of crossings in a $k$-page drawing of $G$. In this paper we investigate the $k$-page crossing numbers of complete graphs. We use semidefinite programming techniques to give improved lower bounds on $\nu_k(K_n)$ for various values of $k$. We also use a maximum satisfiability reformulation to obtain a computer-aided calculation of the exact value of $\nu_k(K_n)$ for several values of $k$ and $n$. Finally, we investigate the best construction known for drawing $K_n$ in $k$ pages, calculate the resulting number of crossings, and discuss this upper bound in light of the new results reported in this paper.

Asymmetric quantum error-correcting codes (AQCs) may offer some advantage over their symmetric counterparts by providing better error-correction for the more frequent error types. The well-known CSS construction of$q$-ary AQCs is extended by removing the$ \BBF _{q}$-linearity requirement as well as the limitation on the type of inner product used. The proposed constructions are called CSS-like constructions and utilize pairs of nested subfield linear codes under one of the Euclidean, trace Euclidean, Hermitian, and trace Hermitian inner products. After establishing some theoretical foundations, best-performing CSS-like AQCs are constructed. Combining some constructions of nested pairs of classical codes and linear programming, many optimal and good pure$q$-ary CSS-like codes for$q \in \{ 2,3,4,5,7,8,9\}$up to reasonable lengths are found. In many instances, removing the$ \BBF _{q}$-linearity and using alternative inner products give us pure AQCs with improved parameters than relying solely on the standard CSS construction.

Weighted voting games (WVG) are coalitional games in which an agent's contribution to a coalition is given by his weight, and a coalition wins if its total weight meets or exceeds a given quota. These games model decision-making in political bodies as well as collaboration and surplus division in multiagent domains. The computational complexity of various solution concepts for weighted voting games received a lot of attention in recent years. In particular, Elkind et al.(2007) studied the complexity of stability-related solution concepts in WVGs, namely, of the core, the least core, and the nucleolus. While they have completely characterized the algorithmic complexity of the core and the least core, for the nucleolus they have only provided an NP-hardness result. In this paper, we solve an open problem posed by Elkind et al. by showing that the nucleolus of WVGs, and, more generally, k-vector weighted voting games with fixed k, can be computed in pseudopolynomial time, i.e., there exists an algorithm that correctly computes the nucleolus and runs in time polynomial in the number of players n and the maximum weight W. In doing so, we propose a general framework for computing the nucleolus, which may be applicable to a wider of class of games.

The problem of minimizing a quadratic form over the standard simplex is known as the standard quadratic optimization problem (SQO). It is NP-hard, and contains the maximum stable set problem in graphs as a special case. In this note, we show that the SQO problem may be reformulated as an (exponentially sized) linear program (LP). This reformulation also suggests a hierarchy of polynomial-time solvable LP’s whose optimal values converge finitely to the optimal value of the SQO problem. The hierarchies of LP relaxations from the literature do not share this finite convergence property for SQO, and we review the relevant counterexamples.

It has been long conjectured that the crossing number $\Cr(K_{m,n})$ of the complete bipartite graph $K_{m,n}$ equals the Zarankiewicz number $Z(m,n):= \floor{\frac{m-1}{2}} \floor{\frac{m}{2}} \floor{\frac{n-1}{2}} \floor{\frac{n}{2}}$. Another longstanding conjecture states that the crossing number $\Cr(K_n)$ of the complete graph $K_n$ equals $Z(n):=\frac{1}{4}\smallfloor{\frac{n}{2}} \smallfloor{\frac{n-1}{2}} \smallfloor{\frac{n-2}{2}}\smallfloor{\frac{n-3}{2}}$. In this paper we show the following improved bounds on the asymptotic ratios of these crossing numbers and their conjectured values: \begin{itemize} \item[(i)] for each fixed $m\ge 9$, $\lim_{n\to\infty} \Cr(K_{m,n})/Z(m,n) \ge 0.83m/(m-1)$; \item[(ii)] $\lim_{n\to\infty} \Cr(K_{n,n})/Z(n,n) \ge 0.83$; and \item[(iii)] $\lim_{n\to\infty} \Cr(K_{n})/Z(n) \ge 0.83$. \end{itemize} The previous best known lower bounds were $0.8m/(m-1), 0.8$, and $0.8$, respectively. These improved bounds are obtained as a consequence of the new bound $\Cr(\ksn) \ge 2.1796n^2 - 4.5n$. To obtain this improved lower bound for $\Cr(\ksn)$, we use some elementary topological facts on drawings of $K_{2,7}$ to set up a quadratic program on $6!$ variables whose minimum p satisfies $\Cr(\ksn) \ge (p/2)n^2 - 4.5n$, and then use state-of-the-art quadratic optimization techniques combined with a bit of invariant theory of permutation groups to show that $p \ge 4.3593$.

Proof systems for polynomial inequalities in 0-1 variables include the well-studied Cutting Planes proof system (CP) and the Lovász- Schrijver calculi (LS) utilizing linear, respectively, quadratic, inequalities. We introduce generalizations LS d of LS involving polynomial inequalities of degree at most d . Surprisingly, the systems LS d turn out to be very strong. We construct polynomial-size bounded degree LS d proofs of the clique-coloring tautologies (which have no polynomial-size CP proofs), the symmetric knapsack problem (which has no bounded degree Positivstellensatz Calculus (PC) proofs), and Tseitin’s tautologies (hard for many known proof systems). Extending our systems with a division rule yields a polynomial simulation of CP with polynomially bounded coefficients , while other extra rules further reduce the proof degrees for the aforementioned examples. Finally, we prove lower bounds on Lovász-Schrijver ranks , demonstrating, in particular, their rather limited applicability for proof complexity.