Semidefinite programming and eigenvalue bounds for the graph partition problem

The graph partition problem is the problem of partitioning the vertex set of a graph into a fixed number of sets of given sizes such that the sum of weights of edges joining different sets is optimized. In this paper we simplify a known matrix-lifting semidefinite programming relaxation of the graph partition problem for several classes of graphs and also show how to aggregate additional triangle and independent set constraints for graphs with symmetry. We present an eigenvalue bound for the graph partition problem of a strongly regular graph, extending a similar result for the equipartition problem. We also derive a linear programming bound of the graph partition problem for certain Johnson and Kneser graphs. Using what we call the Laplacian algebra of a graph, we derive an eigenvalue bound for the graph partition problem that is the first known closed form bound that is applicable to any graph, thereby extending a well-known result in spectral graph theory. Finally, we strengthen a known semidefinite programming relaxation of a specific quadratic assignment problem and the above-mentioned matrix-lifting semidefinite programming relaxation by adding two constraints that correspond to assigning two vertices of the graph to different parts of the partition. This strengthening performs well on highly symmetric graphs when other relaxations provide weak or trivial bounds

Symmetry in RLT cuts for the quadratic assignment and standard quadratic optimization problems

The reformulation-linearization technique (RLT), introduced in [W.P. Adams, H.D. Sher-ali, A tight linearization and an algorithm for zero-one quadratic programming problems, Management Science, 32(10):1274{1290, 1986], provides a way to compute linear program-ming bounds on the optimal values of NP-hard combinatorial optimization problems. In this paper we show that, in the presence of suitable algebraic symmetry in the original problem data, it is sometimes possible to compute level two RLT bounds with additional linear matrix inequality constraints. As an illustration of our methodology, we compute the best-known bounds for certain graph partitioning problems on strongly regular graphs

Commutative association schemes

Association schemes were originally introduced by Bose and his co-workers in the design of statistical experiments. Since that point of inception, the concept has proved useful in the study of group actions, in algebraic graph theory, in algebraic coding theory, and in areas as far afield as knot theory and numerical integration. This branch of the theory, viewed in this collection of surveys as the "commutative case," has seen significant activity in the last few decades. The goal of the present survey is to discuss the most important new developments in several directions, including Gelfand pairs, cometric association schemes, Delsarte Theory, spin models and the semidefinite programming technique. The narrative follows a thread through this list of topics, this being the contrast between combinatorial symmetry and group-theoretic symmetry, culminating in Schrijver's SDP bound for binary codes (based on group actions) and its connection to the Terwilliger algebra (based on combinatorial symmetry). We propose this new role of the Terwilliger algebra in Delsarte Theory as a central topic for future work.Comment: 36 page

Positive discrepancy, MaxCut, and eigenvalues of graphs

The positive discrepancy of a graph $G$ of edge density $p=e(G)/\binom{v(G)}{2}$ is defined as \mbox{disc}^{+}(G)=\max_{U\subset V(G)}e(G[U])-p\binom{|U|}{2}. In 1993, Alon proved (using the equivalent terminology of minimum bisections) that if $G$ is $d$-regular on $n$ vertices, and $d=O(n^{1/9})$, then \mbox{disc}^{+}(G)=\Omega(d^{1/2}n). We greatly extend this by showing that if $G$ has average degree $d$, then \mbox{disc}^{+}(G)=\Omega(d^{\frac{1}{2}}n) if $d\in [0,n^{\frac{2}{3}}]$, $\Omega(n^2/d)$ if $d\in [n^{\frac{2}{3}},n^{\frac{4}{5}}]$, and $\Omega(d^{\frac{1}{4}}n/\log n)$ if $d\in \left[n^{\frac{4}{5}},(\frac{1}{2}-\varepsilon)n\right]$. These bounds are best possible if $d\ll n^{3/4}$, and the complete bipartite graph shows that \mbox{disc}^{+}(G)=\Omega(n) cannot be improved if $d\approx n/2$. Our proofs are based on semidefinite programming and linear algebraic techniques. An interesting corollary of our results is that every $d$-regular graph on $n$ vertices with ${\frac{1}{2}+\varepsilon\leq \frac{d}{n}\leq 1-\varepsilon}$ has a cut of size $\frac{nd}{4}+\Omega(n^{5/4}/\log n)$. This is not necessarily true without the assumption of regularity, or the bounds on $d$. The positive discrepancy of regular graphs is controlled by the second eigenvalue $\lambda_2$, as \mbox{disc}^{+}(G)\leq \frac{\lambda_2}{2} n+d. As a byproduct of our arguments, we present lower bounds on $\lambda_2$ for regular graphs, extending the celebrated Alon-Boppana theorem in the dense regime.Comment: 29 page

Minimizing the number of independent sets in triangle-free regular graphs

Recently, Davies, Jenssen, Perkins, and Roberts gave a very nice proof of the result (due, in various parts, to Kahn, Galvin-Tetali, and Zhao) that the independence polynomial of a $d$-regular graph is maximized by disjoint copies of $K_{d,d}$. Their proof uses linear programming bounds on the distribution of a cleverly chosen random variable. In this paper, we use this method to give lower bounds on the independence polynomial of regular graphs. We also give new bounds on the number of independent sets in triangle-free regular graphs

Lecture notes: Semidefinite programs and harmonic analysis

Lecture notes for the tutorial at the workshop HPOPT 2008 - 10th International Workshop on High Performance Optimization Techniques (Algebraic Structure in Semidefinite Programming), June 11th to 13th, 2008, Tilburg University, The Netherlands.Comment: 31 page