Adjacency labeling schemes and induced-universal graphs

We describe a way of assigning labels to the vertices of any undirected graph on up to $n$ vertices, each composed of $n/2+O(1)$ bits, such that given the labels of two vertices, and no other information regarding the graph, it is possible to decide whether or not the vertices are adjacent in the graph. This is optimal, up to an additive constant, and constitutes the first improvement in almost 50 years of an $n/2+O(\log n)$ bound of Moon. As a consequence, we obtain an induced-universal graph for $n$-vertex graphs containing only $O(2^{n/2})$ vertices, which is optimal up to a multiplicative constant, solving an open problem of Vizing from 1968. We obtain similar tight results for directed graphs, tournaments and bipartite graphs

Gating of memory encoding of time-delayed cross-frequency MEG networks revealed by graph filtration based on persistent homology

To explain gating of memory encoding, magnetoencephalography (MEG) was analyzed over multi-regional network of negative correlations between alpha band power during cue (cue-alpha) and gamma band power during item presentation (item-gamma) in Remember (R) and No-remember (NR) condition. Persistent homology with graph filtration on alpha-gamma correlation disclosed topological invariants to explain memory gating. Instruction compliance (R-hits minus NR-hits) was significantly related to negative coupling between the left superior occipital (cue-alpha) and the left dorsolateral superior frontal gyri (item-gamma) on permutation test, where the coupling was stronger in R than NR. In good memory performers (R-hits minus false alarm), the coupling was stronger in R than NR between the right posterior cingulate (cue-alpha) and the left fusiform gyri (item-gamma). Gating of memory encoding was dictated by inter-regional negative alpha-gamma coupling. Our graph filtration over MEG network revealed these inter-regional time-delayed cross-frequency connectivity serve gating of memory encoding

Inapproximability for Antiferromagnetic Spin Systems in the Tree Non-Uniqueness Region

A remarkable connection has been established for antiferromagnetic 2-spin systems, including the Ising and hard-core models, showing that the computational complexity of approximating the partition function for graphs with maximum degree D undergoes a phase transition that coincides with the statistical physics uniqueness/non-uniqueness phase transition on the infinite D-regular tree. Despite this clear picture for 2-spin systems, there is little known for multi-spin systems. We present the first analog of the above inapproximability results for multi-spin systems. The main difficulty in previous inapproximability results was analyzing the behavior of the model on random D-regular bipartite graphs, which served as the gadget in the reduction. To this end one needs to understand the moments of the partition function. Our key contribution is connecting: (i) induced matrix norms, (ii) maxima of the expectation of the partition function, and (iii) attractive fixed points of the associated tree recursions (belief propagation). The view through matrix norms allows a simple and generic analysis of the second moment for any spin system on random D-regular bipartite graphs. This yields concentration results for any spin system in which one can analyze the maxima of the first moment. The connection to fixed points of the tree recursions enables an analysis of the maxima of the first moment for specific models of interest. For k-colorings we prove that for even k, in the tree non-uniqueness region (which corresponds to k<D) it is NP-hard, unless NP=RP, to approximate the number of colorings for triangle-free D-regular graphs. Our proof extends to the antiferromagnetic Potts model, and, in fact, to every antiferromagnetic model under a mild condition

On Iterated Dominance, Matrix Elimination, and Matched Paths

We study computational problems arising from the iterated removal of weakly dominated actions in anonymous games. Our main result shows that it is NP-complete to decide whether an anonymous game with three actions can be solved via iterated weak dominance. The two-action case can be reformulated as a natural elimination problem on a matrix, the complexity of which turns out to be surprisingly difficult to characterize and ultimately remains open. We however establish connections to a matching problem along paths in a directed graph, which is computationally hard in general but can also be used to identify tractable cases of matrix elimination. We finally identify different classes of anonymous games where iterated dominance is in P and NP-complete, respectively.Comment: 12 pages, 3 figures, 27th International Symposium on Theoretical Aspects of Computer Science (STACS

L(p,q)L(p,q)-Labeling of Graphs with Interval Representations

We provide upper bounds on the $L(p,q)$-labeling number of graphs which have interval (or circular-arc) representations via simple greedy algorithms. We prove that there exists an $L(p,q)$-labeling with span at most $\max\{2(p+q-1)\Delta-4q+2, (2p-1)\mu+(2q-1)\Delta-2q+1\}$ for interval $k$-graphs, $\max\{p,q\}\Delta$ for interval graphs, $\max\{p,q\}\Delta+p\omega$ for circular arc graphs, $2(p+q-1)\Delta-2q+1$ for permutation graphs and $(2p-1)\Delta+(2q-1)(\mu-1)$ for cointerval graphs. In particular, these improve existing bounds on $L(p,q)$-labeling of interval and circular arc graphs and $L(2,1)$-labeling of permutation graphs. Furthermore, we provide upper bounds on the coloring of the squares of aforementioned classes