Faster Graph Coloring in Polynomial Space

We present a polynomial-space algorithm that computes the number independent sets of any input graph in time $O(1.1387^n)$ for graphs with maximum degree 3 and in time $O(1.2355^n)$ for general graphs, where n is the number of vertices. Together with the inclusion-exclusion approach of Bj\"orklund, Husfeldt, and Koivisto [SIAM J. Comput. 2009], this leads to a faster polynomial-space algorithm for the graph coloring problem with running time $O(2.2355^n)$. As a byproduct, we also obtain an exponential-space $O(1.2330^n)$ time algorithm for counting independent sets. Our main algorithm counts independent sets in graphs with maximum degree 3 and no vertex with three neighbors of degree 3. This polynomial-space algorithm is analyzed using the recently introduced Separate, Measure and Conquer approach [Gaspers & Sorkin, ICALP 2015]. Using Wahlstr\"om's compound measure approach, this improvement in running time for small degree graphs is then bootstrapped to larger degrees, giving the improvement for general graphs. Combining both approaches leads to some inflexibility in choosing vertices to branch on for the small-degree cases, which we counter by structural graph properties

Crossing the Logarithmic Barrier for Dynamic Boolean Data Structure Lower Bounds

This paper proves the first super-logarithmic lower bounds on the cell probe complexity of dynamic boolean (a.k.a. decision) data structure problems, a long-standing milestone in data structure lower bounds. We introduce a new method for proving dynamic cell probe lower bounds and use it to prove a $\tilde{\Omega}(\log^{1.5} n)$ lower bound on the operational time of a wide range of boolean data structure problems, most notably, on the query time of dynamic range counting over $\mathbb{F}_2$ ([Pat07]). Proving an $\omega(\lg n)$ lower bound for this problem was explicitly posed as one of five important open problems in the late Mihai P\v{a}tra\c{s}cu's obituary [Tho13]. This result also implies the first $\omega(\lg n)$ lower bound for the classical 2D range counting problem, one of the most fundamental data structure problems in computational geometry and spatial databases. We derive similar lower bounds for boolean versions of dynamic polynomial evaluation and 2D rectangle stabbing, and for the (non-boolean) problems of range selection and range median. Our technical centerpiece is a new way of "weakly" simulating dynamic data structures using efficient one-way communication protocols with small advantage over random guessing. This simulation involves a surprising excursion to low-degree (Chebychev) polynomials which may be of independent interest, and offers an entirely new algorithmic angle on the "cell sampling" method of Panigrahy et al. [PTW10]