Petteri Kaski

Exact exponential algorithms

Discovering surprises in the face of intractability.

Fourier meets möbius: fast subset convolution

We present a fast algorithm for the subset convolution problem:given functions <i>f</i> and <i>g</i> defined on the lattice of subsets of an<i>n</i>-element set <i>n</i>, compute their subset convolution f*g, defined for S⊆ N by [ (f * g)(S) = [T ⊆ S] f(T) g(S/T),,]where addition and multiplication is carried out in an arbitrary ring. Via Möbius transform and inversion, our algorithm evaluates the subset convolution in O(n² 2ⁿ) additions and multiplications, substanti y improving upon the straightforward O(3ⁿ) algorithm. Specifically, if the input functions have aninteger range [-M,-M+1,...,M], their subset convolution over the ordinary sum--product ring can be computed in Õ(2ⁿ log M) time; the notation Õ suppresses polylogarithmic factors.Furthermore, using a standard embedding technique we can compute the subset convolution over the max--sum or min--sum semiring in Õ(2ⁿ M) time. To demonstrate the applicability of fast subset convolution, wepresent the first Õ(2^k n² + n m) algorithm for the Steiner tree problem in graphs with <i>n</i> vertices, <i>k</i> terminals, and <i>m</i> edges with bounded integer weights, improving upon the Õ(3^kn + 2^k n² + n m) time bound of the classical Dreyfus-Wagner algorithm. We also discuss extensions to recent Õ(2ⁿ)-time algorithms for covering and partitioning problems (Björklund and Husfeldt, FOCS 2006; Koivisto, FOCS 2006).

The Travelling Salesman Problem in Bounded Degree Graphs

We show that the travelling salesman problem in bounded-degreegraphs can be solved in time

Computing the Tutte Polynomial in Vertex-Exponential Time

O\bigl((2-\epsilon)^n\bigr)

Algebraic Methods in the Congested Clique

, wheree> 0 depends only on the degree bound but noton the number of cities, n. The algorithm is a variant ofthe classical dynamic programming solution due to Bellman, and,independently, Held and Karp. In the case of bounded integerweights on the edges, we also present a polynomial-space algorithmwith running time

The Steiner triple systems of order 19

O\bigl((2-\epsilon)^n\bigr)

Algorithm Theory – SWAT 2012

on bounded-degreegraphs.

Balanced Data Gathering in Energy-Constrained Sensor Networks

The deletion-contraction algorithm is perhaps the most popular method for computing a host of fundamental graph invariants such as the chromatic, flow, and reliability polynomials in graph theory, the Jones polynomial of an alternating link in knot theory, and the partition functions of the models of Ising, Potts, and Fortuin-Kasteleyn in statistical physics. Prior to this work, deletion-contraction was also the fastest known general-purpose algorithm for these invariants, running in time roughly proportional to the number of spanning trees in the input graph.Here, we give a substantially faster algorithm that computes the Tutte polynomial-and hence, all the aforementioned invariants and more-of an arbitrary graph in time within a polynomial factor of the number of connected vertex sets. The algorithm actually evaluates a multivariate generalization of the Tutte polynomial by making use of an identity due to Fortuin and Kasteleyn. We also provide a polynomial-space variant of the algorithm and give an analogous result for Chung and Grahams cover polynomial.

Almost Stable Matchings by Truncating the Gale–Shapley Algorithm

In this work, we use algebraic methods for studying distance computation and subgraph detection tasks in the congested clique model. Specifically, we adapt parallel matrix multiplication implementations to the congested clique, obtaining an O(n1-2/ω) round matrix multiplication algorithm, where ω < 2.3728639 is the exponent of matrix multiplication. In conjunction with known techniques from centralised algorithmics, this gives significant improvements over previous best upper bounds in the congested clique model. The highlight results include: triangle and 4-cycle counting in O(n0.158) rounds, improving upon the O(n1/3) triangle counting algorithm of Dolev et al. [DISC 2012], a (1 + o(1))-approximation of all-pairs shortest paths in O(n0.158) rounds, improving upon the ~O (n1/2)-round (2 + o(1))-approximation algorithm of Nanongkai [STOC 2014], and computing the girth in O(n0.158) rounds, which is the first non-trivial solution in this model. In addition, we present a novel constant-round combinatorial algorithm for detecting 4-cycles.

Counting paths and packings in halves

Using an orderly algorithm, the Steiner triple systems of order 19 are classied; there are 11;084;874;829 pairwise nonisomorphic such designs. For each design, the order of its automorphism group and the number of Pasch congurations it contains are recorded; 2;591 of the designs are anti-Pasch. There are three main parts of the classication: constructing an initial set of blocks, the seeds; completing the seeds to triple systems with an algorithm for exact cover; and carrying out isomorph rejection of the nal triple systems. Isomorph rejection is based on the graph canonical labeling software nauty supplemented with a vertex invariant based on Pasch congurations. The possibility of using the (strongly regular) block graphs of these designs in the isomorphism tests is utilized. The aforementioned value is in fact a lower bound on the number of pairwise nonisomorphic strongly regular graphs with parameters (57; 24; 11; 9).