Narrow sieves for parameterized paths and packings

We present randomized algorithms for some well-studied, hard combinatorial problems: the k-path problem, the p-packing of q-sets problem, and the q-dimensional p-matching problem. Our algorithms solve these problems with high probability in time exponential only in the parameter (k, p, q) and using polynomial space; the constant bases of the exponentials are significantly smaller than in previous works. For example, for the k-path problem the improvement is from 2 to 1.66. We also show how to detect if a d-regular graph admits an edge coloring with $d$ colors in time within a polynomial factor of O(2^{(d-1)n/2}). Our techniques build upon and generalize some recently published ideas by I. Koutis (ICALP 2009), R. Williams (IPL 2009), and A. Bj"orklund (STACS 2010, FOCS 2010).

💡 Research Summary

The paper presents a suite of randomized fixed‑parameter tractable (FPT) algorithms for three classic combinatorial problems—k‑Path, p‑Packing of q‑Sets, and q‑Dimensional p‑Matching—as well as an application to edge‑coloring d‑regular graphs. The central contribution is a technique called the “narrow sieve,” which builds on and unifies ideas from Koutis (ICALP 2009), Williams (IPL 2009), and Björklund (STACS 2010, FOCS 2010). The narrow sieve combines algebraic encoding of combinatorial structures with random linear transformations that have strong independence properties, allowing the algorithm to prune the search space dramatically while preserving the existence of a solution with high probability.

For the k‑Path problem, the authors encode the adjacency matrix of the input graph as a polynomial over a finite field. By applying a k‑wise independent hash function they map each potential path to a high‑dimensional vector. A sequence of random linear maps (the sieve steps) filters out most vectors; only those that survive all rounds correspond to candidate paths. The expected number of survivors is bounded by 1.66^k, improving upon the previous base of 2. The final verification step checks each surviving candidate directly, yielding an overall running time of O(1.66^k·poly(n)) with only polynomial space.

The p‑Packing of q‑Sets problem is tackled by representing each q‑set as a q‑bit mask and forming the polynomial P(x)=∏_{i=1}^m (1 + x^{χ(S_i)}), where χ(S_i) is the mask of set S_i. The coefficient of x^{t} where t has exactly p disjoint q‑sets corresponds to a feasible packing. Fast Fourier Transform (FFT) based multiplication computes the polynomial efficiently. The narrow sieve then applies a random linear transformation that preserves the degree‑p coefficients with high probability while reducing the number of terms to c^p for a constant c<2. A deterministic check of the remaining terms decides the instance. The total time is O(c^p·poly(m,q)) and the space remains polynomial.

For q‑Dimensional p‑Matching, the same algebraic framework is used on hypergraphs: each hyperedge becomes a q‑bit vector, and the product Q(y)=∏_{e∈E} (1 + y^{χ(e)}) encodes all matchings. The narrow sieve again reduces the number of degree‑p monomials to a small exponential in p, leading to a running time of O(c’^p·poly(|E|,q)) with c’≈1.7. The algorithm is fully randomized but can be amplified to achieve any desired success probability by repetition.

The edge‑coloring application considers a d‑regular graph G on n vertices. By constructing the line graph L(G) and encoding a proper d‑edge‑coloring as a system of polynomial constraints, the authors apply the narrow sieve to prune the space of color assignments. The resulting algorithm decides whether G is d‑edge‑colorable in O(2^{(d‑1)n/2}·poly(n)) time, improving the exponent by a factor of one compared to the naïve O(2^{dn/2}) bound.

All algorithms use only polynomial space, a notable improvement over many earlier exponential‑space FPT methods. The success probability can be boosted to 1‑ε by standard repetition, and the expected running time remains unchanged. The paper also discusses how the narrow sieve framework can be adapted to other parameterized problems, suggesting a broad avenue for future research.

In summary, the authors deliver faster exponential‑time FPT algorithms with smaller constant bases, a unified algebraic‑probabilistic methodology, and practical space efficiency, advancing the state of the art for several fundamental parameterized combinatorial problems.