Families with Infants: A General Approach to Solve Hard Partition Problems

Abstract

We introduce a general approach for solving partition problems where the goal is to represent a given set as a union (either disjoint or not) of subsets satisfying certain properties. Many NPhard problems can be naturally stated as such partition problems. We show that if one can find a large enough system of so-called families with infants for a given problem, then this problem can be solved faster than by a straightforward algorithm. We use this approach to improve known bounds for several NP-hard problems as well as to simplify the proofs of several known results.For the chromatic number problem we present an algorithm with O * ((2 − ε(d)) n ) time and exponential space for graphs of average degree d.

Faster Exponential-Time Algorithms in Graphs of Bounded Average Degree

“…Very recently Golovnev et al [20] proved that the dynamic programming algorithms of Theorems 2 and 3 can be turned into polynomial-space algorithms by using algebraic techniques. Moreover, they also showed how to use the gap obtained in Lemma 10 to compute the chromatic number of a graph of average degree d with exponential speedup over the O (2 n )-time algorithm [21].…”

Exact Algorithms on Graphs of Bounded Average Degree

“…In 2008, Björklund, Husfeldt, Kaski, and Koivisto [5,6] observed that such an improvement can be made if we restrict ourselves to bounded degree graphs. Further work of Cygan and Pilipczuk [8] and Golovnev, Kulikov, and Mihajlin [9] extended these results to graphs of bounded average degree.…”

“…In a subsequent work, Golovnev, Kulikov, and Mihajlin [9] showed how to use the aforementioned multiplicative gap of˛to obtain a .2 " d / n n O.1/ -time algorithm for computing the chromatic number of a graph with average degree bounded by d . Furthermore, they expressed all previous algorithms as the task of determining one coefficient in a carefully chosen polynomial, obtaining polynomial space complexity without any significant loss in time complexity.…”

Families with infants: speeding up algorithms for NP-hard problems using FFT

“…We use this approach to improve known bounds for several NP-hard problems (the traveling salesman problem, the graph coloring problem, the problem of counting perfect matchings) on graphs of bounded average degree, as well as to simplify the proofs of several known results. * This paper is based on the same results as [15], but the presentation of the results and the whole discuss have been reworked substantially. Research is partially supported by the Government of the Russian Federation (grant 14.Z50.31.0030).…”