TY - GEN

T1 - 0/1/all CSPs, half-integral A-path packing, and linear-time FPT algorithms

AU - Iwata, Yoichi

AU - Yamaguchi, Yutaro

AU - Yoshida, Yuichi

N1 - Funding Information:
ACKNOWLEDGMENT We would like to thank Yusuke Kobayashi for pointing out to us the simulation of two-fan constraints by permutation constraints. Yoichi Iwata is supported by JSPS KAKENHI Grant Number JP17K12643. YutaroYamaguchiis supported by JSPS KAKENHI Grant Number JP16H06931 and JST ACT-I Grant Number JPMJPR16UR. Yuichi Yoshida is supported by JST ERATO Grant Number JPMJER1305 and JSPS KAKENHI Grant Number JP17H04676.

PY - 2018/11/30

Y1 - 2018/11/30

N2 - A recent trend in the design of FPT algorithms is exploiting the half-integrality of LP relaxations. In other words, starting with a half-integral optimal solution to an LP relaxation, we assign integral values to variables one-by-one by branch and bound. This technique is general and the resulting time complexity has a low dependency on the parameter. However, the time complexity often becomes a large polynomial in the input size because we need to compute half-integral optimal LP solutions. In this paper, we address this issue by providing an O(km)-time algorithm for solving the LPs arising from various FPT problems, where k is the optimal value and m is the number of edges/constraints. Our algorithm is based on interesting connections among 0/1/all constraints, which has been studied in the field of constraints satisfaction, A-path packing, which has been studied in the field of combinatorial optimization, and the LPs used in FPT algorithms. With the aid of this algorithm, we obtain linear-time FPT algorithms for various problems. The obtained running time for each problem is linear in the input size and has the current smallest dependency on the parameter. Most importantly, instead of using problem-specific approaches, we obtain all of these results by a unified approach, i.e., the branch-and-bound framework combined with the efficient computation of half-integral LPs, which demonstrates its generality.

AB - A recent trend in the design of FPT algorithms is exploiting the half-integrality of LP relaxations. In other words, starting with a half-integral optimal solution to an LP relaxation, we assign integral values to variables one-by-one by branch and bound. This technique is general and the resulting time complexity has a low dependency on the parameter. However, the time complexity often becomes a large polynomial in the input size because we need to compute half-integral optimal LP solutions. In this paper, we address this issue by providing an O(km)-time algorithm for solving the LPs arising from various FPT problems, where k is the optimal value and m is the number of edges/constraints. Our algorithm is based on interesting connections among 0/1/all constraints, which has been studied in the field of constraints satisfaction, A-path packing, which has been studied in the field of combinatorial optimization, and the LPs used in FPT algorithms. With the aid of this algorithm, we obtain linear-time FPT algorithms for various problems. The obtained running time for each problem is linear in the input size and has the current smallest dependency on the parameter. Most importantly, instead of using problem-specific approaches, we obtain all of these results by a unified approach, i.e., the branch-and-bound framework combined with the efficient computation of half-integral LPs, which demonstrates its generality.

UR - http://www.scopus.com/inward/record.url?scp=85059817724&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85059817724&partnerID=8YFLogxK

U2 - 10.1109/FOCS.2018.00051

DO - 10.1109/FOCS.2018.00051

M3 - Conference contribution

AN - SCOPUS:85059817724

T3 - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

SP - 462

EP - 473

BT - Proceedings - 59th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2018

A2 - Thorup, Mikkel

PB - IEEE Computer Society

T2 - 59th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2018

Y2 - 6 October 2018 through 8 October 2018

ER -