TY - GEN

T1 - A space-efficient parameterized algorithm for the hamiltonian cycle problem by dynamic algebraization

AU - Belbasi, Mahdi

AU - Fürer, Martin

N1 - Publisher Copyright:
© Springer Nature Switzerland AG 2019.

PY - 2019

Y1 - 2019

N2 - An NP-hard graph problem may be intractable for general graphs but it could be efficiently solvable using dynamic programming for graphs with bounded treewidth. Employing dynamic programming on a tree decomposition usually uses exponential space. In 2010, Lokshtanov and Nederlof introduced an elegant framework to avoid exponential space by algebraization. Later, Fürer and Yu modified the framework in a way that even works when the underlying set is dynamic, thus applying it to tree decompositions. In this work, we design space-efficient algorithms to count the number of Hamiltonian cycles and furthermore solve the Traveling Salesman problem, using polynomial space while the time complexity is only slightly increased. This might be inevitable since we are reducing the space usage from an exponential amount (in dynamic programming solutions) to polynomial. We give an algorithm to count the number of Hamiltonian cycles in time (formula presented) using (formula presented) space, where M(r) is the time complexity to multiply two integers, each of which being represented by at most r bits. Then, we solve the more general Traveling Salesman problem in time (formula presented) using space O(Wkdnlog n), where k and d are the width and the depth of the given tree decomposition and W is the sum of weights. Furthermore, this algorithm counts the number of Hamiltonian Cycles.

AB - An NP-hard graph problem may be intractable for general graphs but it could be efficiently solvable using dynamic programming for graphs with bounded treewidth. Employing dynamic programming on a tree decomposition usually uses exponential space. In 2010, Lokshtanov and Nederlof introduced an elegant framework to avoid exponential space by algebraization. Later, Fürer and Yu modified the framework in a way that even works when the underlying set is dynamic, thus applying it to tree decompositions. In this work, we design space-efficient algorithms to count the number of Hamiltonian cycles and furthermore solve the Traveling Salesman problem, using polynomial space while the time complexity is only slightly increased. This might be inevitable since we are reducing the space usage from an exponential amount (in dynamic programming solutions) to polynomial. We give an algorithm to count the number of Hamiltonian cycles in time (formula presented) using (formula presented) space, where M(r) is the time complexity to multiply two integers, each of which being represented by at most r bits. Then, we solve the more general Traveling Salesman problem in time (formula presented) using space O(Wkdnlog n), where k and d are the width and the depth of the given tree decomposition and W is the sum of weights. Furthermore, this algorithm counts the number of Hamiltonian Cycles.

UR - http://www.scopus.com/inward/record.url?scp=85068601443&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85068601443&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-19955-5_4

DO - 10.1007/978-3-030-19955-5_4

M3 - Conference contribution

AN - SCOPUS:85068601443

SN - 9783030199548

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 38

EP - 49

BT - Computer Science – Theory and Applications - 14th International Computer Science Symposium in Russia, CSR 2019, Proceedings

A2 - van Bevern, René

A2 - Kucherov, Gregory

PB - Springer Verlag

T2 - 14th International Computer Science Symposium in Russia, CSR 2019

Y2 - 1 July 2019 through 5 July 2019

ER -