TY - GEN
T1 - Knapsack
T2 - 16th Latin American Symposium on Theoretical Informatics, LATIN 2042
AU - Dey, Palash
AU - Kolay, Sudeshna
AU - Singh, Sipra
N1 - Publisher Copyright:
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024.
PY - 2024/1/1
Y1 - 2024/1/1
N2 - We study the Knapsack problem with graph-theoretic constraints. That is, there exists a graph structure on the input set of items of Knapsack and the solution also needs to satisfy certain graph theoretic properties on top of the Knapsack constraints. In particular, we study Connected Knapsack where the solution must be a connected subset of items which has maximum value and satisfies the size constraint of the knapsack. We show that this problem is strongly NP-complete even for graphs of maximum degree four and NP-complete even for star graphs. On the other hand, we develop an algorithm running in time O2O(twlogtw)·poly(n)min{s2,d2} where tw,s,d,n are respectively treewidth of the graph, the size of the knapsack, the target value of the knapsack, and the number of items. We also exhibit a (1-ε) factor approximation algorithm running in time O2O(twlogtw)·poly(n,1/ε) for every ε>0. We show similar results for Path Knapsack and Shortest Path Knapsack, where the solution must also induce a path and shortest path, respectively. Our results suggest that Connected Knapsack is computationally the hardest, followed by Path Knapsack and then Shortest Path Knapsack.
AB - We study the Knapsack problem with graph-theoretic constraints. That is, there exists a graph structure on the input set of items of Knapsack and the solution also needs to satisfy certain graph theoretic properties on top of the Knapsack constraints. In particular, we study Connected Knapsack where the solution must be a connected subset of items which has maximum value and satisfies the size constraint of the knapsack. We show that this problem is strongly NP-complete even for graphs of maximum degree four and NP-complete even for star graphs. On the other hand, we develop an algorithm running in time O2O(twlogtw)·poly(n)min{s2,d2} where tw,s,d,n are respectively treewidth of the graph, the size of the knapsack, the target value of the knapsack, and the number of items. We also exhibit a (1-ε) factor approximation algorithm running in time O2O(twlogtw)·poly(n,1/ε) for every ε>0. We show similar results for Path Knapsack and Shortest Path Knapsack, where the solution must also induce a path and shortest path, respectively. Our results suggest that Connected Knapsack is computationally the hardest, followed by Path Knapsack and then Shortest Path Knapsack.
KW - Approximation algorithm
KW - Graph Algorithms
KW - Knapsack
KW - Parameterised Complexity
UR - http://www.scopus.com/inward/record.url?scp=85188689631&partnerID=8YFLogxK
U2 - 10.1007/978-3-031-55601-2_11
DO - 10.1007/978-3-031-55601-2_11
M3 - Conference contribution
AN - SCOPUS:85188689631
SN - 9783031556005
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 162
EP - 176
BT - LATIN 2024
A2 - Soto, José A.
A2 - Wiese, Andreas
PB - Springer Science and Business Media Deutschland GmbH
Y2 - 18 March 2024 through 22 March 2024
ER -