Approximation algorithms for NP-hard problems
Approximation algorithms for NP-hard problems
On paths avoiding forbidden pairs of vertices in a graph
Discrete Applied Mathematics
Some optimal inapproximability results
Journal of the ACM (JACM)
The importance of being biased
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Approximate Max-Flow Min-(Multi)Cut Theorems and Their Applications
SIAM Journal on Computing
The complexity of satisfiability problems
STOC '78 Proceedings of the tenth annual ACM symposium on Theory of computing
Graph decomposition and a greedy algorithm for edge-disjoint paths
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
On the Unique Games Conjecture
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
ON THE HARDNESS OF APPROXIMATING MULTICUT AND SPARSEST-CUT
Computational Complexity
On Two Problems in the Generation of Program Test Paths
IEEE Transactions on Software Engineering
Finding Paths and Cycles of Superpolylogarithmic Length
SIAM Journal on Computing
On a class of totally unimodular matrices
SFCS '80 Proceedings of the 21st Annual Symposium on Foundations of Computer Science
On the complexity of paths avoiding forbidden pairs
Discrete Applied Mathematics
Combination Can Be Hard: Approximability of the Unique Coverage Problem
SIAM Journal on Computing
Interference in cellular networks: the minimum membership set cover problem
COCOON'05 Proceedings of the 11th annual international conference on Computing and Combinatorics
Complexity of the path avoiding forbidden pairs problem revisited
Discrete Applied Mathematics
Hi-index | 0.00 |
In this paper we consider the checkpoint problem. The input consists of an undirected graph G, a set of source-destination pairs {(s1, t1), ..., (sk, tk)}, and a collection P of paths connecting the (si, ti) pairs. A feasible solution is a multicut E′; namely, a set of edges whose removal disconnects every source-destination pair. For each p ∈ P we define cpE′(p) = |p ∩ E′|. In the sum checkpoint (SCP) problem the goal is to minimize Σp∈P cpE′(p), while in the maximum checkpoint (MCP) problem the goal is to minimize maxp∈p cpE′(p). These problem have several natural applications, e.g., in urban transportation and network security. In a sense, they combine the multicut problem and the minimum membership set cover problem. For the sum objective we show that weighted SCP is equivalent, with respect to approximability, to undirected multicut. Thus there exists an O(log n) approximation for SCP in general graphs. Our current approximability results for the max objective have a wide gap: we provide an approximation factor of O(√nlogn/opt) for MCP and a hardness of 2 under the assumption P ≠ NP. The hardness holds for trees, in which case we can obtain an asymptotic approximation factor of 2. Finally we show strong hardness for the well-known problem of finding a path with minimum forbidden pairs, which in a sense can be considered the dual to the checkpoint problem. Despite various works on this problem, hardness of approximation was not known prior to this work. We show that the problem cannot be approximated within cn for some constant c 0, unless P = NP. This is the strongest type of hardness possible. It carries over to directed acyclic graphs and is a huge improvement over the plain NP-hardness of Gabow (SIAM J. Comp 2007, pages 1648-1671).