In this paper, we study the integrality gap of the Knapsack linear program in the SheraliAdams and Lasserre hierarchies. First, we show that an integrality gap of 2 − persists up to a linear number of rounds of Sherali-Adams, despite the fact that Knapsack admits a fully polynomial time approximation scheme [27,33]. Second, we show that the Lasserre hierarchy closes the gap quickly. Specifically, after t rounds of Lasserre, the integrality gap decreases to t/(t − 1). This answers the open question in [10]. Also, to the best of our knowledge, this is the first positive result that uses more than a small number of rounds in the Lasserre hierarchy. Our proof uses a decomposition theorem for the Lasserre hierarchy, which may be of independent interest.

Abstract. This paper studies the algorithmic issues of the spanning star forest problem. We prove the following results: (1) There is a polynomial-time approximation scheme for planar graphs; (2) there is a polynomial-time -approximation algorithm for graphs; (3) it is NP-hard to approximate the problem within ratio 259 260 + for graphs; (4) there is a linear-time algorithm to compute the maximum star forest of a weighted tree; (5) there is a polynomial-time -approximation algorithm for weighted graphs. We also show how to apply this spanning star forest model to aligning multiple genomic sequences over a tandem duplication region.

No abstract

A star graph is a tree of diameter at most two. A star forest is a graph that consists of node-disjoint star graphs. In the spanning star forest problem, given an unweighted graph G, the objective is to find a star forest that contains all the vertices of G and has the maximum number of edges. This problem is the complement of the dominating set problem in the following sense: On a graph with n vertices, the size of the maximum spanning star forest is equal to n minus the size of the minimum dominating set. We present a 0.71-approximation algorithm for this problem, improving upon the approximation factor of 0.6 of Nguyen et al. [9]. We also present a 0.64-approximation algorithm for the problem on node-weighted graphs. Finally, we present improved hardness of approximation results for the weighted versions of the problem.