We consider the capacitated k-center problem. In this problem we are given a finite set of locations in a metric space and each location has an associated non-negative integer capacity. The goal is to choose (open) k locations (called centers) and assign each location to an open center to minimize the maximum, over all locations, of the distance of the location to its assigned center. The number of locations assigned to a center cannot exceed the center's capacity. The uncapacitated k-center problem has a simple tight 2-approximation from the 80's. In contrast, the first constant factor approximation for the capacitated problem was obtained only recently by Cygan, Hajiaghayi and Khuller who gave an intricate LP-rounding algorithm that achieves an approximation guarantee in the hundreds. In this paper we give a simple algorithm with a clean analysis and prove an approximation guarantee of 9. It uses the standard LP relaxation and comes close to settling the integrality gap (after necessary preprocessing), which is narrowed down to either 7, 8 or 9. The algorithm proceeds by first reducing to special tree instances, and then uses our best-possible algorithm to solve such instances. Our concept of tree instances is versatile and applies to natural variants of the capacitated k-center problem for which we also obtain improved algorithms. Finally, we give evidence to show that more powerful preprocessing could lead to better algorithms, by giving an approximation algorithm that beats the integrality gap for instances where all non-zero capacities are the same.
We investigate a model-theoretic property that generalizes the classical notion of "preservation under substructures". We call this property preservation under substructures modulo bounded cores, and present a syntactic characterization via Σ 0 2 sentences for properties of arbitrary structures definable by FO sentences. As a sharper characterization, we further show that the count of existential quantifiers in the Σ 0 2 sentence equals the size of the smallest bounded core. We also present our results on the sharper characterization for special fragments of FO and also over special classes of structures. We present a (not FO-definable) class of finite structures for which the sharper characterization fails, but for which the classical Łoś-Tarski preservation theorem holds. As a fallout of our studies, we obtain combinatorial proofs of the Łoś-Tarski theorem for some of the aforementioned cases.
We study the multiway cut problem in directed graphs and one of its special cases, the node-weighted multiway cut problem in undirected graphs. In DIRECTED MULTIWAY CUT (DIR-MC) the input is an edge-weighted directed graph G = (V, E) and a set of k terminal nodes {s1, s2, . . . , s k } ⊆ V ; the goal is to find a min-weight subset of edges whose removal ensures that there is no path from si to sj for any i = j. In NODE-WEIGHTED MULTIWAY CUT (NODE-WT-MC) the input is a node-weighted undirected graph G and a set of k terminal nodes {s1, s2, . . . , s k } ⊆ V ; the goal is to find a min-weight subset of nodes whose removal ensures that there is no path from si to sj for any i = j. DIR-MC admits a 2-approximation [28] and NODE-WT-MC admits a 2(1 −, both via rounding of LP relaxations. Previous rounding algorithms for these problems, from nearly twenty years ago, are based on careful rounding of an optimum solution to an LP relaxation. This is particularly true for DIR-MC for which the rounding relies on a custom LP formulation instead of the natural distance based LP relaxation [28]. In this paper we describe extremely simple and near linear-time rounding algorithms for DIR-MC and NODE-WT-MC via a natural distance based LP relaxation. The dual of this relaxation is a special case of the maximum multicommodity flow problem. Our algorithms achieve the same bounds as before but have the significant advantage in that they can work with any feasible solution to the relaxation. Consequently, in addition to obtaining "book" proofs of LP rounding for these two basic problems, we also obtain significantly faster approximation algorithms by taking advantage of known algorithms for computing near-optimal solutions for maximum multicommodity flow problems. We also investigate lower bounds for DIR-MC when k = 2 and prove that the integrality gap of the LP relaxation is 2 even in planar directed graphs.
In the minimum Multicut problem, the input is an edgeweighted supply graph G = (V, E) and a demand graph H = (V, F ). Either G and H are directed (Dir-MulC) or both are undirected (Undir-MulC). The goal is to remove a minimum weight set of supply edges E ⊆ E such that in G − E there is no path from s to t for any demand edge (s, t) ∈ F . Undir-MulC admits O(log k)-approximation where k is the number of edges in H while the best known approximation for DirMulC is min{k,Õ(|V | 11/23 )}. These approximations are obtained by proving corresponding results on the multicommodity flow-cut gap. In this paper we consider the role that the structure of the demand graph plays in determining the approximability of Multicut. We obtain several new positive and negative results.In undirected graphs our main result is a 2-approximation in n O(t) time when the demand graph excludes an induced matching of size t. This gives a constant factor approximation for a specific demand graph that motivated this work, and is based on a reduction to uniform metric labeling and not via the flow-cut gap.In contrast to the positive result for undirected graphs, we prove that in directed graphs such approximation algorithms can not exist. We prove that, assuming the Unique Games Conjecture (UGC), that for a large class of fixed demand graphs Dir-MulC cannot be approximated to a factor better than the worstcase flow-cut gap. As a consequence we prove that for any fixed k, assuming UGC, Dir-MulC with k demand pairs is hard to approximate to within a factor better than k. On the positive side, we obtain a k approximation when the demand graph excludes certain graphs as an induced subgraph. This generalizes the known 2 approximation for directed Multiway Cut to a larger class of demand graphs.
We consider subset feedback edge and vertex set problems in undirected graphs. The input to these problems is an undirected graph G = (V, E) and a set S = {s1, s2,. .. , s k } ⊂ V of k terminals. A cycle in G is interesting if it contains a terminal. In the Subset Feedback Edge Set problem (Subset-FES) the input graph is edge-weighted and the goal is to remove a minimum weight set of edges such that no interesting cycle remains. In the Subset Feedback Vertex Set problem (Subset-FVS) the input graph is node-weighted and the goal is to remove a minimum weight set of nodes such that no interesting cycle remains. A 2-approximation is known for Subset-FES [12] and a 8approximation is known for Subset-FVS [13]. The algorithm and analysis for Subset-FVS is complicated. One reason for the difficulty in addressing feedback set problems in undirected graphs has been the lack of LP relaxations with constant factor integrality gaps; the natural LP has an integrality gap of Θ(log n). In this paper, we introduce new LP relaxations for Subset-FES and Subset-FVS and show that their integrality gap is at most 13. Our LP formulation and rounding are simple although the analysis is non-obvious.
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