6 research outputs found
Complexity and Approximation Results for the Min-Sum and Min-Max Disjoint Paths Problems
Given a graph G=(V, E) and k source-sink pairs (s1, t1), …, (sk, tk) with each si, ti V, the Min-Sum Disjoint Paths problem asks to find k disjoint paths connecting all the source-sink pairs with minimized total length, while the Min-Max Disjoint Paths problem asks for k disjoint paths connecting all the source-sink pairs with minimized length of the longest path. We show that the weighted Min-Sum Disjoint Paths problem is FPNP-complete in general graphs, and the unweighted Min-Sum Disjoint Paths problem and the unweighted Min-Max Disjoint Paths problem cannot be approximated within m(m1-1) for any constant > 0 even in planar graphs, assuming P P NP, where m is the number of edges in G. We give for the first time a simple bicriteria approximation algorithm for the unweighted Min-Max Edge-Disjoint Paths problem and the weighted Min-Sum Edge-Disjoint Paths problem, w
Pre-Reduction Graph Products: Hardnesses of Properly Learning DFAs and Approximating EDP on DAGs
The study of graph products is a major research topic and typically concerns
the term , e.g., to show that . In this paper, we
study graph products in a non-standard form where is a
"reduction", a transformation of any graph into an instance of an intended
optimization problem. We resolve some open problems as applications.
(1) A tight -approximation hardness for the minimum
consistent deterministic finite automaton (DFA) problem, where is the
sample size. Due to Board and Pitt [Theoretical Computer Science 1992], this
implies the hardness of properly learning DFAs assuming (the
weakest possible assumption).
(2) A tight hardness for the edge-disjoint paths (EDP)
problem on directed acyclic graphs (DAGs), where denotes the number of
vertices.
(3) A tight hardness of packing vertex-disjoint -cycles for large .
(4) An alternative (and perhaps simpler) proof for the hardness of properly
learning DNF, CNF and intersection of halfspaces [Alekhnovich et al., FOCS 2004
and J. Comput.Syst.Sci. 2008]
On Approximating Node-Disjoint Paths in Grids
In the Node-Disjoint Paths (NDP) problem, the input is an undirected n-vertex graph G, and a collection {(s_1,t_1),...,(s_k,t_k)} of pairs of vertices called demand pairs. The goal is to route the largest possible number of the demand pairs (s_i,t_i), by selecting a path connecting each such pair, so that the resulting paths are node-disjoint. NDP is one of the most basic and extensively studied routing problems. Unfortunately, its approximability is far from being well-understood: the best current upper bound of O(sqrt(n)) is achieved via a simple greedy algorithm, while the best current lower bound on its approximability is Omega(log^{1/2-delta}(n)) for any constant delta. Even for seemingly simpler special cases, such as planar graphs, and even grid graphs, no better approximation algorithms are currently known. A major reason for this impasse is that the standard technique for designing approximation algorithms for routing problems is LP-rounding of the standard multicommodity flow relaxation of the problem, whose integrality gap for NDP is Omega(sqrt(n)) even on grid graphs.
Our main result is an O(n^{1/4} * log(n))-approximation algorithm for NDP on grids. We distinguish between demand pairs with both vertices close to the grid boundary, and pairs where at least one of the two vertices is far from the grid boundary. Our algorithm shows that when all demand pairs are of the latter type, the integrality gap of the multicommodity flow LP-relaxation is at most O(n^{1/4} * log(n)), and we deal with demand pairs of the former type by other methods. We complement our upper bounds by proving that NDP is APX-hard on grid graphs
An Approximation Algorithm for the Disjoint Paths Problem in Even-Degree Planar Graphs
In joint work with Eva Tardos in 1995, we asked whether it was possible to obtain a polynomial-time, polylogarithmic approximation algorithm for the disjoint paths problem in the class of all even-degree planar graphs [19]. This paper answers the question in the affirmative, by providing such an algorithm. The algorithm builds on recent work of Chekuri, Khanna, and Shepherd [7, 8], who considered routing problems in planar graphs where each edge can carry up to two paths
A Polylogarithimic Approximation Algorithm for Edge-Disjoint Paths with Congestion 2
In the Edge-Disjoint Paths with Congestion problem (EDPwC), we are given an
undirected n-vertex graph G, a collection M={(s_1,t_1),...,(s_k,t_k)} of demand
pairs and an integer c. The goal is to connect the maximum possible number of
the demand pairs by paths, so that the maximum edge congestion - the number of
paths sharing any edge - is bounded by c. When the maximum allowed congestion
is c=1, this is the classical Edge-Disjoint Paths problem (EDP).
The best current approximation algorithm for EDP achieves an -approximation, by rounding the standard multi-commodity flow relaxation of
the problem. This matches the lower bound on the integrality
gap of this relaxation. We show an -approximation algorithm for
EDPwC with congestion c=2, by rounding the same multi-commodity flow
relaxation. This gives the best possible congestion for a sub-polynomial
approximation of EDPwC via this relaxation. Our results are also close to
optimal in terms of the number of pairs routed, since EDPwC is known to be hard
to approximate to within a factor of for
any constant congestion c. Prior to our work, the best approximation factor for
EDPwC with congestion 2 was , and the best algorithm
achieving a polylogarithmic approximation required congestion 14