Additive stabilizers for unstable graphs

Abstract

Stabilization of graphs has received substantial attention in recent years due to its connection to game theory. Stable graphs are exactly the graphs inducing a matching game with non-empty core. They are also the graphs that induce a network bargaining game with a balanced solution. A graph with weighted edges is called stable if the maximum weight of an integral matching equals the cost of a minimum fractional weighted vertex cover. If a graph is not stable, it can be stabilized in different ways. Recent papers have considered the deletion or addition of edges and vertices in order to stabilize a graph. In this work, we focus on a fine-grained stabilization strategy, namely stabilization of graphs by fractionally increasing edge weights.We show the following results for stabilization by minimum weight increase in edge weights (min additive stabilizer): (i) Any approximation algorithm for min additive stabilizer that achieves a factor of O(|V | 1/24−ǫ ) for ǫ > 0 would lead to improvements in the approximability of densestk-subgraph. (ii) Min additive stabilizer has no o(log |V |) approximation unless NP=P. Results (i) and (ii) together provide the first super-constant hardness results for any graph stabilization problem. On the algorithmic side, we present (iii) an algorithm to solve min additive stabilizer in factorcritical graphs exactly in poly-time, (iv) an algorithm to solve min additive stabilizer in arbitrary-graphs exactly in time exponential in the size of the Tutte set, and (v) a poly-time algorithm with approximation factor at most |V | for a super-class of the instances generated in our hardness proofs.

Stabilizing Weighted Graphs

“…In 83:5 particular, Ito et al [10] have given polynomial-time algorithms to stabilize an unweighted graph by adding edges and by adding vertices. Chandrasekaran et al [7] have recently studied the problem of stabilizing unweighted graphs by fractionally increasing edge weights. Ahmadian et al [1] have also studied the vertex-stabilizer problem on unweighted graphs, but in the more-general setting where there are (non-uniform) costs for removing vertices, and gave approximation algorithms for this case.…”

“…Motivated by the above connection, in the last few years many researchers have investigated the algorithmic problem of turning a given graph into a stable one, by performing a minimum number of modifications on the input graph [6,1,10,7,14,4,5]. Two natural operations which have a nice network game interpretation, are vertex-deletion and edgedeletion.…”

The stable fixtures problem with payments

“…This immediately raises the question what to do if we find that the core is non-empty. One approach, which was recently considered for the case b ≡ 1, is that of verifying whether the core can become nonempty after a small modification of the graph via adding or deleting some vertices or edges [1,8,26] or via a fractionally increase of the edge weights [13].…”

Stabilizing Weighted Graphs

“…Motivated by the above connection, in the last few years many researchers have investigated the algorithmic problem of turning a given graph into a stable one, by performing a minimum number of modifications on the input graph [7,1,13,8,17,5,6]. Two natural operations which have a nice network game interpretation, are vertex-deletion and edge-deletion.…”

“…In particular, Ito et al [13] have given polynomial-time algorithms to stabilize an unweighted graph by adding edges and by adding vertices. Chandrasekaran et al [8] have recently studied the problem of stabilizing unweighted graphs by fractionally increasing edge weights. Ahmadian et al [1] have also studied the vertex-stabilizer problem on unweighted graphs, but in the more-general setting where there are (non-uniform) costs for removing vertices, and gave approximation algorithms for this case.…”