Stabilizing Weighted Graphs

An edge-weighted graph G=(V,E) is called stable if the value of a maximum-weight matching equals the value of a maximum-weight fractional matching. Stable graphs play an important role in some interesting game theory problems, such as network bargaining games and cooperative matching games, because they characterize instances which admit stable outcomes. Motivated by this, in the last few years many researchers have investigated the algorithmic problem of turning a given graph into a stable one, via edge- and vertex-removal operations. However, all the algorithmic results developed in the literature so far only hold for unweighted instances, i.e., assuming unit weights on the edges of G. We give the first polynomial-time algorithm to find a minimum cardinality subset of vertices whose removal from G yields a stable graph, for any weighted graph G. The algorithm is combinatorial and exploits new structural properties of basic fractional matchings, which are of independent interest. In particular, one of the main ingredients of our result is the development of a polynomial-time algorithm to compute a basic maximum-weight fractional matching with minimum number of odd cycles in its support. This generalizes a fundamental and classical result on unweighted matchings given by Balas more than 30 years ago, which we expect to prove useful beyond this particular application. In contrast, we show that the problem of finding a minimum cardinality subset of edges whose removal from a weighted graph G yields a stable graph, does not admit any constant-factor approximation algorithm, unless P=NP. In this setting, we develop an O(Delta)-approximation algorithm for the problem, where Delta is the maximum degree of a node in G

Structural Parameterizations with Modulator Oblivion

It is known that problems like Vertex Cover, Feedback Vertex Set and Odd Cycle Transversal are polynomial time solvable in the class of chordal graphs. We consider these problems in a graph that has at most k vertices whose deletion results in a chordal graph, when parameterized by k. While this investigation fits naturally into the recent trend of what are called "structural parameterizations", here we assume that the deletion set is not given. One method to solve them is to compute a k-sized or an approximate (f(k) sized, for a function f) chordal vertex deletion set and then use the structural properties of the graph to design an algorithm. This method leads to at least k^O(k)n^O(1) running time when we use the known parameterized or approximation algorithms for finding a k-sized chordal deletion set on an n vertex graph. In this work, we design 2^O(k)n^O(1) time algorithms for these problems. Our algorithms do not compute a chordal vertex deletion set (or even an approximate solution). Instead, we construct a tree decomposition of the given graph in time 2^O(k)n^O(1) where each bag is a union of four cliques and O(k) vertices. We then apply standard dynamic programming algorithms over this special tree decomposition. This special tree decomposition can be of independent interest. Our algorithms are, what are sometimes called permissive in the sense that given an integer k, they detect whether the graph has no chordal vertex deletion set of size at most k or output the special tree decomposition and solve the problem. We also show lower bounds for the problems we deal with under the Strong Exponential Time Hypothesis (SETH)

A Randomized Polynomial Kernelization for Vertex Cover with a Smaller Parameter

In the Vertex Cover problem we are given a graph $G=(V,E)$ and an integer $k$ and have to determine whether there is a set $X\subseteq V$ of size at most $k$ such that each edge in $E$ has at least one endpoint in $X$. The problem can be easily solved in time $O^*(2^k)$, making it fixed-parameter tractable (FPT) with respect to $k$. While the fastest known algorithm takes only time $O^*(1.2738^k)$, much stronger improvements have been obtained by studying parameters that are smaller than $k$. Apart from treewidth-related results, the arguably best algorithm for Vertex Cover runs in time $O^*(2.3146^p)$, where $p=k-LP(G)$ is only the excess of the solution size $k$ over the best fractional vertex cover (Lokshtanov et al.\ TALG 2014). Since $p\leq k$ but $k$ cannot be bounded in terms of $p$ alone, this strictly increases the range of tractable instances. Recently, Garg and Philip (SODA 2016) greatly contributed to understanding the parameterized complexity of the Vertex Cover problem. They prove that $2LP(G)-MM(G)$ is a lower bound for the vertex cover size of $G$, where $MM(G)$ is the size of a largest matching of $G$, and proceed to study parameter $\ell=k-(2LP(G)-MM(G))$. They give an algorithm of running time $O^*(3^\ell)$, proving that Vertex Cover is FPT in $\ell$. It can be easily observed that $\ell\leq p$ whereas $p$ cannot be bounded in terms of $\ell$ alone. We complement the work of Garg and Philip by proving that Vertex Cover admits a randomized polynomial kernelization in terms of $\ell$, i.e., an efficient preprocessing to size polynomial in $\ell$. This improves over parameter $p=k-LP(G)$ for which this was previously known (Kratsch and Wahlstr\"om FOCS 2012).Comment: Full version of ESA 2016 pape

Streaming Verification of Graph Properties

Streaming interactive proofs (SIPs) are a framework for outsourced computation. A computationally limited streaming client (the verifier) hands over a large data set to an untrusted server (the prover) in the cloud and the two parties run a protocol to confirm the correctness of result with high probability. SIPs are particularly interesting for problems that are hard to solve (or even approximate) well in a streaming setting. The most notable of these problems is finding maximum matchings, which has received intense interest in recent years but has strong lower bounds even for constant factor approximations. In this paper, we present efficient streaming interactive proofs that can verify maximum matchings exactly. Our results cover all flavors of matchings (bipartite/non-bipartite and weighted). In addition, we also present streaming verifiers for approximate metric TSP. In particular, these are the first efficient results for weighted matchings and for metric TSP in any streaming verification model.Comment: 26 pages, 2 figure, 1 tabl

Mending Partial Solutions with Few Changes

In this paper, we study the notion of mending: given a partial solution to a graph problem, how much effort is needed to take one step towards a proper solution? For example, if we have a partial coloring of a graph, how hard is it to properly color one more node? In prior work (SIROCCO 2022), this question was formalized and studied from the perspective of mending radius: if there is a hole that we need to patch, how far do we need to modify the solution? In this work, we investigate a complementary notion of mending volume: how many nodes need to be modified to patch a hole? We focus on the case of locally checkable labeling problems (LCLs) in trees, and show that already in this setting there are two infinite hierarchies of problems: for infinitely many values 0 < ? ? 1, there is an LCL problem with mending volume ?(n^?), and for infinitely many values k ? 1, there is an LCL problem with mending volume ?(log^k n). Hence the mendability of LCL problems on trees is a much more fine-grained question than what one would expect based on the mending radius alone