We initiate the study of the Bipartite Contraction problem from the perspective of parameterized complexity. In this problem we are given a graph $G$ and an integer $k$, and the task is to determine whether we can obtain a bipartite graph from $G$ by a sequence of at most $k$ edge contractions. Our main result is an $f(k) n^{O(1)}$ time algorithm for Bipartite Contraction. Despite a strong resemblance between Bipartite Contraction and the classical Odd Cycle Transversal (OCT) problem, the methods developed to tackle OCT do not seem to be directly applicable to Bipartite Contraction. Our algorithm is based on a novel combination of the irrelevant vertex technique, introduced by Robertson and Seymour, and the concept of important separators. Both techniques have previously been used as key components of algorithms for fundamental problems in parameterized complexity. However, to the best of our knowledge, this is the first time the two techniques are applied in unison.
The Planar Contraction problem is to test whether a given graph can be made planar by using at most k edge contractions. This problem is known to be NP-complete. We show that it is fixed-parameter tractable when parameterized by k.
We present a deterministic way of assigning small (log bit) weights to the edges of a bipartite planar graph so that the minimum weight perfect matching becomes unique. The isolation lemma as described in (Mulmuley et al. 1987) achieves the same for general graphs using a randomized weighting scheme, whereas we can do it deterministically when restricted to bipartite planar graphs. As a consequence, we reduce both decision and construction versions of the matching problem to testing whether a matrix is singular, under the promise that its determinant is 0 or 1, thus obtaining a highly parallel…
Bipartite Correlation clustering is the problem of generating a set of disjoint bi-cliques on a set of nodes while minimizing the symmetric difference to a bipartite input graph. The number or size of the output clusters is not constrained in any way. The best known approximation algorithm for this problem gives a factor of 11. This result and all previous ones involve solving large linear or semi-definite programs which become prohibitive even for modestly sized tasks. In this paper we present an improved factor 4 approximation algorithm to this problem using a simple combinatorial algorithm …
Twin-width is a newly introduced graph width parameter that aims at generalizing a wide range of "nicely structured" graph classes. In this work, we focus on obtaining good bounds on twin-width $\text{tww}(G)$ for graphs $G$ from a number of classic graph classes. We prove the following: - $\text{tww}(G) \leq 3\cdot 2^{\text{tw}(G)-1}$, where $\text{tw}(G)$ is the treewidth of $G$, - $\text{tww}(G) \leq \max(4\text{bw}(G),\frac{9}{2}\text{bw}(G)-3)$ for a planar graph $G$ with $\text{bw}(G) \geq 2$, where $\text{bw}(G)$ is the branchwidth of $G$, - $\text{tww}(G) \leq 183$ for a planar graph $…
We develop a technique that we call Conflict Packing in the context of kernelization, obtaining (and improving) several polynomial kernels for editing problems on dense instances. We apply this technique on several well-studied problems: Feedback Arc Set in (Bipartite) Tournaments, Dense Rooted Triplet Inconsistency and Betweenness in Tournaments. For the former, one is given a (bipartite) tournament $T = (V,A)$ and seeks a set of at most $k$ arcs whose reversal in $T$ results in an acyclic (bipartite) tournament. While a linear vertex-kernel is already known for the first problem, using the C…
Sublinear time algorithms for approximating maximum matching size have long been studied. Much of the progress over the last two decades on this problem has been on the algorithmic side. For instance, an algorithm of Behnezhad [FOCS'21] obtains a 1/2-approximation in $\tilde{O}(n)$ time for $n$-vertex graphs. A more recent algorithm by Behnezhad, Roghani, Rubinstein, and Saberi [SODA'23] obtains a slightly-better-than-1/2 approximation in $O(n^{1+ε})$ time. On the lower bound side, Parnas and Ron [TCS'07] showed 15 years ago that obtaining any constant approximation of maximum matching size re…
We consider the maximum vertex-weighted matching problem (MVM) for non-bipartite graphs. In earlier work we have described a 2/3-approximation algorithm for the MVM on bipartite graphs (Dobrian, Halappanavar, Pothen and Al-Herz, SIAM J. Scientific Computing, 2019). Here we show that a 2/3-approximation algorithm for MVM on non-bipartite graphs can be obtained by restricting the length of augmenting paths to at most three. The algorithm has time complexity $O(m \log Δ+ n \log n)$, where $n$ is the number of vertices, $m$ is the number of edges, and $Δ$ is the maximum degree of a vertex. The app…
In this paper we consider the well-studied problem of finding a perfect matching in a d-regular bipartite graph on 2n nodes with m=nd edges. The best-known algorithm for general bipartite graphs (due to Hopcroft and Karp) takes time O(m\sqrt{n}). In regular bipartite graphs, however, a matching is known to be computable in O(m) time (due to Cole, Ost and Schirra). In a recent line of work by Goel, Kapralov and Khanna the O(m) time algorithm was improved first to \tilde O(min{m, n^{2.5}/d}) and then to \tilde O(min{m, n^2/d}). It was also shown that the latter algorithm is optimal up to polylog…
Until recently, LP relaxations have played a limited role in the design of approximation algorithms for the Steiner tree problem. In 2010, Byrka et al. presented a ln(4)+epsilon approximation based on a hypergraphic LP relaxation, but surprisingly, their analysis does not provide a matching bound on the integrality gap. We take a fresh look at hypergraphic LP relaxations for the Steiner tree problem - one that heavily exploits methods and results from the theory of matroids and submodular functions - which leads to stronger integrality gaps, faster algorithms, and a variety of structural insig…
The classical analysis of online algorithms, due to its worst-case nature, can be quite pessimistic when the input instance at hand is far from worst-case. Often this is not an issue with machine learning approaches, which shine in exploiting patterns in past inputs in order to predict the future. However, such predictions, although usually accurate, can be arbitrarily poor. Inspired by a recent line of work, we augment three well-known online settings with machine learned predictions about the future, and develop algorithms that take them into account. In particular, we study the following on…
We prove complex contraction for zero-free regions of counting weighted set cover problem in which an element can appear in an unbounded number of sets, thus obtaining fully polynomial-time approximation schemes(FPTAS) via Barvinok's algorithmic paradigm\cite{barvinok2016combinatorics}. Relying on the computation tree expansion, our approach does not need proof of correlation decay in the real axis. We directly look in the complex plane for a region that contracts into its interior as the tree recursion procedure goes from leaves to the root. For the class of problems under the framework of we…
In this paper we study the classic problem of computing a maximum cardinality matching in general graphs $G = (V, E)$. The best known algorithm for this problem till date runs in $O(m \sqrt{n})$ time due to Micali and Vazirani \cite{MV80}. Even for general bipartite graphs this is the best known running time (the algorithm of Karp and Hopcroft \cite{HK73} also achieves this bound). For regular bipartite graphs one can achieve an $O(m)$ time algorithm which, following a series of papers, has been recently improved to $O(n \log n)$ by Goel, Kapralov and Khanna (STOC 2010) \cite{GKK10}. In this p…
Finding a maximum cut is a fundamental task in many computational settings. Surprisingly, it has been insufficiently studied in the classic distributed settings, where vertices communicate by synchronously sending messages to their neighbors according to the underlying graph, known as the $\mathcal{LOCAL}$ or $\mathcal{CONGEST}$ models. We amend this by obtaining almost optimal algorithms for Max-Cut on a wide class of graphs in these models. In particular, for any $ε> 0$, we develop randomized approximation algorithms achieving a ratio of $(1-ε)$ to the optimum for Max-Cut on bipartite graphs…
We introduce a framework for proving lower bounds on computational problems over distributions against algorithms that can be implemented using access to a statistical query oracle. For such algorithms, access to the input distribution is limited to obtaining an estimate of the expectation of any given function on a sample drawn randomly from the input distribution, rather than directly accessing samples. Most natural algorithms of interest in theory and in practice, e.g., moments-based methods, local search, standard iterative methods for convex optimization, MCMC and simulated annealing can …
We examine the complexity of the online Dictionary Matching with One Gap Problem (DMOG) which is the following. Preprocess a dictionary $D$ of $d$ patterns, where each pattern contains a special gap symbol that can match any string, so that given a text that arrives online, a character at a time, we can report all of the patterns from $D$ that are suffixes of the text that has arrived so far, before the next character arrives. In more general versions the gap symbols are associated with bounds determining the possible lengths of matching strings. Finding efficient algorithmic solutions for (on…
In the parameterized problem \textsc{MaxLin2-AA}[$k$], we are given a system with variables $x_1,...,x_n$ consisting of equations of the form $\prod_{i \in I}x_i = b$, where $x_i,b \in \{-1, 1\}$ and $I\subseteq [n],$ each equation has a positive integral weight, and we are to decide whether it is possible to simultaneously satisfy equations of total weight at least $W/2+k$, where $W$ is the total weight of all equations and $k$ is the parameter (if $k=0$, the possibility is assured). We show that \textsc{MaxLin2-AA}[$k$] has a kernel with at most $O(k^2\log k)$ variables and can be solved in …
Consider the following toy problem. There are $m$ rectangles and $n$ points on the plane. Each rectangle $R$ is a consumer with budget $B_R$, who is interested in purchasing the cheapest item (point) inside R, given that she has enough budget. Our job is to price the items to maximize the revenue. This problem can also be defined on higher dimensions. We call this problem the geometric pricing problem. In this paper, we study a new class of problems arising from a geometric aspect of the pricing problem. It intuitively captures typical real-world assumptions that have been widely studied in ma…
The dynamic shortest paths problem on planar graphs asks us to preprocess a planar graph $G$ such that we may support insertions and deletions of edges in $G$ as well as distance queries between any two nodes $u,v$ subject to the constraint that the graph remains planar at all times. This problem has been extensively studied in both the theory and experimental communities over the past decades and gets solved millions of times every day by companies like Google, Microsoft, and Uber. The best known algorithm performs queries and updates in $\tilde{O}(n^{2/3})$ time, based on ideas of a seminal …
Is matching in NC, i.e., is there a deterministic fast parallel algorithm for it? This has been an outstanding open question in TCS for over three decades, ever since the discovery of randomized NC matching algorithms [KUW85, MVV87]. Over the last five years, the theoretical computer science community has launched a relentless attack on this question, leading to the discovery of several powerful ideas. We give what appears to be the culmination of this line of work: An NC algorithm for finding a minimum-weight perfect matching in a general graph with polynomially bounded edge weights, provided…
For many online problems, it is known that the uniform arrival order enables the design of algorithms with much better performance guarantees than under worst-case. The quintessential example is the secretary problem. If the sequence of elements is presented in uniformly random order there is an algorithm that picks the maximum value with probability 1/e, whereas no non-trivial performance guarantee is possible if the elements arrive in worst-case order. This work initiates an investigation into relaxations of the random-ordering hypothesis in online algorithms, by focusing on the secretary pr…
