We introduce the {\it Bipartite Multi-cut} problem. This is a generalization of the {\it st-Min-cut} problem, is similar to the {\it Multi-cut} problem (except for more stringent requirements) and also turns out to be an immediate generalization of the {\it Min UnCut} problem. We prove that this problem is {\bf NP}-hard and then present LP and SDP based approximation algorithms. While the LP algorithm is based on the Garg-Vazirani-Yannakakis algorithm for {\it Multi-cut}, the SDP algorithm uses the {\it Structure Theorem} of $\ell_2^2$ Metrics.
In a graph, a matching cut is an edge cut that is a matching. Matching Cut is the problem of deciding whether or not a given graph has a matching cut, which is known to be NP-complete even when restricted to bipartite graphs. It has been proved that Matching Cut is polynomially solvable for graphs of diameter two. In this paper, we show that, for any fixed integer $d\ge 3$, Matching Cut is NP-complete in the class of graphs of diameter $d$. This resolves an open problem posed by Borowiecki and Jesse-Józefczyk in [Matching cutsets in graphs of diameter $2$, Theoretical Computer Science 407 (200…
The K-way vertex cut problem} consists in, given a graph G, finding a subset of vertices of a given size, whose removal partitions G into the maximum number of connected components. This problem has many applications in several areas. It has been proven to be NP-complete on general graphs, as well as on split and planar graphs. In this paper, we enrich its complexity study with two new results. First, we prove that it remains NP-complete even when restricted on the class of bipartite graphs. This is unlike what it is expected, given that the K-way vertex cut problem is a generalization of the …
In this paper the problem of scheduling of jobs on parallel machines under incompatibility relation is considered. In this model a binary relation between jobs is given and no two jobs that are in the relation can be scheduled on the same machine. In particular, we consider job scheduling under incompatibility relation forming bipartite graphs, under makespan optimality criterion, on uniform and unrelated machines. We show that no algorithm can achieve a good approximation ratio for uniform machines, even for a case of unit time jobs, under $P \neq NP$. We also provide an approximation algorit…
We settle the complexities of the maximum-cardinality bipartite matching problem (BMM) up to poly-logarithmic factors in five models of computation: the two-party communication, AND query, OR query, XOR query, and quantum edge query models. Our results answer open problems that have been raised repeatedly since at least three decades ago [Hajnal, Maass, and Turan STOC'88; Ivanyos, Klauck, Lee, Santha, and de Wolf FSTTCS'12; Dobzinski, Nisan, and Oren STOC'14; Nisan SODA'21] and tighten the lower bounds shown by Beniamini and Nisan [STOC'21] and Zhang [ICALP'04]. We also settle the communicatio…
A perfect matching cut is a perfect matching that is also a cutset, or equivalently a perfect matching containing an even number of edges on every cycle. The corresponding algorithmic problem, Perfect Matching Cut, is known to be NP-complete in subcubic bipartite graphs [Le & Telle, TCS '22] but its complexity was open in planar graphs and in cubic graphs. We settle both questions at once by showing that Perfect Matching Cut is NP-complete in 3-connected cubic bipartite planar graphs or Barnette graphs. Prior to our work, among problems whose input is solely an undirected graph, only Distance-…
In the simultaneous Max-Cut problem, we are given $k$ weighted graphs on the same set of $n$ vertices, and the goal is to find a cut of the vertex set so that the minimum, over the $k$ graphs, of the cut value is as large as possible. Previous work [BKS15] gave a polynomial time algorithm which achieved an approximation factor of $1/2 - o(1)$ for this problem (and an approximation factor of $1/2 + ε_k$ in the unweighted case, where $ε_k \rightarrow 0$ as $k \rightarrow \infty$). In this work, we give a polynomial time approximation algorithm for simultaneous Max-Cut with an approximation facto…
We study the problem of approximating the quality of a disperser. A bipartite graph $G$ on $([N],[M])$ is a $(ρN,(1-δ)M)$-disperser if for any subset $S\subseteq [N]$ of size $ρN$, the neighbor set $Γ(S)$ contains at least $(1-δ)M$ distinct vertices. Our main results are strong integrality gaps in the Lasserre hierarchy and an approximation algorithm for dispersers. \begin{enumerate} \item For any $α>0$, $δ>0$, and a random bipartite graph $G$ with left degree $D=O(\log N)$, we prove that the Lasserre hierarchy cannot distinguish whether $G$ is an $(N^α,(1-δ)M)$-disperser or not an $(N^{1-α},δ…
Maximum bipartite matching is a fundamental algorithmic problem which can be solved in polynomial time. We consider a natural variant in which there is a separation constraint: the vertices on one side lie on a path or a grid, and two vertices that are close to each other are not allowed to be matched simultaneously. We show that the problem is hard to approximate even for paths, and provide constant-factor approximation algorithms for both paths and grids.
The Small Set Expansion Hypothesis (SSEH) is a conjecture which roughly states that it is NP-hard to distinguish between a graph with a small subset of vertices whose edge expansion is almost zero and one in which all small subsets of vertices have expansion almost one. In this work, we prove inapproximability results for the following graph problems based on this hypothesis: - Maximum Edge Biclique (MEB): given a bipartite graph $G$, find a complete bipartite subgraph of $G$ with maximum number of edges. - Maximum Balanced Biclique (MBB): given a bipartite graph $G$, find a balanced complete …
In this paper, we present two approximation algorithms for the directed multi-multiway cut and directed multicut problems. The so called region growing paradigm \cite{1} is modified and used for these two cut problems on directed graphs. By using this paradigm, we give for each problem an approximation algorithm such that both algorithms have the approximate factor $O(k)$ the same as the previous works done on these problems. However, the previous works need to solve $k$ linear programming, whereas our algorithms require only one linear programming. Therefore, our algorithms improve the runnin…
Approximation algorithms for constraint satisfaction problems (CSPs) are a central direction of study in theoretical computer science. In this work, we study classical product state approximation algorithms for a physically motivated quantum generalization of Max-Cut, known as the quantum Heisenberg model. This model is notoriously difficult to solve exactly, even on bipartite graphs, in stark contrast to the classical setting of Max-Cut. Here we show, for any interaction graph, how to classically and efficiently obtain approximation ratios 0.649 (anti-ferromagnetic XY model) and 0.498 (anti-f…
In this paper we study the MAX-CUT problem on power law graphs (PLGs) with power law exponent $β$. We prove some new approximability results on that problem. In particular we show that there exist polynomial time approximation schemes (PTAS) for MAX-CUT on PLGs for the power law exponent $β$ in the interval $(0,2)$. For $β>2$ we show that for some $ε>0$, MAX-CUT is NP-hard to approximate within approximation ratio $1+ε$, ruling out the existence of a PTAS in this case. Moreover we give an approximation algorithm with improved constant approximation ratio for the case of $β>2$.
