Approximate Hypergraph Coloring under Low-discrepancy and Related Promises

A hypergraph is said to be $\chi$-colorable if its vertices can be colored with $\chi$ colors so that no hyperedge is monochromatic. $2$-colorability is a fundamental property (called Property B) of hypergraphs and is extensively studied in combinatorics. Algorithmically, however, given a $2$-colorable $k$-uniform hypergraph, it is NP-hard to find a $2$-coloring miscoloring fewer than a fraction $2^{-k+1}$ of hyperedges (which is achieved by a random $2$-coloring), and the best algorithms to color the hypergraph properly require $\approx n^{1-1/k}$ colors, approaching the trivial bound of $n$ as $k$ increases. In this work, we study the complexity of approximate hypergraph coloring, for both the maximization (finding a $2$-coloring with fewest miscolored edges) and minimization (finding a proper coloring using fewest number of colors) versions, when the input hypergraph is promised to have the following stronger properties than $2$-colorability: (A) Low-discrepancy: If the hypergraph has discrepancy $\ell \ll \sqrt{k}$, we give an algorithm to color the it with $\approx n^{O(\ell^2/k)}$ colors. However, for the maximization version, we prove NP-hardness of finding a $2$-coloring miscoloring a smaller than $2^{-O(k)}$ (resp. $k^{-O(k)}$) fraction of the hyperedges when $\ell = O(\log k)$ (resp. $\ell=2$). Assuming the UGC, we improve the latter hardness factor to $2^{-O(k)}$ for almost discrepancy-$1$ hypergraphs. (B) Rainbow colorability: If the hypergraph has a $(k-\ell)$-coloring such that each hyperedge is polychromatic with all these colors, we give a $2$-coloring algorithm that miscolors at most $k^{-\Omega(k)}$ of the hyperedges when $\ell \ll \sqrt{k}$, and complement this with a matching UG hardness result showing that when $\ell =\sqrt{k}$, it is hard to even beat the $2^{-k+1}$ bound achieved by a random coloring.Comment: Approx 201

On the complexity of some colorful problems parameterized by treewidth

In this paper,we study the complexity of several coloring problems on graphs, parameterizedby the treewidth of the graph.1. The List Coloring problem takes as input a graph G, togetherwith an assignment to each vertex v of a set of colors Cv. The problem is to determinewhether it is possible to choose a color for vertex v from the set of permitted colors Cv, for each vertex, so that the obtained coloring of G is proper. We show that this problem is W[1]-hard, parameterized by the treewidth of G. The closely related Precoloring Extension problem is also shown to be W[1]-hard, parameterized by treewidth.2. An equitable coloring of a graph G is a proper coloring of the verticeswhere the numbers of vertices having any two distinct colors differs by at most one.We show that the problem is hard forW[1], parameterized by the treewidth plus the number of colors.We also show that a list-based variation, List Equitable Coloring is W[1]-hard for forests, parameterizedby the number of colors on the lists.3. The list chromatic number χl(G) of a graph G is defined to be the smallest positive integer r, such that for every assignment to the vertices v of G, of a list Lv of colors, where each list has length at least r, there is a choice of one color fromeach vertex list Lv yielding a proper coloring of G. We show that the problem of determining whether χl(G) ≤ r, the ListChromatic Number problem, is solvable in linear time on graphs of constant treewidth

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum

On Reconfiguration Problems: Structure and Tractability

Given an n-vertex graph G and two vertices s and t in G, determining whether there exists a path and computing the length of the shortest path between s and t are two of the most fundamental graph problems. In the classical battle of P versus NP or ``easy'' versus ``hard'', both of these problems are on the easy side. That is, they can be solved in poly(n) time, where poly is any polynomial function. But what if our input consisted of a 2^n-vertex graph? Of course, we can no longer assume G to be part of the input, as reading the input alone requires more than poly(n) time. Instead, we are given an oracle encoded using poly(n) bits and that can, in constant or poly(n) time, answer queries of the form ``is u a vertex in G'' or ``is there an edge between u and v?''. Given such an oracle and two vertices of the 2^n-vertex graph, can we still determine if there is a path or compute the length of the shortest path between s and t in poly(n) time? A slightly different, but equally insightful, formulation of the question above is as follows. Given a set S of n objects, consider the graph R(S) which contains one vertex for each set in the power set of S, 2^S, and two vertices are adjacent in R(S) whenever the size of their symmetric difference is equal to one. Clearly, this graph contains 2^n vertices and can be easily encoded in poly(n) bits using the oracle described above. It is not hard to see that there exists a path between any two vertices of R(S). Moreover, computing the length of a shortest path can be accomplished in constant time; it is equal to the size of the symmetric difference of the two underlying sets. If the vertex set of R(S) were instead restricted to a subset of 2^S, both of our problems can become NP-complete or even PSPACE-complete. Therefore, another interesting question is whether we can determine what types of ``restriction'' on the vertex set of R(S) induce such variations in the complexity of the two problems. These two seemingly artificial questions are in fact quite natural and appear in many practical and theoretical problems. In particular, these are exactly the types of questions asked under the reconfiguration framework, the main subject of this thesis. Under the reconfiguration framework, instead of finding a feasible solution to some instance I of a search problem Q, we are interested in structural and algorithmic questions related to the solution space of Q. Naturally, given some adjacency relation A defined over feasible solutions of Q, size of the symmetric difference being one such relation, the solution space can be represented using a graph R_Q(I). R_Q(I) contains one vertex for each feasible solution of Q on instance I and two vertices share an edge whenever their corresponding solutions are adjacent under A. An edge in R_Q(I) corresponds to a reconfiguration step, a walk in R_Q(I) is a sequence of such steps, a reconfiguration sequence, and R_Q(I) is a reconfiguration graph. Studying problems related to reconfiguration graphs has received considerable attention in recent literature, the most popular problem being to determine whether there exists a reconfiguration sequence between two given feasible solutions; for most NP-complete problems, this problem has been shown to be PSPACE-complete. The purpose of our work is to embark on a systematic investigation of the tractability and structural properties of such problems under both classical and parameterized complexity assumptions. Parameterized complexity is another framework which has become an essential tool for researchers in computational complexity during the last two decades or so and one of its main goals is to provide a better explanation of why some hard problems (in a classical sense) can be in fact much easier than others. Hence, we are interested in what separates the tractable instances from the intractable ones and the fixed-parameter tractable instances from the fixed-parameter intractable ones. It is clear from the generic definition of reconfiguration problems that several factors affect their complexity status. Our work aims at providing a finer classification of the complexity of reconfiguration problems with respect to some of these factors, including the definition of the adjacency relation A, structural properties of the input instance I, structural properties of the reconfiguration graph, and the length of a reconfiguration sequence. As most of these factors can be numerically quantified, we believe that the investigation of reconfiguration problems under both parameterized and classical complexity assumptions will help us further understand the boundaries between tractability and intractability. We consider reconfiguration problems related to Satisfiability, Coloring, Dominating Set, Vertex Cover, Independent Set, Feedback Vertex Set, and Odd Cycle Transversal, and provide lower bounds, polynomial-time algorithms, and fixed-parameter tractable algorithms. In doing so, we answer some of the questions left open in recent work and push the known boundaries between tractable and intractable even closer. As a byproduct of our initiating work on parameterized reconfiguration problems, we present a generic adaptation of parameterized complexity techniques which we believe can be used as a starting point for studying almost any such problem

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

Optimization Methods for Cluster Analysis in Network-based Data Mining

This dissertation focuses on two optimization problems that arise in network-based data mining, concerning identification of basic community structures (clusters) in graphs: the maximum edge weight clique and maximum induced cluster subgraph problems. We propose a continuous quadratic formulation for the maximum edge weight clique problem, and establish the correspondence between its local optima and maximal cliques in the graph. Subsequently, we present a combinatorial branch-and-bound algorithm for this problem that takes advantage of a polynomial-time solvable nonconvex relaxation of the proposed formulation. We also introduce a linear-time-computable analytic upper bound on the clique number of a graph, as well as a new method of upper-bounding the maximum edge weight clique problem, which leads to another exact algorithm for this problem. For the maximum induced cluster subgraph problem, we present the results of a comprehensive polyhedral analysis. We derive several families of facet-defining valid inequalities for the IUC polytope associated with a graph. We also provide a complete description of this polytope for some special classes of graphs. We establish computational complexity of the separation problems for most of the considered families of valid inequalities, and explore the effectiveness of employing the corresponding cutting planes in an integer (linear) programming framework for the maximum induced cluster subgraph problem