In 1943, Hadwiger conjectured that every graph with no $K_t$ minor is $(t-1)$-colorable for every $t\ge 1$. In the 1980s, Kostochka and Thomason independently proved that every graph with no $K_t$ minor has average degree $O(t\sqrt{\log t})$ and hence is $O(t\sqrt{\log t})$-colorable. Recently, Norin, Song and the second author showed that every graph with no $K_t$ minor is $O(t(\log t)^{\beta})$-colorable for every $\beta > 1/4$, making the first improvement on the order of magnitude of the $O(t\sqrt{\log t})$ bound. The first main result of this paper is that every graph with no $K_t$ minor is $O(t\log\log t)$-colorable. This is a corollary of our main technical result that the chromatic number of a $K_t$-minor-free graph is bounded by $O(t(1+f(G,t)))$ where $f(G,t)$ is the maximum of $\frac{\chi(H)}{a}$ over all $a\ge \frac{t}{\sqrt{\log t}}$ and $K_a$-minor-free subgraphs $H$ of $G$ that are small (i.e. $O(a\log^4 a)$ vertices). This has a number of interesting corollaries. First as mentioned, using the current best-known bounds on coloring small $K_t$-minor-free graphs, we show that $K_t$-minor-free graphs are $O(t\log\log t)$-colorable. Second, it shows that proving Linear Hadwiger's Conjecture (that $K_t$-minor-free graphs are $O(t)$-colorable) reduces to proving it for small graphs. Third, we prove that $K_t$-minor-free graphs with clique number at most $\sqrt{\log t}/ (\log \log t)^2$ are $O(t)$-colorable. This implies our final corollary that Linear Hadwiger's Conjecture holds for $K_r$-free graphs for every fixed $r$. One key to proving the main theorem is a new standalone result that every $K_t$-minor-free graph of average degree $d=\Omega(t)$ has a subgraph on $O(t \log^3 t)$ vertices with average degree $\Omega(d)$.
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We consider the all pairs all shortest paths (APASP) problem, which maintains all of the multiple shortest paths for every vertex pair in a directed graph $G=(V,E)$ with a positive real weight on each edge. We present two fully dynamic algorithms for this problem in which an update supports either weight increases or weight decreases on a subset of edges incident to a vertex. Our first algorithm runs in amortized $O({\nu^*}^2 \cdot \log^3 n)$ time per update, where $n = |V|$, and $\nu^*$ bounds the number of edges that lie on shortest paths through any single vertex. Our APASP algorithm leads to the same amortized bound for the fully dynamic computation of betweenness centrality (BC), which is a parameter widely used in the analysis of large complex networks. Our method is a generalization and a variant of the fully dynamic algorithm of Demetrescu and Italiano [DI04] for unique shortest path, and it builds on our recent decremental APASP [NPR14]. Our second (faster) algorithm reduces the amortized cost per operation by a logarithmic factor, and uses new data structures and techniques that are extensions of methods in a fully dynamic algorithm by Thorup.
For relational structures A, B of the same signature, the Promise Constraint Satisfaction Problem PCSP(A,B) asks whether a given input structure maps homomorphically to A or does not even map to B. We are promised that the input satisfies exactly one of these two cases. If there exists a structure C with homomorphisms $A\to C\to B$, then PCSP(A,B) reduces naturally to CSP(C). To the best of our knowledge all known tractable PCSPs reduce to tractable CSPs in this way. However Barto showed that some PCSPs over finite structures A, B require solving CSPs over infinite C. We show that even when such a reduction to finite C is possible, this structure may become arbitrarily large. For every integer $n>1$ and every prime p we give A, B of size n with a single relation of arity $n^p$ such that PCSP(A, B) reduces via a chain of homomorphisms $ A\to C\to B$ to a tractable CSP over some C of size p but not over any smaller structure. In a second family of examples, for every prime $p\geq 7$ we construct A, B of size $p-1$ with a single ternary relation such that PCSP(A, B) reduces via $A\to C\to B$ to a tractable CSP over some C of size p but not over any smaller structure. In contrast we show that if A, B are graphs and PCSP(A,B) reduces to tractable CSP(C) for some finite digraph C, then already A or B has a tractable CSP. This extends results and answers a question of Deng et al.
A directed graph is oriented if it can be obtained by orienting the edges of a simple, undirected graph. For an oriented graph $G$, let $\beta(G)$ denote the size of a minimum feedback arc set, a smallest subset of edges whose deletion leaves an acyclic subgraph. A simple consequence of a result of Berger and Shor is that any oriented graph $G$ with $m$ edges satisfies $\beta(G) = m/2 - \Omega(m^{3/4})$. We observe that if an oriented graph $G$ has a fixed forbidden subgraph $B$, the upper bound of $\beta(G) = m/2 - \Omega(m^{3/4})$ is best possible as a function of the number of edges if $B$ is not bipartite, but the exponent $3/4$ in the lower order term can be improved if $B$ is bipartite. We also show that for every rational number $r$ between $3/4$ and $1$, there is a finite collection of digraphs $\mathcal{B}$ such that every $\mathcal{B}$-free digraph $G$ with $m$ edges satisfies $\beta(G) = m/2 - \Omega(m^r)$, and this bound is best possible up to the implied constant factor. The proof uses a connection to Tur\'an numbers and a result of Bukh and Conlon. Both of our upper bounds come equipped with randomized linear-time algorithms that construct feedback arc sets achieving those bounds. Finally, we give a characterization of quasirandom directed graphs via minimum feedback arc sets.
Given a set $P$ of $n$ points in the plane, the $k$-center problem is to find $k$ congruent disks of minimum possible radius such that their union covers all the points in $P$. The $2$-center problem is a special case of the $k$-center problem that has been extensively studied in the recent past \cite{CAHN,HT,SH}. In this paper, we consider a generalized version of the $2$-center problem called \textit{proximity connected} $2$-center (PCTC) problem. In this problem, we are also given a parameter $\delta\geq 0$ and we have the additional constraint that the distance between the centers of the disks should be at most $\delta$. Note that when $\delta=0$, the PCTC problem is reduced to the $1$-center(minimum enclosing disk) problem and when $\delta$ tends to infinity, it is reduced to the $2$-center problem. The PCTC problem first appeared in the context of wireless networks in 1992 \cite{ACN0}, but obtaining a nontrivial deterministic algorithm for the problem remained open. In this paper, we resolve this open problem by providing a deterministic $O(n^2\log n)$ time algorithm for the problem.
A natural way of increasing our understanding of NP-complete graph problems is to restrict the input to a special graph class. Classes of $H$-free graphs, that is, graphs that do not contain some graph $H$ as an induced subgraph, have proven to be an ideal testbed for such a complexity study. However, if the forbidden graph $H$ contains a cycle or claw, then these problems often stay NP-complete. A recent complexity study on the $k$-Colouring problem shows that we may still obtain tractable results if we also bound the diameter of the $H$-free input graph. We continue this line of research by initiating a complexity study on the impact of bounding the diameter for a variety of classical vertex partitioning problems restricted to $H$-free graphs. We prove that bounding the diameter does not help for Independent Set, but leads to new tractable cases for problems closely related to 3-Colouring. That is, we show that Near-Bipartiteness, Independent Feedback Vertex Set, Independent Odd Cycle Transversal, Acyclic 3-Colouring and Star 3-Colouring are all polynomial-time solvable for chair-free graphs of bounded diameter. To obtain these results we exploit a new structural property of 3-colourable chair-free graphs.
Computing a maximum independent set (MaxIS) is a fundamental NP-hard problem in graph theory, which has important applications in a wide spectrum of fields. Since graphs in many applications are changing frequently over time, the problem of maintaining a MaxIS over dynamic graphs has attracted increasing attention over the past few years. Due to the intractability of maintaining an exact MaxIS, this paper aims to develop efficient algorithms that can maintain an approximate MaxIS with an accuracy guarantee theoretically. In particular, we propose a framework that maintains a $(\frac{\Delta}{2} + 1)$-approximate MaxIS over dynamic graphs and prove that it achieves a constant approximation ratio in many real-world networks. To the best of our knowledge, this is the first non-trivial approximability result for the dynamic MaxIS problem. Following the framework, we implement an efficient linear-time dynamic algorithm and a more effective dynamic algorithm with near-linear expected time complexity. Our thorough experiments over real and synthetic graphs demonstrate the effectiveness and efficiency of the proposed algorithms, especially when the graph is highly dynamic.
Computing a dense subgraph is a fundamental problem in graph mining, with a diverse set of applications ranging from electronic commerce to community detection in social networks. In many of these applications, the underlying context is better modelled as a weighted hypergraph that keeps evolving with time. This motivates the problem of maintaining the densest subhypergraph of a weighted hypergraph in a {\em dynamic setting}, where the input keeps changing via a sequence of updates (hyperedge insertions/deletions). Previously, the only known algorithm for this problem was due to Hu et al. [HWC17]. This algorithm worked only on unweighted hypergraphs, and had an approximation ratio of $(1+\epsilon)r^2$ and an update time of $O(\text{poly} (r, \log n))$, where $r$ denotes the maximum rank of the input across all the updates. We obtain a new algorithm for this problem, which works even when the input hypergraph is weighted. Our algorithm has a significantly improved (near-optimal) approximation ratio of $(1+\epsilon)$ that is independent of $r$, and a similar update time of $O(\text{poly} (r, \log n))$. It is the first $(1+\epsilon)$-approximation algorithm even for the special case of weighted simple graphs. To complement our theoretical analysis, we perform experiments with our dynamic algorithm on large-scale, real-world data-sets. Our algorithm significantly outperforms the state of the art [HWC17] both in terms of accuracy and efficiency.
Let $L_{k,\alpha}^{\mathbb{Z}}$ denote the set of all bi-infinite $\alpha$-power free words over an alphabet with $k$ letters, where $\alpha$ is a positive rational number and $k$ is positive integer. We prove that if $\alpha\geq 5$, $k\geq 3$, $v\in L_{k,\alpha}^{\mathbb{Z}}$, and $w$ is a finite factor of $v$, then there are $\widetilde v\in L_{k,\alpha}^{\mathbb{Z}}$ and a letter $x$ such that $w$ is a factor of $\widetilde v$ and $x$ has only a finitely many occurrences in $\widetilde v$.
Most existing works of polar codes focus on the analysis of block error probability. However, in many scenarios, bit error probability is also important for evaluating the performance of channel codes. In this paper, we establish a new framework to analyze the bit error probability of polar codes. Specifically, by revisiting the error event of bit-channel, we first introduce the conditional bit error probability as a metric to evaluate the reliability of bit-channel for both systematic and non-systematic polar codes. Guided by the concept of polar subcode, we then derive an upper bound on the conditional bit error probability of each bit-channel, and accordingly, an upper bound on the bit error probability of polar codes. Based on these, two types of construction metrics aiming at minimizing the bit error probability of polar codes are proposed, which are of linear computational complexity and explicit forms. Simulation results show that the polar codes constructed by the proposed methods can outperform those constructed by the conventional methods.
A palindromic substring $T[i.. j]$ of a string $T$ is said to be a shortest unique palindromic substring (SUPS) in $T$ for an interval $[p, q]$ if $T[i.. j]$ is a shortest one such that $T[i.. j]$ occurs only once in $T$, and $[i, j]$ contains $[p, q]$. The SUPS problem is, given a string $T$ of length $n$, to construct a data structure that can compute all the SUPSs for any given query interval. It is known that any SUPS query can be answered in $O(\alpha)$ time after $O(n)$-time preprocessing, where $\alpha$ is the number of SUPSs to output [Inoue et al., 2018]. In this paper, we first show that $\alpha$ is at most $4$, and the upper bound is tight. Also, we present an algorithm to solve the SUPS problem for a sliding window that can answer any query in $O(\log\log W)$ time and update data structures in amortized $O(\log\sigma)$ time, where $W$ is the size of the window, and $\sigma$ is the alphabet size. Furthermore, we consider the SUPS problem in the after-edit model and present an efficient algorithm. Namely, we present an algorithm that uses $O(n)$ time for preprocessing and answers any $k$ SUPS queries in $O(\log n\log\log n + k\log\log n)$ time after single character substitution. As a by-product, we propose a fully-dynamic data structure for range minimum queries (RmQs) with a constraint where the width of each query range is limited to polylogarithmic. The constrained RmQ data structure can answer such a query in constant time and support a single-element edit operation in amortized constant time.
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