The generalized coloring numbers of Kierstead and Yang (Order 2003) offer an algorithmically-useful characterization of graph classes with bounded expansion. In this work, we consider the hardness and approximability of these parameters. First, we complete the work of Grohe et al. (WG 2015) by showing that computing the weak 2-coloring number is NP-hard. Our approach further establishes that determining if a graph has weak $r$-coloring number at most $k$ is para-NP-hard when parameterized by $k$ for all $r \geq 2$. We adapt this to determining if a graph has $r$-coloring number at most $k$ as well, proving para-NP-hardness for all $r \geq 2$. Para-NP-hardness implies that no XP algorithm (runtime $O(n^{f(k)})$) exists for testing if a generalized coloring number is at most $k$. Moreover, there exists a constant $c$ such that it is NP-hard to approximate the generalized coloring numbers within a factor of $c$. To complement these results, we give an approximation algorithm for the generalized coloring numbers, improving both the runtime and approximation factor of the existing approach of Dvo\v{r}\'{a}k (EuJC 2013). We prove that greedily ordering vertices with small estimated backconnectivity achieves a $(k-1)^{r-1}$-approximation for the $r$-coloring number and an $O(k^{r-1})$-approximation for the weak $r$-coloring number.
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Consider words of length $n$. The set of all periods of a word of length $n$ is a subset of $\{0,1,2,\ldots,n-1\}$. However, any subset of $\{0,1,2,\ldots,n-1\}$ is not necessarily a valid set of periods. In a seminal paper in 1981, Guibas and Odlyzko have proposed to encode the set of periods of a word into an $n$ long binary string, called an autocorrelation, where a one at position $i$ denotes the period $i$. They considered the question of recognizing a valid period set, and also studied the number of valid period sets for length $n$, denoted $\kappa_n$. They conjectured that $\ln(\kappa_n)$ asymptotically converges to a constant times $\ln^2(n)$. If improved lower bounds for $\ln(\kappa_n)/\ln^2(n)$ were proposed in 2001, the question of a tight upper bound has remained opened since Guibas and Odlyzko's paper. Here, we exhibit an upper bound for this fraction, which implies its convergence and closes this long standing conjecture. Moreover, we extend our result to find similar bounds for the number of correlations: a generalization of autocorrelations which encodes the overlaps between two strings.
Given graphs $H$ and $G$, possibly with vertex-colors, a homomorphism is a function $f:V(H)\to V(G)$ that preserves colors and edges. Many interesting counting problems (e.g., subgraph and induced subgraph counts) are finite linear combinations $p(\cdot)=\sum_{H}\alpha_{H}\hom(H,\cdot)$ of homomorphism counts, and such linear combinations are known to be hard to evaluate iff they contain a large-treewidth graph $S$. The hardness can be shown in two steps: First, the problems $\hom(S,\cdot)$ for colorful (i.e., bijectively colored) large-treewidth graphs $S$ are shown to be hard. In a second step, these problems are reduced to finite linear combinations of homomorphism counts that contain the uncolored version $S^{\circ}$ of $S$. This step can be performed via inclusion-exclusion in $2^{|E(S)|}\mathrm{poly}(n,s)$ time, where $n$ is the size of the input graph and $s$ is the maximum number of vertices among all graphs in the linear combination. We show that the second step can be performed even in time $4^{\Delta(S)}\mathrm{poly}(n,s)$, where $\Delta(S)$ is the maximum degree of $S$. Our reduction is based on graph products with Cai-F\"urer-Immerman graphs, a novel technique that is likely of independent interest. For colorful graphs $S$ of constant maximum degree, this technique yields a polynomial-time reduction from $\hom(S,\cdot)$ to linear combinations of homomorphism counts involving $S^{\circ}$. Under certain conditions, it actually suffices that a supergraph $T$ of $S^{\circ}$ is contained in the target linear combination. The new reduction yields $\mathsf{\#P}$-hardness results for several counting problems that could previously be studied only under parameterized complexity assumptions. This includes the problems of counting, on input a graph from a restricted graph class and a general graph $G$, the homomorphisms or (induced) subgraph copies from $H$ in $G$.
Let a polytope $P$ be defined by a system $A x \leq b$. We consider the problem of counting the number of integer points inside $P$, assuming that $P$ is $\Delta$-modular, where the polytope $P$ is called $\Delta$-modular if all the rank sub-determinants of $A$ are bounded by $\Delta$ in the absolute value. We present a new FPT-algorithm, parameterized by $\Delta$ and by the maximal number of vertices in $P$, where the maximum is taken by all r.h.s. vectors $b$. We show that our algorithm is more efficient for $\Delta$-modular problems than the approach of A. Barvinok et al. To this end, we do not directly compute the short rational generating function for $P \cap Z^n$, which is commonly used for the considered problem. Instead, we use the dynamic programming principle to compute its particular representation in the form of exponential series that depends on a single variable. We completely do not rely to the Barvinok's unimodular sign decomposition technique. Using our new complexity bound, we consider different special cases that may be of independent interest. For example, we give FPT-algorithms for counting the integer points number in $\Delta$-modular simplices and similar polytopes that have $n + O(1)$ facets. As a special case, for any fixed $m$, we give an FPT-algorithm to count solutions of the unbounded $m$-dimensional $\Delta$-modular subset-sum problem.
Motivated by an application from geodesy, we introduce a novel clustering problem which is a $k$-center (or k-diameter) problem with a side constraint. For the side constraint, we are given an undirected connectivity graph $G$ on the input points, and a clustering is now only feasible if every cluster induces a connected subgraph in $G$. We call the resulting problems the connected $k$-center problem and the connected $k$-diameter problem. We prove several results on the complexity and approximability of these problems. Our main result is an $O(\log^2{k})$-approximation algorithm for the connected $k$-center and the connected $k$-diameter problem. For Euclidean metrics and metrics with constant doubling dimension, the approximation factor of this algorithm improves to $O(1)$. We also consider the special cases that the connectivity graph is a line or a tree. For the line we give optimal polynomial-time algorithms and for the case that the connectivity graph is a tree, we either give an optimal polynomial-time algorithm or a $2$-approximation algorithm for all variants of our model. We complement our upper bounds by several lower bounds.
We initiate the study of fair distribution of delivery tasks among a set of agents wherein delivery jobs are placed along the vertices of a graph. Our goal is to fairly distribute delivery costs (modeled as a submodular function) among a fixed set of agents while satisfying some desirable notions of economic efficiency. We adopt well-established fairness concepts$\unicode{x2014}$such as envy-freeness up to one item (EF1) and minimax share (MMS)$\unicode{x2014}$to our setting and show that fairness is often incompatible with the efficiency notion of social optimality. Yet, we characterize instances that admit fair and socially optimal solutions by exploiting graph structures. We further show that achieving fairness along with Pareto optimality is computationally intractable. Nonetheless, we design an XP algorithm (parameterized by the number of agents) for finding MMS and Pareto optimal solutions on every instance, and show that the same algorithm can be modified to find efficient solutions along with EF1, when such solutions exist. We complement our theoretical results by experimentally analyzing the price of fairness on randomly generated graph structures.
We study the query version of the approximate heavy hitter and quantile problems. In the former problem, the input is a parameter $\varepsilon$ and a set $P$ of $n$ points in $\mathbb{R}^d$ where each point is assigned a color from a set $C$, and we want to build a structure s.t. given any geometric range $\gamma$, we can efficiently find a list of approximate heavy hitters in $\gamma\cap P$, i.e., colors that appear at least $\varepsilon |\gamma \cap P|$ times in $\gamma \cap P$, as well as their frequencies with an additive error of $\varepsilon |\gamma \cap P|$. In the latter problem, each point is assigned a weight from a totally ordered universe and the query must output a sequence $S$ of $1+1/\varepsilon$ weights s.t. the $i$-th weight in $S$ has approximate rank $i\varepsilon|\gamma\cap P|$, meaning, rank $i\varepsilon|\gamma\cap P|$ up to an additive error of $\varepsilon|\gamma\cap P|$. Previously, optimal results were only known in 1D [WY11] but a few sub-optimal methods were available in higher dimensions [AW17, ACH+12]. We study the problems for 3D halfspace and dominance queries. We consider the real RAM model with integer registers of size $w=\Theta(\log n)$ bits. For dominance queries, we show optimal solutions for both heavy hitter and quantile problems: using linear space, we can answer both queries in time $O(\log n + 1/\varepsilon)$. Note that as the output size is $\frac{1}{\varepsilon}$, after investing the initial $O(\log n)$ searching time, our structure takes on average $O(1)$ time to find a heavy hitter or a quantile! For more general halfspace heavy hitter queries, the same optimal query time can be achieved by increasing the space by an extra $\log_w\frac{1}{\varepsilon}$ (resp. $\log\log_w\frac{1}{\varepsilon}$) factor in 3D (resp. 2D). By spending extra $\log^{O(1)}\frac{1}{\varepsilon}$ factors in time and space, we can also support quantile queries.
We investigate expansions of Presburger arithmetic, i.e., the theory of the integers with addition and order, with additional structure related to exponentiation: either a function that takes a number to the power of $2$, or a predicate for the powers of $2$. The latter theory, denoted $\mathrm{PresPower}$, was introduced by B\"uchi as a first attempt at characterising the sets of tuples of numbers that can be expressed using finite automata; B\"uchi's method does not give an elementary upper bound, and the complexity of this theory has been open. The former theory, denoted as $\mathrm{PresExp}$, was shown decidable by Semenov; while the decision procedure for this theory differs radically from the automata-based method proposed by B\"uchi, the method is also non-elementary. And in fact, the theory with the power function has a non-elementary lower bound. In this paper, we show that while Semenov's and B\"uchi's approaches yield non-elementary blow-ups for $\mathrm{PresPower}$, the theory is in fact decidable in triply exponential time, similar to the best known quantifier-elimination algorithm for Presburger arithmetic. We also provide a $\mathrm{NExpTime}$ upper bound for the existential fragment of $\mathrm{PresExp}$, a step towards a finer-grained analysis of its complexity. Both these results are established by analysing a single parameterized satisfiability algorithm for $\mathrm{PresExp}$, which can be specialized to either the setting of $\mathrm{PresPower}$ or the existential theory of $\mathrm{PresExp}$. Besides the new upper bounds for the existential theory of $\mathrm{PresExp}$ and $\mathrm{PresPower}$, we believe our algorithm provides new intuition for the decidability of these theories, and for the features that lead to non-elementary blow-ups.
This paper categorizes the parameterized complexity of the algorithmic problems Perfect Phylogeny and Triangulating Colored Graphs. We show that they are complete for the parameterized complexity class XALP using a reduction from Tree-chained Multicolor Independent Set and a proof of membership. We introduce the problem Triangulating Multicolored Graphs as a stepping stone and prove XALP-completeness for this problem as well. We also show that, assuming the Exponential Time Hypothesis, there exists no algorithm that solves any of these problems in time $f(k) n^{o(k)}$, where $n$ is the input size, $k$ the parameter, and $f$ any computable function.
For a fixed set ${\cal H}$ of graphs, a graph $G$ is ${\cal H}$-subgraph-free if $G$ does not contain any $H \in {\cal H}$ as a (not necessarily induced) subgraph. A recently proposed framework gives a complete classification on ${\cal H}$-subgraph-free graphs (for finite sets ${\cal H}$) for problems that are solvable in polynomial time on graph classes of bounded treewidth, NP-complete on subcubic graphs, and whose NP-hardness is preserved under edge subdivision. While a lot of problems satisfy these conditions, there are also many problems that do not satisfy all three conditions and for which the complexity ${\cal H}$-subgraph-free graphs is unknown. In this paper, we study problems for which only the first two conditions of the framework hold (they are solvable in polynomial time on classes of bounded treewidth and NP-complete on subcubic graphs, but NP-hardness is not preserved under edge subdivision). In particular, we make inroads into the classification of the complexity of four such problems: $k$-Induced Disjoint Paths, $C_5$-Colouring, Hamilton Cycle and Star $3$-Colouring. Although we do not complete the classifications, we show that the boundary between polynomial time and NP-complete differs among our problems and differs from problems that do satisfy all three conditions of the framework. Hence, we exhibit a rich complexity landscape among problems for ${\cal H}$-subgraph-free graph classes.
A conflict-free open neighborhood coloring of a graph is an assignment of colors to the vertices such that for every vertex there is a color that appears exactly once in its open neighborhood. For a graph G, the smallest number of colors required for such a coloring is called the conflict-free open neighborhood (CFON) chromatic number and is denoted by \chi_{ON}(G). Analogously, we define conflict-free closed neighborhood (CFCN) coloring, and CFCN chromatic number (denoted by \chi_{CN}(G)). First studied in 2002, this problem has received considerable attention. We study the CFON and CFCN coloring problems and show the following results. In what follows, \Delta denotes the maximum degree of the graph. 1. For a K_{1, k}-free graph G, we show that \chi_{ON}(G) = O(k \ln\Delta). This improves the bound of O(k^2 \ln \Delta) from [Bhyravarapu, Kalyanasundaram, Mathew, MFCS 2022]. Since \chi_{CN}(G) \leq 2\chi_{ON}(G), our result implies an upper bound on \chi_{CN}(G) as well. It is known that there exist separate classes of graphs with \chi_{ON}(G) = \Omega(\ln\Delta) and \chi_{ON}(G) = \Omega(k). 2. Let f(\delta) be defined as follows: f(\delta) = max {\chi_{CN} (G) : G is a graph with minimum degree \delta}. It is easy to see that f(\delta') \geq f(\delta) when \delta' < \delta. It is known [Debski and Przybylo, JGT 2021] that f(c \Delta) = \Theta(\log \Delta), for any positive constant c. In this paper, we show that f(c\Delta^{1 - \epsilon}) = \Omega (\ln^2 \Delta), where c, \epsilon are positive constants such that \epsilon < 0.75. Together with the known upper bound \chi_{CN}(G) = O(\ln^2 \Delta), this implies that f(c\Delta^{1 - \epsilon}) = \Theta (\ln^2 \Delta). 3. For a K_{1, k}-free graph G on n vertices, we show that \chi_{CN}(G) = O(\ln k \ln n). This bound is asymptotically tight.
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