Erd\H{o}s and Purdy, and later Agarwal and Sharir, conjectured that any set of $n$ points in $\mathbb R^{d}$ determine at most $Cn^{d/2}$ congruent $k$-simplices for even $d$. We obtain the first significant progress towards this conjecture, showing that this number is at most $C n^{3d/4}$ for $k<d$. As a consequence, we obtain an upper bound of $C n^{3d/4+2}$ for the number of similar $k$-simplices determined by $n$ points in $\mathbb R^d$, which improves the results of Agarwal, Apfelbaum, Purdy and Sharir. This problem is motivated by the problem of exact pattern matching. We also address Zarankiewicz-type questions of finding the maximum number of edges in semi-algebraic graphs with no $K_{u,u}$. Here, we improve the previous result of Fox, Pach, Sheffer, Suk, and Zahl, and Do for $d\le 4$, as well as for any $d$ and moderately large $u$. We get an improvement of their results for any $d$ and $u$ for unit-distance graphs, which was one of the main applications of their results. From a more general prospective, our results are proved using classical cutting techniques. In the recent years, we saw a great development of the polynomial partitioning method in incidence geometry that followed the breakthrough result by Guth and Katz. One consequence of that development is that the attention of the researchers in incidence geometry swayed in polynomial techniques. In this paper, we argue that there is a number of open problems where classical techniques work better.
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Paths $P_1,\ldots, P_k$ in a graph $G=(V,E)$ are mutually induced if any two distinct $P_i$ and $P_j$ have neither common vertices nor adjacent vertices. The Induced Disjoint Paths problem is to decide if a graph $G$ with $k$ pairs of specified vertices $(s_i,t_i)$ contains $k$ mutually induced paths $P_i$ such that each $P_i$ starts from $s_i$ and ends at $t_i$. This is a classical graph problem that is NP-complete even for $k=2$. We introduce a natural generalization, Induced Disjoint Connected Subgraphs: instead of connecting pairs of terminals, we must connect sets of terminals. We give almost-complete dichotomies of the computational complexity of both problems for $H$-free graphs, that is, graphs that do not contain some fixed graph $H$ as an induced subgraph. We also classify the complexity of the second problem (subject to one missing case) if the number of terminal sets is fixed, that is, not part of the input.
Twin-width is a graph width parameter recently introduced by Bonnet, Kim, Thomass\'{e} & Watrigant. Given two graphs $G$ and $H$ and a graph product $\star$, we address the question: is the twin-width of $G\star H$ bounded by a function of the twin-widths of $G$ and $H$ and their maximum degrees? It is known that a bound of this type holds for strong products (Bonnet, Geniet, Kim, Thomass\'{e} & Watrigant; SODA 2021). We show that bounds of the same form hold for Cartesian, tensor/direct, rooted, replacement, and zig-zag products. For the lexicographical product we prove that the twin-width of the product of two graphs is exactly the maximum of the twin-widths of the individual graphs. In contrast, for the modular product we show that no bound can hold. In addition, we provide examples showing many of our bounds are tight, and give improved bounds for certain classes of graphs.
The main theme of this paper is using $k$-dimensional generalizations of the combinatorial Boolean Matrix Multiplication (BMM) hypothesis and the closely-related Online Matrix Vector Multiplication (OMv) hypothesis to prove new tight conditional lower bounds for dynamic problems. The combinatorial $k$-Clique hypothesis, which is a standard hypothesis in the literature, naturally generalizes the combinatorial BMM hypothesis. In this paper, we prove tight lower bounds for several dynamic problems under the combinatorial $k$-Clique hypothesis. For instance, we show that: * The Dynamic Range Mode problem has no combinatorial algorithms with $\mathrm{poly}(n)$ pre-processing time, $O(n^{2/3-\epsilon})$ update time and $O(n^{2/3-\epsilon})$ query time for any $\epsilon > 0$, matching the known upper bounds for this problem. Previous lower bounds only ruled out algorithms with $O(n^{1/2-\epsilon})$ update and query time under the OMv hypothesis. Other examples include tight combinatorial lower bounds for Dynamic Subgraph Connectivity, Dynamic 2D Orthogonal Range Color Counting, Dynamic 2-Pattern Document Retrieval, and Dynamic Range Mode in higher dimensions. Furthermore, we propose the OuMv$_k$ hypothesis as a natural generalization of the OMv hypothesis. Under this hypothesis, we prove tight lower bounds for various dynamic problems. For instance, we show that: * The Dynamic Skyline Points Counting problem in $(2k-1)$-dimensional space has no algorithm with $\mathrm{poly}(n)$ pre-processing time and $O(n^{1-1/k-\epsilon})$ update and query time for $\epsilon > 0$, even if the updates are semi-online. Other examples include tight conditional lower bounds for (semi-online) Dynamic Klee's measure for unit cubes, and high-dimensional generalizations of Erickson's problem and Langerman's problem.
We prove that the number of edges of a multigraph $G$ with $n$ vertices is at most $O(n^2\log n)$, provided that any two edges cross at most once, parallel edges are noncrossing, and the lens enclosed by every pair of parallel edges in $G$ contains at least one vertex. As a consequence, we prove the following extension of the Crossing Lemma of Ajtai, Chv\'atal, Newborn, Szemer\'edi and Leighton, if $G$ has $e \geq 4n$ edges, in any drawing of $G$ with the above property, the number of crossings is $\Omega\left(\frac{e^3}{n^2\log(e/n)}\right)$. This answers a question of Kaufmann et al. and is tight up to the logarithmic factor.
The subset sum problem is known to be an NP-hard problem in the field of computer science with the fastest known approach having a run-time complexity of $O(2^{0.3113n})$. A modified version of this problem is known as the perfect sum problem and extends the subset sum idea further. This extension results in additional complexity, making it difficult to compute for a large input. In this paper, I propose a probabilistic approach which approximates the solution to the perfect sum problem by approximating the distribution of potential sums. Since this problem is an extension of the subset sum, our approximation also grants some probabilistic insight into the solution for the subset sum problem. We harness distributional approximations to model the number of subsets which sum to a certain size. These distributional approximations are formulated in two ways: using bounds to justify normal approximation, and approximating the empirical distribution via density estimation. These approximations can be computed in $O(n)$ complexity, and can increase in accuracy with the size of the input data making it useful for large-scale combinatorial problems. Code is available at //github.com/KristofPusztai/PerfectSum.
We study the problem of assortative and disassortative partitions on random $d$-regular graphs. Nodes in the graph are partitioned into two non-empty groups. In the assortative partition every node requires at least $H$ of their neighbors to be in their own group. In the disassortative partition they require less than $H$ neighbors to be in their own group. Using the cavity method based on analysis of the Belief Propagation algorithm we establish for which combinations of parameters $(d,H)$ these partitions exist with high probability and for which they do not. For $H>\lceil \frac{d}{2} \rceil $ we establish that the structure of solutions to the assortative partition problems corresponds to the so-called frozen-1RSB. This entails a conjecture of algorithmic hardness of finding these partitions efficiently. For $H \le \lceil \frac{d}{2} \rceil $ we argue that the assortative partition problem is algorithmically easy on average for all $d$. Further we provide arguments about asymptotic equivalence between the assortative partition problem and the disassortative one, going trough a close relation to the problem of single-spin-flip-stable states in spin glasses. In the context of spin glasses, our results on algorithmic hardness imply a conjecture that gapped single spin flip stable states are hard to find which may be a universal reason behind the observation that physical dynamics in glassy systems display convergence to marginal stability.
We study the problem of learning a hypergraph via edge detecting queries. In this problem, a learner queries subsets of vertices of a hidden hypergraph and observes whether these subsets contain an edge or not. In general, learning a hypergraph with $m$ edges of maximum size $d$ requires $\Omega((2m/d)^{d/2})$ queries. In this paper, we aim to identify families of hypergraphs that can be learned without suffering from a query complexity that grows exponentially in the size of the edges. We show that hypermatchings and low-degree near-uniform hypergraphs with $n$ vertices are learnable with poly$(n)$ queries. For learning hypermatchings (hypergraphs of maximum degree $ 1$), we give an $O(\log^3 n)$-round algorithm with $O(n \log^5 n)$ queries. We complement this upper bound by showing that there are no algorithms with poly$(n)$ queries that learn hypermatchings in $o(\log \log n)$ adaptive rounds. For hypergraphs with maximum degree $\Delta$ and edge size ratio $\rho$, we give a non-adaptive algorithm with $O((2n)^{\rho \Delta+1}\log^2 n)$ queries. To the best of our knowledge, these are the first algorithms with poly$(n, m)$ query complexity for learning non-trivial families of hypergraphs that have a super-constant number of edges of super-constant size.
The rooted subtree prune and regraft (rSPR) distance between two rooted binary phylogenetic trees is a well-studied measure of topological dissimilarity that is NP-hard to compute. Here we describe an improved linear kernel for the problem. In particular, we show that if the classical subtree and chain reduction rules are augmented with a modified type of chain reduction rule, the resulting trees have at most 9k-3 leaves, where k is the rSPR distance; and that this bound is tight. The previous best-known linear kernel had size O(28k). To achieve this improvement we introduce cyclic generators, which can be viewed as cyclic analogues of the generators used in the phylogenetic networks literature. As a corollary to our main result we also give an improved weighted linear kernel for the minimum hybridization problem on two rooted binary phylogenetic trees.
We present a polynomial-time quantum algorithm for the Hidden Subgroup Problem over $\mathbb{D}_{2^n}$. The usual approach to the Hidden Subgroup Problem relies on harmonic analysis in the domain of the problem, and the best known algorithm using this approach has time complexity in $2^{\mathcal{O}(\sqrt{n})}$. By focusing on structure encoded in the codomain of the problem, we develop a polynomial-time algorithm which uses this structure to direct a "walk" down the subgroup lattice of $\mathbb{D}_{2^n}$ terminating at the hidden subgroup.
Consider a univariate polynomial f in Z[x] with degree d, exactly t monomial terms, and coefficients in {-H,...,H}. Solving f over the reals, R, in polynomial-time can be defined as counting the exact number of real roots of f and then finding (for each such root z) an approximation w of logarithmic height (log(dH))^{O(1)} such that the Newton iterates of w have error decaying at a rate of O((1/2)^{2^i}). Solving efficiently in this sense, using (log(dH))^{O(1)} deterministic bit operations, is arguably the most honest formulation of solving a polynomial equation over R in time polynomial in the input size. Unfortunately, deterministic algorithms this fast are known only for t=2, unknown for t=3, and provably impossible for t=4. (One can of course resort to older techniques with complexity (d\log H)^{O(1)} for t>=4.) We give evidence that polynomial-time real-solving in the strong sense above is possible for t=3: We give a polynomial-time algorithm employing A-hypergeometric series that works for all but a fraction of 1/Omega(log(dH)) of the input f. We also show an equivalence between fast trinomial solving and sign evaluation at rational points of small height. As a consequence, we show that for "most" trinomials f, we can compute the sign of f at a rational point r in time polynomial in log(dH) and the logarithmic height of r. (This was known only for binomials before.) We also mention a related family of polynomial systems that should admit a similar speed-up for solving.
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