arXiv subjects

Illya V. Hicks

Publications and source records attributed to Illya V. Hicks.

17 recordsLinked to original sources

arXiv2026

Characterizing QUBO Reformulations of the Max-k-Cut Problem for Quantum Computing

Quantum computing offers significant potential for solving NP-hard combinatorial (optimization) problems that are beyond the reach of classical computers. One way to tap into this potential is by reformulating combinatorial problems as a quadratic unconstrained binary optimization (QUBO) problem. The solution of the QUBO reformulation can then be addressed using adiabatic quantum computing devices or appropriate quantum computing algorithms on gate-based quantum computing devices. In general, QUBO reformulations of combinatorial problems can be readily obtained by properly penalizing the violation of the problem's constraints in the original problem's objective. However, characterizing tight (i.e., minimal but sufficient) penalty coefficients for this purpose is important and non-trivial for enabling the solution of the resulting QUBO in current and near-term quantum computing devices. Along these lines, we present closed-form characterizations of tight penalty coefficients for two distinct QUBO reformulations of the max $k$-cut problem whose values depend on the (weighted) degree of the vertices of the graph defining the problem. These findings contribute to the ongoing effort to make quantum computing a viable tool for solving combinatorial problems at scale. We support our theoretical results with illustrative examples and simple numerical results.

quant-ph↗

arXiv2025

Optimizing Cost Hamiltonian Compilation for Max-Cut QAOA on Unweighted Graphs Using Global Controls and Qubit Bit Flips

We study a cost Hamiltonian compilation problem for the quantum approximate optimization algorithm (QAOA) applied to the Max-Cut problem, focusing on trapped-ion quantum computers. Instead of standard compilation with CNOT and Rz gates, we employ global coupling operations and single-qubit bit flips. Prior work by Rajakumar et al. established that such a compilation is always possible. Minimizing operational error requires short operation sequences. The problem reduces to a low-rank semi-discrete decomposition of the graph's adjacency matrix, where the minimum achievable rank, the graph coupling number gc(G), represents the number of global control layers. Rajakumar et al. introduced the Union of Stars construction, proving gc(G) <= 3n - 2 for unweighted graphs with n vertices, and gave an O(m)-rank construction for weighted graphs. We concentrate on unweighted graphs. We derive structural properties of the compilation problem and show the Union of Stars method is order-optimal by proving a lower bound of gc(G) >= n - 1 for a family of graphs. We also improve the general upper bound to 2.5n + 2. For particular graph families -- cliques, perfect matchings, paths, and cycles -- we provide sharper bounds. Further, we reveal a link between the problem and Hadamard matrix theory. Finally, we introduce a compact mixed-integer programming (MIP) formulation that outperforms the previously studied exponential-size MIP.

quant-ph↗

arXiv2024

Optimal Mixed Integer Linear Optimization Trained Multivariate Classification Trees

Multivariate decision trees are powerful machine learning tools for classification and regression that attract many researchers and industry professionals. An optimal binary tree has two types of vertices, (i) branching vertices which have exactly two children and where datapoints are assessed on a set of discrete features and (ii) leaf vertices at which datapoints are given a prediction, and can be obtained by solving a biobjective optimization problem that seeks to (i) maximize the number of correctly classified datapoints and (ii) minimize the number of branching vertices. Branching vertices are linear combinations of training features and therefore can be thought of as hyperplanes. In this paper, we propose two cut-based mixed integer linear optimization (MILO) formulations for designing optimal binary classification trees (leaf vertices assign discrete classes). Our models leverage on-the-fly identification of minimal infeasible subsystems (MISs) from which we derive cutting planes that hold the form of packing constraints. We show theoretical improvements on the strongest flow-based MILO formulation currently in the literature and conduct experiments on publicly available datasets to show our models' ability to scale, strength against traditional branch and bound approaches, and robustness in out-of-sample test performance. Our code and data are available on GitHub.

cs.LG↗

arXiv2023

Combinatorial Disjunctive Constraints for Obstacle Avoidance in Path Planning

We present a new approach for modeling avoidance constraints in 2D environments, in which waypoints are assigned to obstacle-free polyhedral regions. Constraints of this form are often formulated as mixed-integer programming (MIP) problems employing big-M techniques -- however, these are generally not the strongest formulations possible with respect to the MIP's convex relaxation (so called ideal formulations), potentially resulting in larger computational burden. We instead model obstacle avoidance as combinatorial disjunctive constraints and leverage the independent branching scheme to construct small, ideal formulations. As our approach requires a biclique cover for an associated graph, we exploit the structure of this class of graphs to develop a fast subroutine for obtaining biclique covers in polynomial time. We also contribute an open-source Julia library named ClutteredEnvPathOpt to facilitate computational experiments of MIP formulations for obstacle avoidance. Experiments have shown our formulation is more compact and remains competitive on a number of instances compared with standard big-M techniques, for which solvers possess highly optimized procedures.

math.OC↗

arXiv2023

On relaxations of the max $k$-cut problem formulations

A tight continuous relaxation is a crucial factor in solving mixed integer formulations of many NP-hard combinatorial optimization problems. The (weighted) max $k$-cut problem is a fundamental combinatorial optimization problem with multiple notorious mixed integer optimization formulations. In this paper, we explore four existing mixed integer optimization formulations of the max $k$-cut problem. Specifically, we show that the continuous relaxation of a binary quadratic optimization formulation of the problem is: (i) stronger than the continuous relaxation of two mixed integer linear optimization formulations and (ii) at least as strong as the continuous relaxation of a mixed integer semidefinite optimization formulation. We also conduct a set of experiments on multiple sets of instances of the max $k$-cut problem using state-of-the-art solvers that empirically confirm the theoretical results in item (i). Furthermore, these numerical results illustrate the advances in the efficiency of global non-convex quadratic optimization solvers and more general mixed integer nonlinear optimization solvers. As a result, these solvers provide a promising option to solve combinatorial optimization problems. Our codes and data are available on GitHub.

math.OC↗

arXiv2023

Mixed integer linear optimization formulations for learning optimal binary classification trees

Decision trees are powerful tools for classification and regression that attract many researchers working in the burgeoning area of machine learning. One advantage of decision trees over other methods is their interpretability, which is often preferred over other higher accuracy methods that are relatively uninterpretable. A binary classification tree has two types of vertices: (i) branching vertices which have exactly two children and where datapoints are assessed on a set of discrete features; and (ii) leaf vertices at which datapoints are given a discrete prediction. An optimal binary classification tree can be obtained by solving a biobjective optimization problem that seeks to (i) maximize the number of correctly classified datapoints and (ii) minimize the number of branching vertices. In this paper, we propose four mixed integer linear optimization (MILO) formulations for designing optimal binary classification trees: two flow-based formulations and two-cut based formulations. We provide theoretical comparisons between our proposed formulations and the strongest flow-based MILO formulation of Aghaei et al. (2021). We conduct experiments on 13 publicly available datasets to show the models' ability to scale and the strength of a biobjective approach using Pareto frontiers. Our code and data are available on GitHub.

cs.LG↗

arXiv2023

Building Formulations for Piecewise Linear Relaxations of Nonlinear Functions

We study mixed-integer programming formulations for the piecewise linear lower and upper bounds (in other words, piecewise linear relaxations) of nonlinear functions that can be modeled by a new class of combinatorial disjunctive constraints (CDCs), generalized $n$D-ordered CDCs. We first introduce a general formulation technique to model piecewise linear lower and upper bounds of univariate nonlinear functions concurrently so that it uses fewer binary variables than modeling bounds separately. Next, we propose logarithmically sized ideal non-extended formulations to model the piecewise linear relaxations of univariate and higher-dimensional nonlinear functions under the CDC and independent branching frameworks. We also perform computational experiments for the approaches modeling the piecewise linear relaxations of univariate nonlinear functions and show significant speed-ups of our proposed formulations. Furthermore, we demonstrate that piecewise linear relaxations can provide strong dual bounds of the original problems with less computational time in order of magnitude.

math.OC↗

arXiv2023

Maximal Clique and Edge-Ranking Bounds of Biclique Cover Number

The biclique cover number $(\text{bc})$ of a graph $G$ denotes the minimum number of complete bipartite (biclique) subgraphs to cover all the edges of the graph. In this paper, we show that $\text{bc}(G) \geq \lceil \log_2(\text{mc}(G^c)) \rceil \geq \lceil \log_2(χ(G)) \rceil$ for an arbitrary graph $G$, where $χ(G)$ is the chromatic number of $G$ and $\text{mc}(G^c)$ is the number of maximal cliques of the complementary graph $G^c$, i.e., the number of maximal independent sets of $G$. We also show that $\lceil \log_2(\text{mc}(G^c)) \rceil$ could be a strictly tighter lower bound of the biclique cover number than other existing lower bounds. We can also provide a bound of $\text{bc}(G)$ with respect to the biclique partition number ($\text{bp}$) of $G$: $\text{bc}(G) \geq \lceil \log_2(\text{bp}(G) + 1) \rceil$ or $\text{bp}(G) \leq 2^{\text{bc}(G)} - 1$ if $G$ is co-chordal. Furthermore, we show that $\text{bc}(G) \leq χ_r'(T_{{K}^c})$, where $G$ is a co-chordal graph such that each vertex is in at most two maximal independent sets and $χ_r'({T}_{{K}^c})$ is the optimal edge-ranking number of a clique tree of $G^c$.

math.CO↗

arXiv2023

Finding Biclique Partitions of Co-Chordal Graphs

The biclique partition number $(\text{bp})$ of a graph $G$ is referred to as the least number of complete bipartite (biclique) subgraphs that are required to cover the edges of the graph exactly once. In this paper, we show that the biclique partition number ($\text{bp}$) of a co-chordal (complementary graph of chordal) graph $G = (V, E)$ is less than the number of maximal cliques ($\text{mc}$) of its complementary graph: a chordal graph $G^c = (V, E^c)$. We first provide a general framework of the ``divide and conquer" heuristic of finding minimum biclique partitions of co-chordal graphs based on clique trees. Furthermore, a heuristic of complexity $O[|V|(|V|+|E^c|)]$ is proposed by applying lexicographic breadth-first search to find structures called moplexes. Either heuristic gives us a biclique partition of $G$ with size $\text{mc}(G^c)-1$. In addition, we prove that both of our heuristics can solve the minimum biclique partition problem on $G$ exactly if its complement $G^c$ is chordal and clique vertex irreducible. We also show that $\text{mc}(G^c) - 2 \leq \text{bp}(G) \leq \text{mc}(G^c) - 1$ if $G$ is a split graph.

math.CO↗

arXiv2022

Modeling Combinatorial Disjunctive Constraints via Junction Trees

We introduce techniques to build small ideal mixed-integer programming (MIP) formulations of combinatorial disjunctive constraints (CDCs) via the independent branching scheme. We present a novel pairwise IB-representable class of CDCs, CDCs admitting junction trees, and provide a combinatorial procedure to build MIP formulations for those constraints. Generalized special ordered sets ($\text{SOS} k$) can be modeled by CDCs admitting junction trees and we also obtain MIP formulations of $\text{SOS} k$. Furthermore, we provide a novel ideal extended formulation of any combinatorial disjunctive constraints with fewer auxiliary binary variables with an application in planar obstacle avoidance.

math.OC↗

arXiv2021

An integer program and new lower bounds for computing the strong rainbow connection numbers of graphs

We present an integer programming model to compute the strong rainbow connection number, $src(G)$, of any simple graph $G$. We introduce several enhancements to the proposed model, including a fast heuristic, and a variable elimination scheme. Moreover, we present a novel lower bound for $src(G)$ which may be of independent research interest. We solve the integer program both directly and using an alternative method based on iterative lower bound improvement, the latter of which we show to be highly effective in practice. To our knowledge, these are the first computational methods for the strong rainbow connection problem. We demonstrate the efficacy of our methods by computing the strong rainbow connection numbers of graphs containing up to $379$ vertices.

math.CO↗

arXiv2020

Optimal Sensor Placement in Power Grids: Power Domination, Set Covering, and the Neighborhoods of Zero Forcing Forts

To monitor electrical activity throughout the power grid and mitigate outages, sensors known as phasor measurement units can installed. Due to implementation costs, it is desirable to minimize the number of sensors deployed while ensuring that the grid can be effectively monitored. This optimization problem motivates the graph theoretic power dominating set problem. In this paper, we propose a novel integer program for identifying minimum power dominating sets by formulating a set cover problem. This problem's constraints correspond to neighborhoods of zero forcing forts; we study their structural properties and show they can be separated, allowing the proposed model to be solved via row generation. The proposed and existing methods are compared in several computational experiments in which the proposed method consistently exhibits an order of magnitude improvement in runtime performance.

cs.DS↗

arXiv2019

A Polynomial Time Algorithm for Computing the Strong Rainbow Connection Numbers of Odd Cacti

We consider the problem of computing the strong rainbow connection number $src(G)$ for cactus graphs $G$ in which all cycles have odd length. We present a formula to calculate $src(G)$ for such odd cacti which can be evaluated in linear time, as well as an algorithm for computing the corresponding optimal strong rainbow edge coloring, with polynomial worst case run time complexity. Although computing $src(G)$ is NP-hard in general, previous work has demonstrated that it may be computed in polynomial time for certain classes of graphs, including cycles, trees and block clique graphs. This work extends the class of graphs for which $src(G)$ may be computed in polynomial time.

math.CO↗

arXiv2018

Power domination throttling

A power dominating set of a graph $G=(V,E)$ is a set $S\subset V$ that colors every vertex of $G$ according to the following rules: in the first timestep, every vertex in $N[S]$ becomes colored; in each subsequent timestep, every vertex which is the only non-colored neighbor of some colored vertex becomes colored. The power domination throttling number of $G$ is the minimum sum of the size of a power dominating set $S$ and the number of timesteps it takes $S$ to color the graph. In this paper, we determine the complexity of power domination throttling and give some tools for computing and bounding the power domination throttling number. Some of our results apply to very general variants of throttling and to other aspects of power domination.

math.CO↗

arXiv2018

Computational Approaches for Zero Forcing and Related Problems

In this paper, we propose computational approaches for the zero forcing problem, the connected zero forcing problem, and the problem of forcing a graph within a specified number of timesteps. Our approaches are based on a combination of integer programming models and combinatorial algorithms, and include formulations for zero forcing as a dynamic process, and as a set-covering problem. We explore several solution strategies for these models, test them on various types of graphs, and show that they are competitive with the state-of-the-art algorithm for zero forcing. Our proposed algorithms for connected zero forcing and for controlling the number of zero forcing timesteps are the first general-purpose computational methods for these problems, and are superior to brute force computation.

cs.DM↗

arXiv2017

Graphs with Extremal Connected Forcing Numbers

Zero forcing is an iterative graph coloring process where at each discrete time step, a colored vertex with a single uncolored neighbor forces that neighbor to become colored. The zero forcing number of a graph is the cardinality of the smallest set of initially colored vertices which forces the entire graph to eventually become colored. Connected forcing is a variant of zero forcing in which the initially colored set of vertices induces a connected subgraph; the analogous parameter of interest is the connected forcing number. In this paper, we characterize the graphs with connected forcing numbers 2 and $n-2$. Our results extend existing characterizations of graphs with zero forcing numbers 2 and $n-2$; we use combinatorial and graph theoretic techniques, in contrast to the linear algebraic approach used to obtain the latter. We also present several other structural results about the connected forcing sets of a graph.

cs.DM↗

arXiv2015

Chromatic and flow polynomials of generalized vertex join graphs and outerplanar graphs

A generalized vertex join of a graph is obtained by joining an arbitrary multiset of its vertices to a new vertex. We present a low-order polynomial time algorithm for finding the chromatic polynomials of generalized vertex joins of trees, and by duality we find the flow polynomials of arbitrary outerplanar graphs. We also present closed formulas for the chromatic and flow polynomials of vertex joins of cliques and cycles, otherwise known as "generalized wheel" graphs.

cs.DM↗