Chao Xu | 许(xǔ) 超(chāo)

A subset B of the ring \Z_n is referred to as a \ell-covering set if \{ ab \pmod n \mid 0\leq a \leq \ell, b\in B\} = \Z_n. We show that there exists a \ell-covering set of \Z_n of size O(\frac{n}{\ell}\log n) for all n and \ell, and how to construct such a set. We also provide examples where any \ell-covering set must have a size of \Omega(\frac{n}{\ell}\frac{\log n}{\log \log n}). The proof employs a refined bound for the relative totient function obtained through sieve theory and the existence of a large divisor with a linear divisor sum. The result can be used to simplify a modular subset sum algorithm.

Nested ticketing represents a prevalent practice in airline bookings, employed to bypass certain airline ticketing regulations with the aim of reducing costs on multiple round-trip tickets. We consider the computational complexity of nested ticketing, and it is 'covert' version, which we call interleaved ticketing. Consider multiple agents living in different locations and a single client who demands that one agent be present each week. The objective for the company is to schedule these agents and arrange their flight itineraries to minimize the cost. We show that when nested ticketing is allowed, the problem is NP-hard when there are at least two agents. We also show there exists a polynomial-time algorithm if only interleaved ticketing is allowed, and the number of airlines is bounded.

Let \mathbb{F} be any field, we consider solving Ax=b for a matrix A\in\mathbb{F}^{n\times n} of m non-zero elements and b\in\mathbb{F}^{n}. If we are given a zero forcing set of A of size k, we can solve the linear equation in O(mk+k^\omega) time, where \omega is the matrix multiplication exponent. As an application, we show how the lights out game in an n\times n grid is solved in O(n^3) time, and then improve the running time to O(n^\omega\log n) by exploiting the repeated structure in grids.

In this paper, we study the minimum k-partition problem of submodular functions, i.e., given a finite set V and a submodular function f:2^V\to \R, computing a k-partition \{ V_1, \ldots, V_k \} of V with minimum \sum_{i=1}^k f(V_i). The problem is a natural generalization of the minimum k-cut problem in graphs and hypergraphs. It is known that the problem is NP-hard for general k, and solvable in polynomial time for k \leq 3. In this paper, we construct the first polynomial-time algorithm for the minimum 4-partition problem.

Decision trees are well-known due to their ease of interpretability. To improve accuracy, we need to grow deep trees or ensembles of trees. These are hard to interpret, offsetting their original benefits. Shapley values have recently become a popular way to explain the predictions of tree-based machine learning models. It provides a linear weighting to features independent of the tree structure. The rise in popularity is mainly due to TreeShap, which solves a general exponential complexity problem in polynomial time. Following extensive adoption in the industry, more efficient algorithms are required. This paper presents a more efficient and straightforward algorithm: Linear TreeShap. Like TreeShap, Linear TreeShap is exact and requires the same amount of memory.

Submodular functions are fundamental to combinatorial optimization. Many interesting problems can be formulated as special cases of problems involving submodular functions. In this work, we consider the problem of approximating symmetric submodular functions everywhere using hypergraph cut functions. Devanur, Dughmi, Schwartz, Sharma, and Singh showed that symmetric submodular functions over n-element ground sets cannot be approximated within (n/8)-factor using a graph cut function and raised the question of approximating them using hypergraph cut functions. Our main result is that there exist symmetric submodular functions over n-element ground sets that cannot be approximated within a o(n^{1/3}/\log^2 n)-factor using a hypergraph cut function. On the positive side, we show that symmetrized concave linear functions and symmetrized rank functions of uniform matroids and partition matroids can be constant-approximated using hypergraph cut functions.

The Traveling Tournament Problem (TTP) is a well-known benchmark problem in the field of tournament timetabling, which asks us to design a double round-robin schedule such that each pair of teams plays one game in each other's home venue, minimizing the total distance traveled by all n teams (n is even). TTP-k is the problem with one more constraint that each team can have at most k consecutive home games or away games. The case where k=3, TTP-3, is one of the most investigated cases. In this paper, we improve the approximation ratio of TTP-3 from (1.667+\epsilon) to (1.598+\epsilon), for any \epsilon>0. Previous schedules were constructed based on a Hamiltonian cycle of the graph. We propose a novel construction based on triangle packing. Then, by combining our triangle packing schedule with the Hamiltonian cycle schedule, we obtain the improved approximation ratio. The idea of our construction can also be extended to k\geq 4. We demonstrate that the approximation ratio of TTP-4 can be improved from (1.750+\epsilon) to (1.700+\epsilon) by the same method. As an additional product, we also improve the approximation ratio of LDTTP-3 (TTP-3 where all teams are allocated on a straight line) from 4/3 to (6/5+\epsilon).

Street parking spots for automobiles are a scarce commodity in most urban environments. The heterogeneity of car sizes makes it inefficient to rigidly define fixed-sized spots. Instead, unmarked streets in cities like New York leave placement decisions to individual drivers, who have no direct incentive to maximize street utilization. In this paper, we explore the effectiveness of two different behavioral interventions designed to encourage better parking, namely (1) educational campaigns to encourage parkers to "kiss the bumper" and reduce the distance between themselves and their neighbors, or (2) painting appropriately-spaced markings on the street and urging drivers to "hit the line". Through analysis and simulation, we establish that the greatest densities are achieved when lines are painted to create spots roughly twice the length of average-sized cars. Kiss-the-bumper campaigns are in principle more effective than hit-the-line for equal degrees of compliance, although we believe that the visual cues of painted lines induce better parking behavior.

We address counting and optimization variants of multicriteria global min-cut and size-constrained min-k-cut in hypergraphs.

1. For an r-rank n-vertex hypergraph endowed with t hyperedge-cost functions, we show that the number of multiobjective min-cuts is O(r2^{tr}n^{3t−1}). In particular, this shows that the number of parametric min-cuts in constant rank hypergraphs for a constant number of criteria is strongly polynomial, thus resolving an open question by Aissi, Mahjoub, McCormick, and Queyranne (Math Programming, 2015). In addition, we give randomized algorithms to enumerate all multiobjective min-cuts and all pareto-optimal cuts in strongly polynomial-time.
2. We also address node-budgeted multiobjective min-cuts: For an n-vertex hypergraph endowed with t vertex-weight functions, we show that the number of node-budgeted multiobjective min-cuts is O(r2^{r}nt+2), where r is the rank of the hypergraph, and the number of node-budgeted b-multiobjective min-cuts for a fixed budget-vector b is O(n^2).
3. We show that min-k-cut in hypergraphs subject to constant lower bounds on part sizes is solvable in polynomial-time for constant k, thus resolving an open problem posed by Queyranne. Our technique also shows that the number of optimal solutions is polynomial. All of our results build on the random contraction approach of Karger (SODA, 1993). Our techniques illustrate the versatility of the random contraction approach to address counting and algorithmic problems concerning multiobjective min-cuts and size-constrained k-cuts in hypergraphs.

Karger used spanning tree packings to derive a near linear-time randomized algorithm for the global minimum cut problem as well as a bound on the number of approximate minimum cuts. This is a different approach from his well-known random contraction algorithm. Thorup developed a fast deterministic algorithm for the minimum k-cut problem via greedy recursive tree packings. In this paper we revisit properties of an LP relaxation for k-cut proposed by Naor and Rabani, and analyzed by Chekuri, Guha and Naor. We show that the dual of the LP yields a tree packing, that when combined with an upper bound on the integrality gap for the LP, easily and transparently extends Karger's analysis for mincut to the k-cut problem. In addition to the simplicity of the algorithm and its analysis, this allows us to improve the running time of Thorup's algorithm by a factor of n. We also improve the bound on the number of \alpha-approximate k-cuts. Second, we give a simple proof that the integrality gap of the LP is 2(1-1/n). Third, we show that an optimum solution to the LP relaxation, for all values of k, is fully determined by the principal sequence of partitions of the input graph. This allows us to relate the LP relaxation to the Lagrangean relaxation approach of Barahona and Ravi and Sinha; it also shows that the idealized recursive tree packing considered by Thorup gives an optimum dual solution to the LP. This work arose from an effort to understand and simplify the results of Thorup.

In the hypergraph k-cut problem, the input is a hypergraph, and the goal is to find a smallest subset of hyperedges whose removal ensures that the remaining hypergraph has at least k connected components. This problem is known to be at least as hard as the densest k-subgraph problem when k is part of the input (Chekuri-Li, 2015). We present a randomized polynomial time algorithm to solve the hypergraph k-cut problem for constant k. Our algorithm solves the more general hedge k-cut problem when the subgraph induced by every hedge has a constant number of connected components. In the hedge k-cut problem, the input is a hedgegraph specified by a vertex set and a disjoint set of hedges, where each hedge is a subset of edges defined over the vertices. The goal is to find a smallest subset of hedges whose removal ensures that the number of connected components in the remaining underlying (multi-)graph is at least k. Our algorithm is based on random contractions akin to Karger's min cut algorithm. Our main technical contribution is a distribution over the hedges (hyperedges) so that random contraction of hedges (hyperedges) chosen from the distribution succeeds in returning an optimum solution with large probability.

The computational complexity of multicut-like problems may vary significantly depending on whether the terminals are fixed or not. In this work we present a comprehensive study of this phenomenon in two types of cut problems in directed graphs: double cut and bicut.

• The fixed-terminal edge-weighted double cut is known to be solvable efficiently. We show a tight approximability factor of 2 for the fixed-terminal node-weighted double cut. We show that the global node-weighted double cut cannot be approximated to a factor smaller than \frac{3}{2} under the Unique Games Conjecture (UGC).
• The fixed-terminal edge-weighted bicut is known to have a tight approximability factor of 2. We show that the global edge-weighted bicut is approximable to a factor strictly better than 2, and that the global node-weighted bicut cannot be approximated to a factor smaller than \frac{3}{2} under UGC.
• In relation to these investigations, we also prove two results on undirected graphs which are of independent interest. First, we show NP-completeness and a tight inapproximability bound of \frac{4}{3} for the node-weighted 3-cut problem. Second, we show that for constant k, there exists an efficient algorithm to solve the minimum \{s,t\}-separating k-cut problem.

Our techniques for the algorithms are combinatorial, based on LPs and based on enumeration of approximate min-cuts. Our hardness results are based on combinatorial reductions and integrality gap instances.

Given a multiset S of n positive integers and a target integer t, the subset sum problem is to decide if there is a subset of S that sums up to t. We present a new divide-and-conquer algorithm that computes all the realizable subset sums up to an integer u in \tilde{O}\left(\min\{n\sqrt{u},u^{4/3},\sigma\}\right), where \sigma is the sum of all elements in S and \tilde{O} hides polylogarithmic factors. This result improves upon the standard dynamic programming algorithm that runs in O(nu) time. To the best of our knowledge, the new algorithm is the fastest general algorithm for this problem. We also present a modified algorithm for cyclic groups, which computes all the realizable subset sums within the group in \tilde{O}\left(\min\{n\sqrt{m},m^{5/4}\}\right) time, where m is the order of the group.

We study algorithmic and structural aspects of connectivity in hypergraphs. Given a hypergraph H=(V,E) with n=|V|, m=|E| and p=\sum_{e\in E}|e| the best known algorithm to compute a global minimum cut in H runs in time O(np) for the uncapacitated case and in O(np+n^2\log n) time for the capacitated case. We show the following new results.

1. Given an uncapacitated hypergraph H and an integer k we describe an algorithm that runs in O(p) time to find a subhypergraph H' with sum of degrees O(kn) that preserves all edge-connectivities up to k (a k-sparsifier). This generalizes the corresponding result of Nagamochi and Ibaraki from graphs to hypergraphs. Using this sparsification we obtain an O(p+\lambda n^2) time algorithm for computing a global minimum cut of H where \lambda is the minimum cut value.
2. We generalize Matula's argument for graphs to hypergraphs and obtain a (2+\e)-approximation to the global minimum cut in a capacitated hypergraph in O(\frac{1}{\e}(p+n \log n)\log n) time.
3. We show that a hypercactus representation of all the global minimum cuts of a capacitated hypergraph can be computed in O(np+n^2\log n) time and O(p) space. We utilize vertex ordering based ideas to obtain our results. Unlike graphs we observe that there are several different orderings for hypergraphs which yield different insights.

The notion of element-connectivity has found several important applications in network design and routing problems. We focus on a reduction step that preserves the element-connectivity, which when applied repeatedly allows one to reduce the original graph to a simpler one. This pre-processing step is a crucial ingredient in several applications. In this paper we revisit this reduction step and provide a new proof via the use of setpairs. Our main contribution is algorithmic results for several basic problems on element-connectivity including the problem of achieving the aforementioned graph simplification. We utilize the underlying submodularity properties of element-connectivity to derive faster algorithms.

A closed curve in the plane is weakly simple if it is the limit (in the Fréchet metric) of a sequence of simple closed curves. We describe an algorithm to determine whether a closed walk of length n in a simple plane graph is weakly simple in O(n \log n) time, improving an earlier O(n^3)-time algorithm of Cortese et al.. As an immediate corollary, we obtain the first efficient algorithm to determine whether an arbitrary n-vertex polygon is weakly simple; our algorithm runs in O(n^2 \log n) time. We also describe algorithms that detect weak simplicity in O(n \log n) time for two interesting classes of polygons. Finally, we discuss subtle errors in several previously published definitions of weak simplicity.

We describe an algorithm for the high multiplicity asymmetric traveling salesman problem with feedback vertex set of size k (HMTSPFVS-kFVS) where each vertex can be visited a certain number of times and each cycle in a solution contains at least one vertex from the feedback vertex set. We show how it can be used to improve algorithms in automated storage and retrieval systems. An automated storage and retrieval system includes storage blocks and storage and retrieval machines that either move to retrieve unit loads from their current locations in the system to a depot or take unit loads from a depot and store them to specific locations in the system. Given n storage and retrieval requests in a system with k depots and one storage and retrieval machine, we show that our algorithm for HMTSPFVS-kFVS can solve the problem of minimizing total traveling time of the storage and retrieval machine in O(n^k+n^3) time when all depots are specialized (each depot fulfills one type of requests) and in O(n^{2k}+n^3) time when depots are regular (each depot fulfills both types of requests). The best previous algorithm only solves the special case of the problem with 2 regular depots in O(n^6) time. The applicability of our algorithm for several generalizations and special cases of the problem is also discussed. Furthermore, to evaluate the performance of our solution method, we perform extensive numerical experiments.

We consider a problem in descriptive kinship systems, namely finding the shortest sequence of terms that describes the kinship between a person and his/her relatives. The problem reduces to finding the minimum weight path in a labeled graph where the label of the path comes from a regular language. The running time of the algorithm is O(n^3+s), where n and s are the input size and the output size of the algorithm, respectively.

For a simple undirected graph with n vertices and m edges, we consider a data structure that given a query of a pair of vertices u, v and an integer k\geq 1, it returns k edge-disjoint uv-paths. The data structure takes \tilde{O}(n^{3.375}) time to build, using O(mn^{1.5}\log n) space, and each query takes O(kn) time, which is optimal and beats the previous query time of O(kn\alpha(n)).

In the game of Graph Nim, players take turns removing one or more edges incident to a chosen vertex in a graph. The player that removes the last edge in the graph wins. A spider graph is a champion if it has a Sprague-Grundy number equal to the number of edges in the graph. We investigate the the Sprague-Grundy numbers of various spider graphs when the number of paths or length of paths increase.

The capacitated vehicle routing problem (CVRP) is one of the most well known NP-hard combinatorial optimization problems. Single depot CVRP with a general metric is NP-hard even for fixed capacity 3, while polynomial time solvable for fixed capacity 2. We consider the variant of CVRP where restocking is available. We show that if there is a constant number of depots, then the problem can be solved in polynomial time when capacity is 2.

In this thesis, we consider cut and connectivity problems on graphs, digraphs, hypergraphs and hedgegraphs. The main results are the following:

• We introduce a faster algorithm for finding the reduced graph in element-connectivity computations. We also show its application to node separation.
• We present several results on hypergraph cuts, including (a) a near linear time algorithm for finding a (2 + ε)-approximate min-cut, (b) an algorithm to find a representation of all min-cuts in the same time as finding a single min-cut, (c) a sparse subgraph that preserves connectivity for hypergraphs and (d) a near linear-time hypergraph cut sparsifier.
• We design the first randomized polynomial time algorithm for the hypergraph k-cut problem whose complexity has been open for over 20 years. The algorithm generalizes to hedgegraphs with constant span.
• We address the complexity gap between global vs. fixed-terminal cuts problems in digraphs by presenting a 2-\frac{1}{448} approximation algorithm for the global bicut problem.