The problem of rectangle tiling binary arrays is defined as follows. Given an $n \times n$ array $A$ of zeros and ones and a natural number $p$, our task is to partition $A$ into at most $p$ rectangular tiles, so that the maximal weight of a tile is minimized. A tile is any rectangular subarray of $A$. The weight of a tile is the sum of elements that fall within it. We present a linear $(O(n^2))$ time $(\frac{3}{2}+\frac{p^2}{w(A)})$-approximation algorithm (where $\frac{p^2}{w(A)} < \frac{1}{2}$) for this problem, where $w(A)$ denotes the weight of the whole array $A$. This improves on the previously known approximation with the ratio $2$. The result is best possible in the following sense. The algorithm employs the lower bound of $L=\lceil \frac{w(A)}{p} \rceil$, which is the only known and used bound on the optimum in all algorithms for rectangle tiling. We prove that a better approximation factor for the binary \RTILE cannot be achieved using $L$, because there exist arrays, whose every partition contains a tile with weight at least $(\frac{3}{2}+\frac{p^2}{w(A)})L$. We also consider the dual problem of rectangle tiling for binary arrays, where we are given an upper bound on the weight of the tiles, and we have to cover the array $A$ with the minimum number of non-overlapping tiles. Both problems have natural extensions to $d$-dimensional versions, for which we provide analogous results.

We give faster and simpler approximation algorithms for the (1,2)-TSP problem, a well-studied variant of the traveling salesperson problem where all distances between cities are either 1 or 2. Our main results are two approximation algorithms for (1,2)-TSP, one with approximation factor 8/7 and run time O(n^3) and the other having an approximation guarantee of 7/6 and run time O(n^{2.5}). The 8/7-approximation matches the best known approximation factor for (1,2)-TSP, due to Berman and Karpinski (SODA 2006), but considerably improves the previous best run time of O(n^9). Thus, ours is the first improvement for the (1,2)-TSP problem in more than 10 years. The algorithm is based on combining three copies of a minimum-cost cycle cover of the input graph together with a relaxed version of a minimum weight matching, which allows using "half-edges". The resulting multigraph is then edge-colored with four colors so that each color class yields a collection of vertex-disjoint paths. The paths from one color class can then be extended to an 8/7-approximate traveling salesperson tour. Our algorithm, and in particular its analysis, is simpler than the previously best 8/7-approximation. The 7/6-approximation algorithm is similar and even simpler, and has the advantage of not using Hartvigsen's complicated algorithm for computing a minimum-cost triangle-free cycle cover.

An instance of a strongly stable matching problem (SSMP) is an undirected bipartite graph G = (A ∪ B, E), with an adjacency list of each vertex being a linearly ordered list of ties, which are subsets of vertices equally good for a given vertex. Ties are disjoint and may contain one vertex. A matching M is a set of vertex-disjoint edges. An edge (x, y) ∊ E\M is a blocking edge for M if x is either unmatched or strictly prefers y to its current partner in M, and y is either unmatched or strictly prefers x to its current partner in M or is indifferent between them. A matching is strongly stable if there is no blocking edge with respect to it. We present a characterisation of the set of all strongly stable matchings, thus solving an open problem already stated in the book by Gusfield and Irving [7]. It has previously been shown that strongly stable matchings form a distributive lattice [8] and although the number of strongly stable matchings can be exponential in the number of vertices, we show that there exists a partial order with O(m) elements representing all strongly stable matchings, where m denotes the number of edges in the graph. We give two algorithms that construct two such representations: one in O(nm2) time and the other in O(nm) time, where n denotes the number of vertices in the graph. Note that the construction of the second representation has the same time complexity as that of computing a single strongly stable matching.

We give a very simple approximation algorithm for the maximum asymmetric traveling salesman problem. The approximation guarantee of our algorithm is 2/3, which matches the best known approximation guarantee by Kaplan, Lewenstein, Shafrir and Sviridenko. Our algorithm is simple to analyze, and contrary to previous approaches, which need an optimal solution to a linear program, our algorithm is combinatorial and only uses maximum weight perfect matching algorithm.

We consider the problem of computing a minimum cycle basis of an undirected non-negative edge-weighted graph G with m edges and n vertices. In this problem, a {0,1} incidence vector is associated with each cycle and the vector space over $\mathbb{F}_{2}$ generated by these vectors is the cycle space of G. A set of cycles is called a cycle basis of G if it forms a basis for its cycle space. A cycle basis where the sum of the weights of the cycles is minimum is called a minimum cycle basis of G. Minimum cycle basis are useful in a number of contexts, e.g. the analysis of electrical networks and structural engineering. The previous best algorithm for computing a minimum cycle basis has running time O(m ω n), where ω is the best exponent of matrix multiplication. It is presently known that ω<2.376. We exhibit an O(m 2 n+mn 2log n) algorithm. When the edge weights are integers, we have an O(m 2 n) algorithm. For unweighted graphs which are reasonably dense, our algorithm runs in O(m ω ) time. For any ε>0, we also design an 1+ε approximation algorithm. The running time of this algorithm is O((m ω /ε)log (W/ε)) for reasonably dense graphs, where W is the largest edge weight.

An instance of the stable marriage problem is an undirected bipartite graph G = ( X ∪ W , E ) with linearly ordered adjacency lists with ties allowed in the ordering. A matching M is a set of edges, no two of which share an endpoint. An edge e = ( a , b ) ∈ E ∖ M is a blocking edge for M if a is either unmatched or strictly prefers b to its partner in M , and b is unmatched, strictly prefers a to its partner in M , or is indifferent between them. A matching is strongly stable if there is no blocking edge with respect to it. We give an O ( nm ) algorithm for computing strongly stable matchings, where n is the number of vertices and m the number of edges. The previous best algorithm had running time O ( m 2 ). We also study this problem in the hospitals-residents setting, which is a many-to-one extension of the aforementioned problem. We give an O ( m ∑ h∈H p h ) algorithm for computing a strongly stable matching in the hospitals-residents problem, where p h is the quota of a hospital h . The previous best algorithm had running time O ( m 2 ).

Suppose that each member of a set A of applicants ranks a subset of a set P of posts in an order of preference, possibly involving ties. A matching is a set of (applicant, post) pairs such that each applicant and each post appears in at most one pair. A rank-maximal matching is one in which the maximum possible number of applicants are matched to their first choice post, and subject to that condition, the maximum possible number are matched to their second choice post, and so on. This is a relevant concept in any practical matching situation and it was first studied by Irving [2003].We give an algorithm to compute a rank-maximal matching with running time O (min( n + C , C √ n ) m ), where C is the maximal rank of an edge used in a rank-maximal matching, n is the number of applicants and posts and m is the total size of the preference lists.