We show that the problem of finding the simplex of largest volume in the convex hull of n points in ℚd can be approximated with a factor of O(log d) d/2 in polynomial time. This improves upon the previously best known approximation guarantee of d(d–1)/2 by Khachiyan. On the other hand, we show that there exists a constant c > 1 such that this problem cannot be approximated with a factor of cd, unless P = NP. Our hardness result holds even if n = O(d), in which case there exists a d-approximation algorithm that relies on recent sampling techniques, where is again a constant. We show that similar results hold for the problem of finding the largest absolute value of a subdeterminant of a d × n matrix.

The classic stable set problem asks to find a maximum cardinality set of pairwise non-adjacent vertices in an undirected graph G. This problem is NP-hard to approximate with factor n^{1-epsilon} for any constant epsilon>0 [Hastad/Acta Mathematica/1996; Zuckerman/STOC/2006], where n is the number of vertices, and therefore there is no hope for good approximations in the general case. We study the stable set problem when restricted to graphs with bounded odd cycle packing number ocp(G), possibly by a function of n. This is the largest number of vertex-disjoint odd cycles in G. Equivalently, it is the logarithm of the largest absolute value of a sub-determinant of the edge-node incidence matrix A_G of G. Hence, if A_G is totally unimodular, then ocp(G)=0. Therefore, ocp(G) is a natural distance measure of A_G to the set of totally unimodular matrices on a scale from 1 to n/3. When ocp(G)=0, the graph is bipartite and it is well known that stable set can be solved in polynomial time. Our results imply that the odd cycle packing number indeed strongly influences the approximability of stable set. More precisely, we obtain a polynomial-time approximation scheme for graphs with ocp(G)=o(n/log(n)), and an alpha-approximation algorithm for any graph where alpha smoothly increases from a constant to n as ocp(G) grows from O(n/log(n)) to n/3. On the hardness side, we show that, assuming the exponential-time hypothesis, stable set cannot be solved in polynomial time if ocp(G)=Omega(log^{1+epsilon}(n)) for some epsilon>0. Finally, we generalize a theorem by Györi et al. [Györi et al./Discrete Mathematics/1997] and show that graphs without odd cycles of small weight can be made bipartite by removing a small number of vertices. This allows us to extend some of our above results to the weighted stable set problem.

We consider the discrepancy problem of coloring n intervals with k colors such that at each point on the line, the maximal difference between the number of intervals of any two colors is minimal. Somewhat surprisingly, a coloring with maximal difference at most one always exists. Furthermore, we give an algorithm with running time O(n log n + kn log k) for its construction. This is in particular interesting because many known results for discrepancy problems are non-constructive. This problem naturally models a load balancing scenario, where $n$~tasks with given start- and endtimes have to be distributed among $k$~servers. Our results imply that this can be done ideally balanced. When generalizing to $d$-dimensional boxes (instead of intervals), a solution with difference at most one is not always possible. We show that for any d >= 2 and any k >= 2 it is NP-complete to decide if such a solution exists, which implies also NP-hardness of the respective minimization problem. In an online scenario, where intervals arrive over time and the color has to be decided upon arrival, the maximal difference in the size of color classes can become arbitrarily high for any online algorithm.

On the surface, bidirectional search (BDS) is an attractive idea with the potential for significant asymptotic reductions in search effort. However, the results in practice often fall far short of expectations. We introduce a new bidirectional search algorithm, Single-Frontier Bidirectional Searc (SFBDS). Unlike traditional BDS which keeps two frontiers, SFBDS uses a single frontier. Each node in the tree can be seen as an independent task of finding the shortest path between the current start and current goal. At a particular node we can decide to search from start to goal or from goal to start, choosing the direction with the highest potential for minimizing the total work done. Theoretical results give insights as to when this approach will work and experimental data validates the algorithm for a broad range of domains.

We introduce a new bidirectional search algorithm, Single-Frontier Bidirectional Search (SFBDS). Unlike traditional BDS which keeps two frontiers, SFBDS uses a single frontier. At a particular node we can decide to search from start to goal or from goal to start, choosing the direction with the highest potential for minimizing the total work done. We provide theoretical analysis that explains when SFBDS will work validated by experimental results.